lectures on the orbit method

Lectures on theOrbit Method

A.A. Kirillov

Graduate Studiesin MathematicsVolume 64

American Mathematical SocietY


A.A. Kirillov

Graduate Studiesin Mathematics

Volume 64

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEEWalter Craig

Nikolai IvanovSteven G. Krantz

David Saltman (Chair)

2000 Mathematics Subject Classification. Primary 22-02.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-64

Library of Congress Cataloging-in-Publication DataKirillov, A. A. (Aleksandr Aleksandrovich). 1936-

Lectures on the orbit method / A.A. Kirillov.p. cm. - (Graduate studies in mathematics, ISSN 1065-7339; v. 64)

Includes bibliographical references and index.ISBN 0-8218-3530.0 (alk. paper)1. Orbit method. 2. Lie groups. I. Title. II. Series

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To Kirill, Vanya, Lena, and Andrei

Contents

Preface xv

Introduction xvii

Chapter I. Geometry of Coadjoint Orbits 1

V. Basic definitions 1

1.1. Coadjoint representation 1

1.2. Canonical form on 4

§2. Symplectic structure on coadjoint orbits 5

2.1. The first (original) approach 6

2.2. The second (Poisson) approach 7

2.3. The third (symplectic reduction) approach 9

2.4. Integrality condition 11

§3. Coadjoint invariant functions 14

3.1. General properties of invariants 14

3.2. Examples 15

§4. The moment map 16

4.1. The universal property of coadjoint orbits 16

4.2. Some particular cases 19

§5. Polarizations 23

5.1. Elements of symplectic geometry 23

5.2. Invariant polarizations on homogeneous symplectic man-ifolds 26

vii

viii Contents

Chapter 2. Representations and Orbits of the Heisenberg Group 31

§1. Heisenberg Lie algebra and Heisenberg Lie group 32

1.1. Some realizations 32

1.2. Universal enveloping algebra U(I)) 35

1.3. The Heisenberg Lie algebra as a contraction 37

§2. Canonical commutation relations 39

2.1. Creation and annihilation operators 39

2.2. Two-sided ideals in U(I)) 41

2.3. H. Weyl reformulation of CCR 41

2.4. The standard realization of CCR 43

2.5. Other realizations of CCR 45

2.6. Uniqueness theorem 49

§3. Representation theory for the Heisenberg group 57

3.1. The unitary dual H 57

3.2. The generalized characters of H 59

3.3. The infinitesimal characters of H 60

3.4. The tensor product of unirreps 60

§4. Coadjoint orbits of the Heisenberg group 61

4.1. Description of coadjoint orbits 61

4.2. Symplectic forms on orbits and the Poisson structureonb* 62

4.3. Projections of coadjoint orbits 63

§5. Orbits and representations 63

5.1. Restriction-induction principle and construction ofunirreps 64

5.2. Other rules of the User's Guide 68


6.1. Real polarizations 68

6.2. Complex polarization 69

6.3. Discrete polarizations 69

Chapter 3. The Orbit Method for Nilpotent Lie Groups 71

§1. Generalities on nilpotent Lie groups 71

§2. Comments on the User's Guide 73

2.1. The unitary dual 73

Contents ix

2.2. The construction of unirreps 73

2.3. Restriction-induction functors 74

2.4. Generalized characters 74

2.5. Infinitesimal characters 75

2.6. Functional dimension 75

2.7. Plancherel measure 76

§3. Worked-out examples 77

3.1. The unitary dual 78

3.2. Construction of unirreps 80

3.3. Restriction functor 84

3.4. Induction functor 86

3.5. Decomposition of a tensor product of two unirreps 88



3.8. Functional dimension 91

3.9. Plancherel measure 92

3.10. Other examples 93

§4. Proofs 95

4.1. Nilpotent groups with 1-dimensional center 95

4.2. The main induction procedure 98

4.3. The image of U(g) and the functional dimension 103

4.4. The existence of generalized characters 104

4.5. Homeomorphism of G and 0(G) 106

Chapter 4. Solvable Lie Groups 109

§1. Exponential Lie groups 109

1.1. Generalities 109

1.2. Pukanszky condition 111

U. Restriction-induction functors 113



§2. General solvable Lie groups 118

2.1. Tame and wild Lie groups 118

2.2. Tame solvable Lie groups 123

x Contents

§3. Example: The diamond Lie algebra g 126

3.1. The coadjoint orbits for g 126

3.2. Representations corresponding to generic orbits 128

3.3. Representations corresponding to cylindrical orbits 131

§4. Amendments to other rules 132

4.1. Rules 3 - 5 132

4.2. Rules 6, 7, and 10 134

Chapter 5. Compact Lie Groups 135

§1. Structure of semisimple compact Lie groups 136

1.1. Compact and complex semisimple groups 137

1.2. Classical and exceptional groups 144

§2. Coadjoint orbits for compact Lie groups 147

2.1. Geometry of coadjoint orbits 147

2.2. Topology of coadjoint orbits 155


3.1. Overlook 161

3.2. Weights of a unirrep 164

3.3. Functors Ind and Res 168

3.4. Borel-Weil-Bott theorem 170

3.5. The integral formula for characters 173


§4. Intertwining operators 176

Chapter 6. Miscellaneous 179

§1. Semisimple groups 179

1.1. Complex semisimple groups 179

1.2. Real semisimple groups 180

§2. Lie groups of general type 180

2.1. Poincare group 181

2.2. Odd symplectic groups 182

§3. Beyond Lie groups 184

3.1. Infinite-dimensional groups 184

3.2. p-adic and adelic groups 188

3.3. Finite groups 189

Contents

3.4. Supergroups§4. Why the orbit method works4.1. Mathematical argument4.2. Physical argument

§5. Byproducts and relations to other domains

5.1. Moment map5.2. Integrable systems

§6. Some open problems and subjects for meditation

6.1. Functional dimension

6.2. Infinitesimal characters6.3. Multiplicities and geometry

6.4. Complementary series6.5. Finite groups6.6. Infinite-dimensional groups

Appendix I. Abstract Nonsense§1. Topology

1.1. Topological spaces

1.2. Metric spaces and metrizable topological spaces

§2. Language of categories

2.1. Introduction to categories2.2. The use of categories

2.3. Application: Homotopy groups

§3. Cohomology

3.1. Generalities3.2. Group cohomology3.3. Lie algebra cohomology

3.4. Cohomology of smooth manifolds

Appendix II. Smooth Manifolds§1. Around the definition

1.1. Smooth manifolds. Geometric approach

1.2. Abstract smooth manifolds. Analytic approach1.3. Complex manifolds

1.4. Algebraic approach

xi

194

194

194

196

198

198

199

201

201

203

203

204

205

205

207

207

207

208

211

211

214

215

216

216

217

219

220

227

227

227

230

235

236

xii Contents

§2. Geometry of manifolds 238

2.1. Fiber bundles 238

2.2. Geometric objects on manifolds 243

2.3. Natural operations on geometric objects 247

2.4. Integration on manifolds 253

§3. Symplectic and Poisson manifolds 256

3.1. Symplectic manifolds 256

3.2. Poisson manifolds 263

3.3. Mathematical model of classical mechanics 264

3.4. Symplectic reduction 265

Appendix III. Lie Groups and Homogeneous Manifolds 269

§1. Lie groups and Lie algebras 269

1.1. Lie groups 269

1.2. Lie algebras 270

1.3. Five definitions of the functor Lie: G g 274

1.4. Universal enveloping algebras 286

§2. Review of the set of Lie algebras 288

2.1. Sources of Lie algebras 288

2.2. The variety of structure constants 291

2.3. Types of Lie algebras 297

§ 3. Semisimple Lie algebras 298

3.1. Abstract root systems 298

3.2. Lie algebra sl(2, C) 308

3.3. Root system related to (g, h) 310

3.4. Real forms 315

§ 4. Homogeneous manifolds 318

4.1. G-sets 318

4.2. G-manifolds 323

4.3. Geometric objects on homogeneous manifolds 325

Appendix IV. Elements of Functional Analysis 333

§ 1. Infinite-dimensional vector spaces 333

1.1. Banach spaces 333

1.2. Operators in Banach spaces 335

Contents

1.3. Vector integrals

1.4. Hilbert spaces§2. Operators in Hilbert spaces

2.1. Types of bounded operators2.2. Hilbert-Schmidt and trace class operators

2.3. Unbounded operators2.4. Spectral theory of self-adjoint operators2.5. Decompositions of Hilbert spaces

2.6. Application to representation theory§3. Mathematical model of quantum mechanics

Appendix V. Representation Theory§1. Infinite-dimensional representations of Lie groups

1.1. Generalities on unitary representations1.2. Unitary representations of Lie groups

1.3. Infinitesimal characters

1.4. Generalized and distributional characters1.5. Non-commutative Fourier transform

§2. Induced representations2.1. Induced representations of finite groups

2.2. Induced representations of Lie groups2.3. *-representations of smooth G-manifolds

2.4. Mackey Inducibility Criterion

References

336

337

339

340

340

343

345

350

353

355

357

357

357

363

368

369

370

371

371

379

384

389

395

Index 403

Preface

The goal of these lectures is to describe the essence of the orbit methodfor non-experts and to attract the younger generation of mathematiciansto some old and still unsolved problems in representation theory where Ibelieve the orbit method could help.

It is said that to become a scientist is the same as to catch a train at fullspeed. Indeed, while you are learning well-known facts and theories, manynew important achievements happen. So, you are always behind the presentstate of the science. The only way to overcome this obstacle is to "jump",that is, to learn very quickly and thoroughly some relatively small domain.and have only a general idea about all the rest.

So, in my exposition I deliberately skip many details that are not ab-solutely necessary for understanding the main facts and ideas. The mostpersistent readers can try to reconstruct these details using other sources.I hope, however, that for the majority of users the book will be sufficientlyself-contained.

The level of exposition is different in different chapters so that bothexperts and beginners can find something interesting and useful for them.Some of this material is contained in my book [Ki2] and in the surveys[Ki5], [Ki6], and [Ki9]. But a systematic and reasonably self-containedexposition of the orbit method is given here for the first time.

I wrote this book simultaneously in English and in Russian. For severalreasons the English edition appears later than the Russian one and differsfrom it in the organization of material.

Sergei Gelfand was the initiator of the publication of this book andpushed me hard to finish it in time.

xv

xvi Preface

Craig Jackson read the English version of the book and made manyuseful corrections and remarks.

The final part of the work on the book was done during my visits to theInstitut des Hautes Etudes Scientifiques (Bures-sur-Yvette, France) and theMax Planck Institute of Mathematics (Bonn, Germany). I am very gratefulto both institutions for their hospitality.

In conclusion I want to thank my teachers, friends, colleagues, and es-pecially my students, from whom I learned so much.

Introduction

The idea behind the orbit method is to unite harmonic analysis with sym-plectic geometry. This can be considered as a part of the more general ideaof the unification of mathematics and physics.

In fact, this is a post factum formulation. Historically, the orbit methodwas proposed in (Kill for the description of the unitary dual (i.e. the setof equivalence classes of unitary irreducible representations) of nilpotentLie groups. It turned out that the method not only solves this problembut also gives simple and visual solutions to all other principal questionsin representation theory: topological structure of the unitary dual, the ex-plicit description of the restriction and induction functors, the formulae forgeneralized and infinitesimal characters, the computation of the Plancherelmeasure, etc.

Moreover, the answers make sense for general Lie groups and even be-yond, although sometimes with more or less evident corrections. I alreadymentioned in [Kill the possible applications of the orbit method to othertypes of Lie groups, but the realization of this program has taken a longtime and is still not accomplished despite the efforts of many authors.

I cannot mention here all those who contributed to the developmentof the orbit method, nor give a complete bibliography: Mathematical Re-views now contains hundreds of papers where coadjoint orbits are mentionedand thousands of papers on geometric quantization (which is the physicalcounterpart of the orbit method). But I certainly ought to mention theoutstanding role of Bertram Kostant and Michel Duflo.

As usual, the faults of the method are the continuations of its advantages.I quote briefly the most important ones.

xvii

xviii Introduction

MERITS VERSUS DEMERITS

1. Universality: the method works

for Lie groups of any type

over any field.

1. The recipes are not accuratelyand precisely developed.

2. The rules are visual,and are easy to memorize

and illustrate by a picture.

3. The method explainssome facts which otherwise

look mysterious.

4. It provides a great amount ofsymplectic manifolds and

Poisson commuting families

of functions.

5. The method introduces newfundamental notions: coadjointorbit and moment map.

2. Sometimes they are wrongand need corrections

or modifications.

3. It could be difficult

to transform this explanationinto a rigorous proof.

4. Most of the completely integrabledynamical systems were

discovered earlier

by other methods.

5. The description of coadjointorbits and their structuresis sometimes not an easy problem.

For the reader's convenience we formulate the ideology of the orbitmethod here in the form of a "User's Guide" where practical instructionsare given as to how to get answers to ten basic questions in representationtheory.

These simple rules are applicable literally for all connected and simplyconnected nilpotent groups. For groups of general type we formulate the"ten amendments" to these rules in the main text of the book.

Throughout the User's Guide we use the following notation:G - a connected simply connected Lie group;H C G - a closed connected subgroup;g, 4 - Lie algebras of G, H, respectively;g', f)' - the dual spaces to g, F , respectively;p: g' h' - the canonical projection;a - the canonical 2-form (symplectic structure) on a coadjoint orbit;nn - the unirrep of G corresponding to the orbit SZ C g';

Introduction xix

pF,H - the 1-dimensional unirrep of H given by pF,Ei (exp X) = e2'"`(F x) .

PA - the G-invariant polynomial on g" related to A E Z(g), the centerof U(g).

For other notation, when it is not self-explanatory, the reader must con-sult the Index and look for definitions given in the main text or in theAppendices.

USER'S GUIDE

What you want What you have to do

1. Describe the unitary dual G Take the space 0(G) of coadjointas a topological space. orbits with the quotient topology.

2. Construct the unirrep 7rn Choose a point F E 12, take

associated to the orbit 12 E g`. a subalgebra lj of maximal

dimension subordinate to F,

and put 7rS2 = IndHpF,H.

3. Describe the spectrum

of Res H 7rn.

Take the projection p(1l) and

split it into H-orbits.

4. Describe the spectrum

of Ind H 1r,r..

5. Describe the spectrum ofthe tensor product irsal ®1T122

6. Compute the generalized

character of iro.

7. Compute the infinitesimal

character of irn.

8. What is the relation betweenirn and ir_52?

9. Find the functional

Take the G-saturation of p-1 (w)

and split it into G-orbits.

Take the arithmetic sum 01 + 122

and split it into orbits.

tr irn(exp X )/_ e21r£(Fx)+o or

(X11 , 40) = JZp- (F) ea.

n

For A E Z(g) take the value of

PA E Pol(g*)G on the orbit Q.

They are contragredient (dual)representations.

It is equal to 2 dim Q.

dimension of 7rn.

xx Introduction

10. Compute the Plancherel measure The measure on O(G) arising when

p on G. the Lebesgue measure on g'is decomposed into canonical

measures on coadjoint orbits.

These short instructions are developed in Chapter 3 and illustrated inthe worked-out examples in the main text.

Finally, a technical remark. I am using the standard sign to signalthe end of a proof (or the absence of proof). I also use less standard notation:

the end of an example;the end of a remark:the end of an exercise;

the end of a warning about a possible mistake or misunderstand-

The most difficult exercises and parts of the text are marked by anasterisk (*).

Chapter 1

Geometry ofCoadjoint Orbits

We start our book with the study of coadjoint orbits. This notion is themain ingredient of the orbit method. It is also the most important newmathematical object that has been brought into consideration in connectionwith the orbit method.

1. Basic definitionsBy a coadjoint orbit we mean an orbit of a Lie group G in the space g`dual to g = Lie(G). The group G acts on g* via the coadjoint representation,dual to the adjoint one (see the definition below and also Appendix I11.1.1).

In this chapter we consider the geometry of coadjoint orbits and discussthe problem of their classification.

1.1. Coadjoint representation.Let G be a Lie group. It is useful to have in mind the particular case

when G is a matrix group, i.e. a subgroup and at the same time a smoothsubmanifold of GL(n. R).

Let g = Lie(G) be the tangent space Te(G) to G at the unit point e.The group G acts on itself by inner automorphisms: A(g) : x --, g xg-1.The point e is a fixed point of this action, so we can define the derived map(A(g)).(e): g --+ g. This map is usually denoted by Ad(g).

The map g " Ad(g) is called the adjoint representation of G. In thecase of a matrix group G the Lie algebra g is a subspace of Mat,, (R) and

1

2 1. Geometry of Coadjoint Orbits

the adjoint representation is simply the matrix conjugation:

(1) Ad(g)X = g X g-', X E g, g E G.

The same formula holds in the general case if we accept the matrix notationintroduced in Appendix III.1.1.

Consider now the vector space dual to g. We shall denote it by g`. Recallthat for any linear representation (ir, V) of a group G one can define a dualrepresentation (7r*, V*) in the dual space V*:

x*(g) := 7r(g-1)*

where the asterisk in the right-hand side means the dual operator in V'defined by

(A*f,v) := (f, Av) for any v E V, f E V.

In particular, we have a representation of a Lie group G in g' that is dualto the adjoint representation in g. This representation is called coacllvjoint.

Since this notion is very important, and also for brevity, we use thespecial notation' K(g) for it instead of the full notation Ad`(g) = Ad(g-')'.So, by definition,

(2) (K(g)F, X) = (F, Ad(g-')X)

where X E g, F E g', and by (F, X) we denote the value of a linear func-tional F on a vector X.

For matrix groups we can use the fact that the space Mat (R) has abilinear form

(3) (A, B) = tr (AB),

which is non-degenerate and invariant under conjugation. So, the spaceg`, dual to the subspace g C Matn(R), can be identified with the quotientspace Mat,,(R)/gl. Here the sign -L means the orthogonal complement withrespect to the form ( , ):

gl = {A E I (A, B) = b for all B E g}.

In practice the quotient space is often identified with a subspace V CMath (R) that is transversal to gl and has the complementary dimension.Therefore, we can write Mat,, (R) = V ® gl. Let pv be the projection of

I In Russian the word "coadjoint" starts with k.

§ 1. Basic definitions 3

Math(]R) onto V parallel to g-i-. Then the coadjoint representation K canbe written in a simple form

(4) K(g): F .- pv (gFg-1).

Remark 1. If we could choose V invariant under Ad (G) (which wecan always assume for g semisimple or reductive), then we can omit theprojection pv in (4). C7

Example 1. Denote by G the group of all (non-strictly) upper trian-gular matrices g E GL(n, IR), i.e. such that gig = 0 for i > j. Then theLie algebra g consists of all upper triangular matrices from Mat (R). Thespace g1 is the space of strictly upper triangular matrices X satisfying thecondition xi, = 0 for i > j.

We can take for V the space of all lower triangular matrices.The projection pv in this case sends any matrix to its "lower part" (i.e.

replaces all entries above the main diagonal by zeros). Hence, the coadjointrepresentation takes the form

K(g) : F H (g Fg-1)lowcr part.

Although this example has been known for a long time and has beenthoroughly studied by many authors, we still do not know how to classifythe coadjoint orbits for general n. 0

Example 2. Let G = SO(n, R). Then g consists of all skew-symmetricmatrices X = -Xt from Matn(1R).

Here we can put V = g and omit the projection pv in (4) (cf. Remark1):

K(g)X g-'.

Thus, the coadjoint representation is equivalent to the adjoint one and co-incides with the standard action of the orthogonal group on the space ofantisymmetric bilinear forms. It is well known that a coadjoint orbit passingthrough X is determined by the spectrum of X, which can be any multisetin iR, symmetric with respect to the complex conjugation. Another conve-nient set of parameters labelling the orbits is the collection of real numbers{tr X2, tr X4, ..., tr X2k } where k = [2]. Q

We also give the formula for the infinitesimal version of the coadjointaction, i.e. for the corresponding representation K. of the Lie algebra g ing':

(5) (K.(X)F, Y) = (F, -ad(X) Y) = (F, [Y,X]).


For matrix groups it takes the form

(5') K.(X)F=pv([X,F]) forXE9, FEV-g'.

Remark 2. The notions of coadjoint representation and coadjoint orbitcan be defined beyond the realm of Lie groups in the ordinary sense.

Three particular cases are of special interest: infinite-dimensional groups,algebraic groups over arbitrary fields and quantum groups (which are notgroups at all). In all three cases the ideology of the orbit method seems tobe very useful and often suggests the right formulations of important results.

We discuss below some examples of such results, although this subjectis outside the main scope of the book. O

1.2. Canonical form au.One feature of coadjoint orbits is eye-catching when you consider a few

examples: they always have an even dimension. This is not accidental, buthas a deep geometric reason.

All coadjoint orbits are symplectic manifolds. Moreover, each coad-joint orbit possesses a canonical G-invariant symplectic structure. Thismeans that on each orbit Il C g' there is a canonically defined closed non-degenerate G-invariant differential 2-form an.

In the next sections we give several explanations of this phenomenonand here just give the definition of an.

We use the fact that a G-invariant differential form w on a homogeneousG-manifold M = G/H is uniquely determined by its value at the initialpoint mo and this value can be any H-invariant antisymmetric polylinearform on the tangent space Tm0 M.

Thus, to define an it is enough to specify its value at some point F E Q,which must be an antisymmetric bilinear form on Tp1 invariant under theaction of the group Stab F, the stabilizer of F.

Let stab(F) be the Lie algebra of Stab(F). We can consider the groupG as a fiber bundle over the base St G/Stab(F) with projection

PF : G -+ St, PF(9) = K(g)F.

It is clear that the fiber above the point F is exactly Stab(F). Consider theexact sequence of vector spaces

0 -+ stab(F) g(- TF(1l) -+ 0

that comes from the above interpretation of G as a fiber bundle over Q. Itallows us to identify the tangent space TF(St) with the quotient g/stab(F).


Now we introduce the antisymmetric bilinear form BF on g by the for-mula

(6) BF(X.Y) = (F. [X,Y])

Lemma 1. The kernel of BF is exactly stab(F).

Proof.

kerBF={XEg I BF(X,Y)=0VYEg}={XEg I (K.(X)F, Y)=0VYEg}= {X E g I K.(X)F = 0} = stab(F).

0

Lemma 2. The form BF is invariant under Stab(F).

Proof.

(F, [AdhX, AdhY]) = (F, Adh[X, Y]) = (K(h-1)F, [X, Y]) = (F, [X, Y])

for any h E Stab(F). 0

Now we are ready to introduce

Definition 1. Let Il be a coadjoint orbit in g'. We define the differential2-form an on St by

(7) BF(X, Y)

The correctness of this definition, as well as the non-degeneracy and G-invariance of the constructed form follows immediately from the discussionabove.

2. Symplectic structure on coadjoint orbitsThe goal of this section is to prove

Theorem 1. The form an is closed, hence defines on S2 a G-invariantsymplectic structure.

There exist several proofs of this theorem that use quite different ap-proaches. This can be considered as circumstantial evidence of the depthand importance of the theorem. Three and a half of these proofs are pre-sented below.


2.1. The first (original) approach.We use the explicit formula (19) in Appendix 11.2.3 for the differential

of a 2-form:da((, n, () =0 (a(rl, ()- v a([(, 771, ()

where the sign 0 denotes the summation over cyclic permutations of (, 77, (.

Let (, 77, (be the vector fields on Il which correspond to elements X, Y,Z of the Lie algebra 9.2 Then ((F) = K.(X)F, g(F) = K.(Y)F, ((F) _K.(Z)F, and we obtain

(F, [Y, Z]), (a(g, () = (K.(X)F, [Y, Z]) = -(F, [X, [Y, Z]]),rl] = Y])F, a([f, 7)], () = -(F, [[X, Y], Z]).

Therefore,

da((, g, () = 2 U (F, [X, [Y, Z]j) = 0 via the Jacobi identity.

Since G acts transitively on St, the vectors K. (X )F, X E g, span the wholetangent space TFSI. Thus, da = 0.

This proof of Theorem 1, being short enough, can be however not quitesatisfactory for a geometric-minded reader. So, we give a variant of it whichis based on more geometric observations. This variant of the proof is alsoin accordance with the general metamathematical homology principle men-tioned in Remark 1 in Appendix 1.3.

Consider again the fibration pF : G - f and introduce the form EFp-(a) on the group G. By the very construction, EF is a left-invariant 2-form on G with initial value EF(e) = BF. We intend to show that this formis not only closed but exact.

To see this, we shall use the so-called Maurer-Cartan form 6. Bydefinition, it is a g-valued left-invariant 1-form on G defined by the condition6(e)(X) = X. Since the left action of G on itself is simply transitive(i.e. there is exactly one left shift which sends a given point g1 to anothergiven point g2), to define a left-invariant form on G we only need to specifyarbitrarily its value at one point. The explicit formula for A in matrixnotations is

(8) 6(g)(X) = g-1 X for X E TgG.

Often, especially in physics papers, this form is denoted by g-'dg be-cause for any smooth curve g = g(t) we have g'9 = g(t)-1g(t)dt = g-1dg.

2Recall that for left G-manifolds the vector field C corresponding to X E g is defined byfi(x) = d(exptX) x It_o. The map X ,-+{ is an antihomomorphism of g to Vect M.

§2. Symplectic structure on coadjoint orbits

Proposition 1. The 2-form EF. = pF((7) on G is the exterior derivative ofthe left-invariant real-valued 1-form OF given by

(9) OF=-(F, E)).

Proof. We shall use the formula for the exterior derivative of a 1-form (see(19) from Appendix 11.2.3):

d9(t;,'9) =te(e) -0(0 - O([C rl]).

Let X and k be left-invariant vector fields on G (see the fourth definitionof a Lie algebra in Appendix 111.1.3). Putting 0 = OF, C = X, 77 = Y, weget

dOF(X, k) = XOF(Y) - YOF(X) - OF([X, Y])

The_first and second terms in the right-hand side vanish because OF(X)and OF(Y) are constant functions. We can rewrite the last term as

-OF([X, Y]) = -OF([X, Y]) = (F, [X, Y]) = pF(a)(X, Y)-

Now we return to the form a. Since pF is a submersion, the linear map(pF)* is surjective. Therefore, the dual map pF is injective. But pFda =dpp(a) = dEF = d2OF = 0. Hence, a is closed.

Note that in general OF cannot be written as p`F(¢) for some 1-form 0on 11, so we cannot claim that a is exact (and actually it is not in general).

2.2. The second (Poisson) approach.We now discuss another way to introduce the canonical symplectic struc-

ture on coadjoint orbits. It is based on the notion of Poisson manifold (seeAppendix 11.3.2).

Consider the real n-dimensional vector space V and, making an excep-tion to the general rules, denote the coordinates (X1, ..., on V usinglower indices. Let c be a bivector field on V with linear coefficients:

(10) c=ckjXk8'(9 8)

where 8'=8/BX,,O

Lemma 3. The bivector (10) defines a Poisson structure on V if and onlyif the coefficients cki form a collection of structure constants for some Liealgebra g.


Proof. Consider the bracket operation defined by c:

aft{fl, f2} = k Xah

Cij k OXi OX;

Since the bivector c has linear coefficients, the space V' of linear functionson V is closed under this operation.

If c defines a Poisson structure on V, then V' is a Lie subalgebra inC°°(V) that we denote by g. Therefore, V itself can be identified with thedual space g', which justifies the labelling of coordinates by lower indices.

In the natural basis in V' formed by coordinates (X1, ... , the coef-ficients ck are precisely the structure constants of g:

{Xi, X,} = c= Xk.k

Conversely, if ck. are structure constants of a Lie algebra g, then the brackets(11) satisfy the Jacobi identity for any linear functions fl, f2, f3. But thisidentity involves only the first partial derivatives of fi. Therefore, it is truefor all functions.

Remark 3. The existence of a Poisson bracket on g' was already knownto Sophus Lie in 1890, as was pointed out recently by A. Weinstein. It seemsthat Lie made no use of it. F. A. Berezin rediscovered this bracket in 1967in connection with his study of universal enveloping algebras [Bell. Therelation of this fact to coadjoint orbits was apparently first noted in [Ki4].c7

This relation can be formulated as follows.

Theorem 2. The symplectic leaves of the Poisson manifold (g', c) are ex-actly the coadjoint orbits.

Proof. Let LF be the leaf that contains the point F E 9'. The tangentspace to LF at F by definition (see Theorem 6 in Appendix 11.3.2) is spannedby vectors vi = But vi is exactly the value at F of the vector field ong' corresponding to the infinitesimal coadjoint action of Xi E g. Therefore,the coadjoint orbit Q F and the leaf LF have the same tangent space. Sinceit is true for every point F E g', we get LF = SlF.

Theorem 2 gives an alternative approach to the construction of thecanonical symplectic structure on coadjoint orbits.

§2. Symplectic structure on coadjoint orbits

2.3.* The third (symplectic reduction) approach.We apply the symplectic reduction procedure described in Appendix

11.3.2 to the special case when the symplectic manifold M is the cotangentbundle T*G over a Lie group G. This case has its peculiarities.

First, the bundle T*G is trivial. Namely, we shall use the left action ofG on itself to make the identification T*(G) ^- G x g*. In matrix notationthe covector g F E T9_1G corresponds to the pair (g, F) E G x g*.3

Further, the set T*(G) is itself a group with respect to the law

(91, Fi)(92, F2) = (9192. K(92)-1F1 + F2).

If we identify (g, 0) E T*(G) with g E G and (e, F) E T*(G) with F E g*,then T* (G) becomes a semidirect product G a g*. So, we can write (g, F) _g F both in matrix notation and in the sense of the group law in T*G.

Note also the identity g F g-1 = K(g)F.Since T*G is a Lie group, the tangent bundle T(T*(G)) is also trivial.

We identify T(9,F)T *G with Lie (T*G) g,@g* using the left shift and obtain

T(T*(G)) ^ (G x g*) x (g a g*).

Theorem 3. The canonical symplectic structure on T* (G) in the trivial-ization above is given by the bilinear form a:

(12) a(g,F)(X1 ® F1, X2 E) F2) = (F1, X2) - (F2, X1) - (F, [X1, X2]).

Proof. Let us compute first the canonical 1-form Bon T' (G). For a tangentvector v = (g, F; X, F') E T(9.F)T*G the projection to TqG equals X g-1.Therefore, 0(v) = (g F, X g-1) = (F, X).

Now we can compute a as the exterior derivative of 0:

a(6, 42) = X10(42) - 420(1) - 9([41, 6])

for any vector fields 1, 6 on T* G.

We choose 1;1,2 as the left-invariant fields on the Lie group T*G withinitial values

e1(e, 0) = (X1, F1), We, 0) = (X2, F2).

3Recall that g F is defined by (g F. F) = (F, i; g'1).


Then [c1, 1;2] (e, 0) = ([X1, X2], K(XI)F2 - K(X2)F1) and we also have

fiO(W = S1(F, X2) = (Fl, X2), 6e(b) _ 2(F, X1) _ (F2, XI),0QC1, e2]) = (F, [X1, X2]).

The desired formula (12) follows from these computations. O

The group G x G acts on G by left and right shifts: (91, g2) -g = glgg21.

This action extends to a Hamiltonian action on T*G. Actually, the extensionis again given by left and right shifts on elements G viewed as elements ofthe group T'G. Using the above identification we can write it in the form

(13) (91, g2) - (g, F) = (91992 1, K(g2)F).

Recall (see Appendix 111. 1. 1) that to any X E g there correspond two vectorfields on G:

- the infinitesimal right shift k, which is a unique left-invariant fieldsatisfying X(e) = X, and

the infinitesimal left shift X, which is a unique right-invariant fieldsatisfying X(e) = X.

Exercise 1. Using the above identifications, write explicitly

a) the vector fields X and k on G;b) their Hamiltonian lifts X' and k* on T*(G).Answer:

(14)a) X(g) = X; X(g) = Adg-1X;

b) Xs(9, F) = (X, 0). X'(g, F) = (Adg-'X, K.(X)F).

Hint. Use formula (13).To construct the reduction of T*G with respect to the left, right, or

two-sided action of G (see Appendix II.3.4), we have to know the momentmap.

Lemma 4. For the action of G x G on T*G the moment map t t: T*Gg' ED g' is given by the formula

(15) µ (g, F) = (F (D K(g)F)


Proof._ The proof follows from Theorem 3 and the explicit formulae abovefor B, X `, X `. 0

For the left action the moment map looks especially simple (because weare using the left trivialization): it (g, F) = F.

The fiber of the moment map over F E g' is G x {F}. So, the reducedmanifold is G/Stab(F) OF. We omit the (rather tautological) verificationof the equality ao = ail.

We could get the same result using the right action of G on T*G. Indeed,here p (g, F) = K(g)F and µ-1(12F) = G x 11F. Therefore, the reducedmanifold is (G x IhF)/G ^_-12F.

2.4. Integrality condition.In Appendix II.2.4 we explain how to integrate a differential k-form on

M over an oriented smooth k-dimensional submanifold.More generally, define a real (resp. integral) singular k-cycle in a

manifold M as a linear combination C = Ei c, Vi(Mi) of images of smoothk-manifolds Mi under smooth maps pj : Mi , M with real (resp. integer)coefficients ci. Then we can define the integral of the form w E f2k(M) overa singular k-cycle C as

f=(C.f(W)).l,

It is known that the integral of a closed k-form over a singular k-dimensional cycle C depends only on the homology class (C] E Hsiny(M).4Moreover, a k-form is exact if its integral over any k-cycle vanishes.

For future use we make the following

Definition 2. A coadjoint orbit Sl is integral if the canonical form a hasthe property

(16) f a E Z for every integral singular 2-cycle C in Q.C

In particular, it is true when C is any smooth 2-dimensional submanifoldS c f2.

4 We will not give the accurate definitions hereof the groups Hein9(,'l1, R) and H,,,,9 (M, Z) ofsingular homologies of a manifold M. For the applications in representation theory the informationgiven in this section is quite enough.


The integrality condition has important geometric and representation-theoretic interpretations. They are revealed by the following

Proposition 2. Assume that G is a simply connected Lie group. The fol-lowing are equivalent:

(i) St C g' is integral.(ii) There exists a G-equivariant complex line bundle over 12 with a G-

invariant Hermitian connection V such that

(17) curv (V) = 2iria.

(iii) For any F E ft there exists a unitary 1-dimensional representationX of the connected Lie group Stab°(F) such that

(18) X(eXP X) = e2ai(F. X).

Observe that condition (i) is automatically true for homotopically trivial(i.e. contractible) orbits. It is also true when the canonical form a is exact.

Proof. (i) .(ii). Let L be a complex line bundle over Q. Choose a cov-ering of S2 by open sets {UQ}QEA such that for any a E A there exists anon-vanishing section sQ of L over UQ. Then we can specify a section s bythe collection of functions fQ E A(UQ) given by

A connection V in a line bundle L is given by a family of differential1-forms 9Q. Namely, define oQ by OQ(v) sQ for any v E Vect (UQ).In terms of these forms the covariant derivative is

Ov = v + 9Q(v), i.e. s --* {f,,} = V o H {vfQ + BQ(v)fQ}.

The connection is Hermitian if a scalar product is defined in all fibersso that

v - (31, 32) = (Vvs1, 32) + (sl, V s2)

If we normalize sQ by the condition (se, sQ) = 1, this condition becomesoQ = -FQ.

Let cQ, Q be the transition functions, so that fQ = cQ, p fp on U,,,,3. Thenthe forms oQ satisfy

Bp - oQ = d log cQ, p.

Therefore, the form doQ coincides with doQ on U,,,,3. Hence, the collection{d0,,} defines a single 2-form 9 on Q. This form is called the curvature


form of the connection V and is denoted curvy. (In some books anothernormalization is used and the real form 2na9 is called the curvature form.)

Exercise 2. Prove that

(curv V) (v, w) = [Vu, V,,,] - V[t,,w].

Hint. Use the formula for V,, above and the formula for d9. 4We will not bother the reader with the verification of the following fact:

Lemma 5. The Cech cocycle corresponding to the 2 -form cure V has theform

Ca..,3,y = log ca, 0 + log C3, -f + log C.y, a.

0

From this formula and from the relation ca,,3 c3,y Cy,a = 1 it followsimmediately that the cohomology class of cure V belongs to 2iri H2(Sl, Z).Hence, the form a = --i-curvV belongs to an integral cohomology class.

Conversely, let a be a real 2-form on SZ, and let 9a be the real antideriv-ative of or on Ua. Then we can define the functions c,,,p on Ua,o so thatdca,i3 = 27ri(90 - 9a). If a has the property [a] E H2(1l, Z), these functionssatisfy ca, 3 ci3, 7 cy. a = 1. Hence, they can be considered as the transitionfunctions of some complex line bundle L over Q. Since the 9a are real, wecan assume that Ica,,31 = 1. Therefore, L admits a scalar product in fiberssuch that V, = v + 2iriOa(v) is a Hermitian connection on L.

(ii)e==*(iii). Since stab(F) = kerBF, the representation (18) is in fact arepresentation of an abelian Lie group A = Stab°(F)/[Stab°(F), Stab°(F)].This group has the form Tk x Rt. The representation (18) corresponds to aunitary 1-dimensional representation of A if it takes integer values on thek-dimensional lattice A = exp-1(e).

Let X E A and denote by 'y the loop in Stab°(F) which is the image ofthe segment [0, X] C stab(F) under the exponential map.

Since G is supposed to be simply connected, the loop 'y is the boundary ofsome 2-dimensional surface S in G, whose projection to Sl we denote by p(S).The correspondence [7] w [S] is precisely the isomorphism ir1(Stab(F))7r2(Sl) (see formula (6) in Appendix 1.2.3). Finally we have

(F, X)=J 9F=JdO = Jy r(S)

0


3. Coadjoint invariant functions

3.1. General properties of invariants.The ideology of the orbit method attaches significant importance to the

classification problem for coadjoint orbits. The first step in the solution ofthis problem is to find all invariants of the group action. Polynomial andrational invariants are especially interesting.

We also need the notion of relative invariants defined as follows. LetX he a left G-set, and let A be a multiplicative character of G. We say thata function f is a relative invariant of type \ if

f(9-x)=\(9)f(x) for allgE C.

It is known that for algebraic actions of complex algebraic groups onaffine algebraic manifolds there are enough rational invariants to separatethe orbits in the following sense.

Proposition 3 (see [Bor, R]). The common level set of all rational in-variants consists of a finite number of orbits, and a generic level is just oneorbit.

Moreover, each rational invariant can be written in the form R = Pwhere P and Q are relative polynomial invariants of the same type. 0

For real algebraic groups the common level sets of invariants can splitinto a finite number of connected components that are not separated byrational invariants. (Compare with the geometry of quadrics in a real affineplane. The two branches of a hyperbola is the most visual example.)

A useful scheme for the construction of invariants of a group G acting ona space X is the following. Suppose we can construct a subset S C X whichintersects all (or almost all) orbits in a single point. Any invariant functionon X defines, by restriction, a function on S. Conversely, any function onS can be canonically extended to a G-invariant function defined on X (oralmost everywhere on X).

If S is smooth (resp. algebraic, rational, etc.), then we get informationabout smooth (resp. algebraic, rational, etc.) invariants.

Warning. While the restriction to S usually preserves the nice prop-erties of invariants, the extending procedure does not. For example, theextension of a polynomial function could only be rational (see Example 5below). 6

§3. Coadjoint invariant functions 15

3.2. Examples.Example 3. Let G = GL(n,R) act on X = Matn(]R) by conjugation.

Let S be the affine subset consisting of matrices of the form

0 1 0 .. 0

0 0 1 ... 0

0 0 0 ... 1

Cn Cn-1 Cn-2 " ' C1

One can check that S intersects almost all conjugacy classes in exactlyone point. Geometrically this means that for almost all operators A on IItnthere exists a cyclic vector t;, i.e. such that the vectors ,An-1C form a basis in R. It is clear that in this basis the matrix of A hasthe above form.

Using the section S we show that in this case polynomial invariants forman algebra 1R[c1, c2, ... , c ). Indeed, every polynomial invariant restrictedto S becomes a polynomial in C1, c2, ... , e,i. On the other hand, all theci's admit extensions as invariant polynomials on Matn(1R). Namely, theycoincide up to sign with the coefficients of the characteristic polynomialPA(A) = det(A - A 1).

There is a nice generalization of this example to all semisimple Lie alge-bras due to Kostant (see [Ko2] ). 0

Example 4. Let N+ (resp. N_) be the subgroup of strictly upper (resp.lower) triangular matrices from GL(n, ]R). The group C = N+ x N_ acts onX = Mat,, (]R):

g=(n+,n-):Take the subspace of diagonal matrices as S. Then almost all G-orbitsintersect S in a single point (the Gauss Lemma in linear algebra). But inthis case polynomial functions on S extend to rational invariant functionson X.

Namely, let Ok(A) denote the principal minor of order k for a matrix A.It is a G-invariant polynomial on X. Denote by fk the function on S thatis equal to the k-th diagonal element. Then the restriction of Ok to S is theproduct f1 f2 fk. We see that the function fk extends to X as a rationalfunction Ok/Ok-1. 0

In the case of the coadjoint action the polynomial and rational invariantsplay an important role in representation theory due to their connection withinfinitesimal characters (see the next chapters). Here we remark only thatsmooth K(G)-invariants on g' form the center of the Lie algebra C°°(g`)with respect to Poisson brackets.


Indeed, this center consists of functions f such that

But this means exactly that f is annihilated by all Lie vector fields K,(X3),1 < j < n, hence is K(G)-invariant.

4. The moment map

4.1. The universal property of coadjoint orbits.We have seen that any coadjoint orbit is a homogeneous symplectic man-

ifold. The converse is "almost true" : up to some algebraic and topologicalcorrections (see below for details) any homogeneous symplectic manifold isa coadjoint orbit.

This theorem looks more natural in the context of Poisson G-manifolds(see Appendix 11.3.2 for the introduction to Poisson manifolds).

In this section we always assume that G is connected.Let us define a Poisson G-manifold as a pair (M, ft') where M is

a Poisson manifold with an action of G and f : g -. C°O(M) : X F-fX is a homomorphism of Lie algebras such that the following diagram iscommutative:

LVect(M)

(19) 1s-grad

C- (M)

where LX is the Lie field on M associated with X E g and s-grad(f) denotesthe skew gradient of a function f, i.e. the vector field on M such thats-grad(f)g = { f, g} for all g E C°°(M).

For a given Lie group G the collection of all Poisson G-manifolds formsthe category P(G) where a morphism a : (M, f (M)) (N, is a smoothmap from Al to N which preserves the Poisson brackets: {&(O), a'(&)} _a * ({ 0, v }) and makes the following diagram commutative:

C°°(N) ° C°°(M)(20) fcN>

id

9 9

Observe that the last condition implies that a commutes with the G-action.


An important example of a Poisson G-manifold is the space (g', c)

considered in Section 2.2 with the map g -' C°°(g') defined by fX (F) _(F, X).

Theorem 4. The Poisson G-manifold (g`, c) is a universal (final) objectin the category P(G).

This means that for any object (Al, f(f) there exists a unique morphism

p from (M, f() to (g`. f" ), namely, the so-called moment map definedby

(21) (A (M), X) = fx (m)

Proof. A direct corollary of the property (20) of a morphism in the categoryP(G).

Lemma 6. Let M be a homogeneous Poisson G-manifold. Then the mo-ment map is a covering of a coadjoint orbit.

Proof. First, the image µ(M) is a homogeneous submanifold in g`, i.e. acoadjoint orbit ci C g'.

Second, the transitivity of the G-action on M implies that the rank of theJacobi matrix for It is equal to dim M. Hence, p is locally a diffeomorphismand globally a covering of Q. 0

It is worthwhile to discuss here the relation between homogeneous Pois-son G-manifolds and homogeneous symplectic G-manifolds.

From Lemma 6 we see that all homogeneous Poisson G-manifolds are ho-mogeneous symplectic G-manifolds. We show here that the converse is alsotrue when coadjoint orbits are simply connected. In general, the conversestatement becomes true after minor corrections.

This I call the universal property of coadjoint orbits.Any symplectic manifold (M, a) has a canonical Poisson structure c

(Appendix 11.3.2). But if a Lie group G acts on M and preserves a, itdoes not imply that (Al, c) is a Poisson G-manifold. Indeed, there are twoobstacles for this:

1. Topological obstacle. The Lie field LX, X E g, is locally a skewgradient of some function fX, but this function may not be defined globally.

To overcome this obstacle, we can consider an appropriate covering Mof M where all fX, X E g, are single-valued. It may happen that the initialgroup G is not acting on M and must be replaced by some covering groupG.


The simply connected coverings of M and G are always sufficient.2. Algebraic obstacle. The map X '--' fx of g to C°°(M) can always be

chosen to be linear. Indeed, let {Xi} be a basis in g. We choose functionsfi such that Lx, = s-grad fi and put fx = F_i cifi for X = Ei ciXi. But ingeneral this is not a Lie algebra homomorphism: f[x, yJ and {fx, fy} candiffer by a constant c(X, Y).

Exercise 3. a) Show that the map c : g x g R : (X, Y) '--' c(X, Y)is a 2-cocycle on g, i.e. satisfies the cocycle equation

0 c([X, Y], Z) = 0 for all X, Y, Z E g.

b) Check that the freedom in the choice of fx is not essential for thecohomology class of c. In other words, the cocycle corresponding to a dif-ferent choice of fx differs from the original one by a trivial cocycle (or acoboundary of a 1-cycle b):

db(X, Y) = (b, [X, Y])

where b E g' is a linear functional on g.To cope with the algebraic obstacle we have to pass from the initial Lie

algebra g to its central extension g given by the cocycle c. By definition,'g = g ® JR as a vector space. The commutator in j is defined by

(22) [(X, a), (Y, b)] = ([X, Y], c(X, Y)).We define the action of g on M by L(x.a) := Lx. So, practically it is theaction of g = g/Ilt, which is of no surprise because the center acts triviallyin the adjoint and coadjoint representations.

We claim that this action is Poisson. Namely, we put

f(x, a) fx + a

and a simple computation shows that f[(x, a). (Y, b)) = {f(x, a), f(Y, b) }.The final conclusion is

Proposition 4. Any symplectic action of a connected Lie group G on asymplecticmanifold (M, v) can be modified to a Poisson action of a centralextension G of G on some covering M of M so that the following diagramis commutative:

GxM-M(23)

GxM- MHere the horizontal arrows denote the actions and the vertical arrows denotethe natural projections. 0


4.2. Some particular cases.For most "classical" (or "natural") groups the classification of coadjoint

orbits is equivalent to one or another already known problem. In somecases, especially for infinite-dimensional groups, new interesting geometricand analytic problems arise. We discuss here only a few examples. Someothers will appear later.

Example 5. Let G = GL(n, R). This group is neither connectednor simply connected. So, we introduce G° = GL+(n, R), the connectedcomponent of unity in G, and denote by G° the universal cover of G°. Itis worthwhile to note that G is homotopically equivalent to its maximalcompact subgroup O(n, IR), while G° is equivalent to SO(n, R) and G°is equivalent to Spin(n, IR) for n > 3. This follows from the well-knownunique decomposition g = kp, g E G, k E K, p E P, where G = GL(n, IIY),K = O(n, R) and P is the set of symmetric positive definite matrices.

The case n = 2 is a sort of exception. Here G° is diffeomorphic to Sl x 1R3and G° is diffeomorphic to R4.

As we mentioned in Section 1.1, the Lie algebra g = Matn(lR) possessesan Ad(G)-invariant bilinear form

(A, B) = tr(AB).

Thus, the coadjoint representation is equivalent to the adjoint one. More-over, because the center acts trivially, the coadjoint action of G° factorsthrough G° and even through G°/center ^_- PSL(n, R). Therefore, coad-joint orbits for G° are just G°-conjugacy classes in Matn(lR).

Since the Lie algebra g = gl(n, I(g) IR ® 51(n, Iii) has no non-trivial1-cocycles, the algebraic obstacle is absent. So, all homogeneous symplecticG-manifolds are coverings of the G°-conjugacy classes.

Exercise 4.* Show that every orbit is homotopic to one of the Stiefelmanifolds O(nl + +nk)/(O(nl) x . . . x O(nk)).

Hint. Use the information from Appendix 1.2.3.Note that the fundamental group of an orbit is not necessarily commu-

tative, e.g. for n = 3 there are orbits homotopic to

0(3)/(0(1) x 0(1) x 0(1)) - U(1, H)/{f1, ±i, ±j, ±k}.

The fundamental group of these orbits is the so-called quaternionic groupQ of order 8. Q

Example 6. Let G = SO(n, R). Here again, the group is not simplyconnected and we denote its universal cover by G. It is known (and can be


derived from results in Appendix 1.2.3) that

7rI (SO(n, R)) =Z for n = 2.

7L/2Z for n > 3.

For small values of n the group Gn is isomorphic to one of the followingclassical groups:

n

Gn

2 3 4 5 6

R SU(2) SU(2) x SU(2) SU(2, 1HI) SU(4)

For general n > 3 the group Gn is the so-called spinor group Spin n. Themost natural realization of this group can be obtained using the differentialoperators with polynomial coefficients on a supermanifold Roan (see, e.g.,[QFS], vol. 1).

The point is that the operations Afk of left multiplication by an oddcoordinate G and the operations D1, of left differentiation with respect toG satisfy the canonical anticommutation relations (CAR in short):

A14Mj+M3Mi=D;Dj +D3Di=0:

The Lie algebra g = so(n, R) is the set Asymn(R) of all antisymmetricmatrices X satisfying Xt = -X. The coadjoint action of G factors throughG.

The restriction of the bilinear form above to g is non-degenerate andAd(G)-invariant. So, the coadjoint representation is again equivalent tothe adjoint one. The description of coadjoint orbits here is the problem ofclassification of antisymmetric matrices up to orthogonal conjugacy. In thiscase the orbits are simply connected and have the form

S2 = SO(2n1 + + 2nk + m)/U(nl) x ... x U(nk) X SO(7)-

0Example 7. Let G = Sp(2n, R). It consists of matrices y satisfying

=J whereJn On -1nn9 ln On

The Lie algebra g consists of matrices X satisfying

XtJn + JnX = 0, or S = J"X is symmetric : St = S.


The classification of coadjoint orbits reduces in this case to the problem ofclassification of symmetric matrices up to transformations

S - gtSg, g E Sp(2n. R).

0Remark 4. The problems arising in Examples 7 and 8 are particular

cases of the following general problem: classification of a pair (S, A) whereS is a symmetric matrix and A is an antisymmetric matrix with respect tosimultaneous linear transformations:

(S, A)'-' (gtSg, g'Ag), g E GL(n, ]R).

ci

Example 8.` Let M be a compact smooth simply connected 3-dimen-sional manifolds with a given volume form vol. Let G = Diff(M, vol) be thegroup of volume preserving diffeomorphisms of M.

The role of the Lie algebra g = Lie(G) is played by the space Vert(M, vol)of all divergence-free vector fields on M. We recall that the divergence of avector field t; with respect to a volume form vol is a function div 1; on Msuch that L&ol) = div l; vol. Here L is the Lie derivative along the fieldi;. Using the identity (see formula (16) in Appendix 11.2.3)

we obtain that dOC where 8C is some 1-form on M defined moduloexact forms (differentials of functions). Now any smooth map K : S' . Mdefines a linear functional FK on g:

(24)J

K'(OO).st

(It is clear that adding a differential of a function to O does not changethe value of the integral.) Moreover, the functional FK does not change ifwe reparametrize Sl so that the orientation is preserved. In other words, itdepends only on the oriented curve K(S!).

We see that the classification of coadjoint orbits in this particular casecontains as a subproblem the classification of oriented knots in M up to avolume preserving isotopy. Q

Example 9.` Let G = Diff+(Sl) denote the group of orientation pre-serving diffeomorphisms of the circle, and let G be its simply connected

5The famous Poincard conjecture claims that such a manifold is diffeomorphic to .S3 but itis still unknown. We use only the equalities 112(M) = X1(M) = {0).


cover. This group has a unique non-trivial central extension G, the so-calledVirasoro-Bott group.

This example will be discussed in Chapter 6. Here we show only thatthe classification of coadjoint orbits for this group is equivalent to each ofthe following apparently non-related problems.

1. Consider the ordinary differential equation of the second order

(25) Ly = cy" + p(x)y = 0.

If we change the independent variable: x ' 0(t), then equation (25) changesits form: the term with y' appears.

But if, at the same time, we change the unknown function: y ' y o 42 , then the unwanted term with y' disappears and equation (25) goes

to the equation Ly = 0 of the same form but with a new coefficient

0111 - 2 2(26) p = p o O (01)2 + cS(t) where S(0) = (_)Assume now that the coefficient p(x) is 21r-periodic and the function 0(t)has the property O(t + 21r) = 0(t) + 2a. The problem is to classify theequations (25) with respect to the transformations (26).

2. Let G be the simply connected covering of the group SL(2, R), andlet A be the group of all automorphisms of G. The problem is to classifyelements of G up to the action of A.

3. The locally projective structure on the oriented circle S' is de-fined by a covering of S' by charts {UQ}QEA with local parameter tQ on UQsuch that the transition functions OQp are fractional-linear and orientation

ato+preserving. (This means that tQ = ctp+ with ad - be > 0.)

The problem is to classify the locally projective structures on S' up tothe action of Diff+(S' ). 0

Let us make a general observation about the relation between the coad-joint orbits of a group G and of its central extension G by a 1-dimensionalsubgroup A. This observation will explain the relation between the coadjointorbits of the Virasoro-Bott group and problem 1 in Example 9.

Let g and g be the Lie algebras of G and G. As a vector space, g can beidentified with g ® Ilt so that the commutator looks like

(27) [(X, a), (Y, b)] = ([X, Y], c(X, Y))

where c(X, Y) is the cocycle defining the central extension. It is an anti-symmetric bilinear map from g x g to R satisfying the cocycle equation:

(28) 0 c([X, Y], Z) = 0.


Here as usual the sign 0 denotes the sum over cyclic permutations of threevariables.

We identify g' with g' ® R and denote its general element by (F, a).The coadjoint action of G reduces to an action of G because the centralsubgroup A acts trivially.

Lemma 7. The coadjoint actions of G on g' and on g' are related by theformula

(29) K(g) (F, a) = (K(g)F + a S(g), a)

where S is a 1-cocycle on the group G with values in g', i.e. a solution tothe cocycle equation

(30) S(gig2) = S(gl) + K(gl)S(g2).

Proof. Since the extension 'g is central, the action of G on the quotientspace g'/g* is trivial. Therefore, k(g) preserves hyperplanes a = const andon the hyperplanea = 0 coincides with the ordinary coadjoint action ofG on g'. Hence, k(g) has the form (29) for some map S : G g'. Thecocycle property (30) follows directly from multiplicativity of the map K.O

Exercise 5. Show that for any connected Lie group G the map S in (29)can be reconstructed from the cocycle c(X, Y) entering in (27) as follows.

For any g E G the cocycles c(X, Y) and c'(X, Y) = c(Ad g X, Ad g Y)are equivalent.6 Thus, we can write

(31) c(AdgX, AdgY) = c(X, Y) + (4) (g), [X, YJ).

From this we derive that

Ad g(X, a) = (Ad g X, a + (4(g), X) )

and, consequently, (29) follows with S(g) =

5. Polarizations

5.1. Elements of symplectic geometry.We shall use here the general facts about symplectic manifolds from

Appendix 11.3: the notions of skew gradient, Poisson brackets, etc.

In the general scheme of geometric quantization (which is a quantummechanical counterpart of the construction of unirreps from coadjoint orbits)the notion of a polarization plays an important role.

6The infinitesimal version of this statement follows directly from (28).


Definition 3. Let (M, a) be a symplectic manifold. A real polarizationof (M, a) is an integrable subbundle P of the tangent bundle TM such thateach fiber P(m) is a maximal isotropic subspace in the symplectic vectorspace (T,,,M, a(m)). In particular, the dimension of P is equal to

zdim M.

Recall that a subbundle P is called integrable if there exists a foliationof Al, i.e. a decomposition of M into disjoint parts, the so-called leaves,such that the tangent space to a leaf at any point m E Al is exactly P(m).

To formulate the necessary and sufficient conditions for the integrabilityof P we need some notation.

Let us call a vector field t; on M P-admissible if t;(m) E P(m) for allm E M. The space of all P-admissible vector fields is denoted by Vectp(M).

The dual object is the ideal S2p(M) of all P-admissible differential formsw on M which have the property:

w(1;1 i ... , l;k) = 0 for any P-admissible vector fields k = deg w.

Frobenius Integrability Criterion. The following are equivalent:

a) A subbundle P C TM is integrable.b) The vector space Vectp(M) is a Lie subalgebra in Vect(M).c) The vector space lp(M) is a differential ideal in the algebra S2(M).0

In practice only those polarizations that are actually fibrations of M areused. In this case the set of leaves is itself a smooth manifold B and M isa fibered space over B with leaves as fibers. These leaves are Lagrangian(i.e. maximal isotropic) submanifolds of M.

Let Cp (M) denote the space of smooth functions on M which are con-stant along the leaves. In fact it is a subalgebra in COO(M) which can alsobe defined as the set of functions annihilated by all admissible vector fields.

Lemma 8. A subbundle P C TM of dimension 2 dim M is a polarizationip'Cp(M) is a maximal abelian subalgebra in the Lie algebra C°O(M) withrespect to Poisson brackets.

Proof. Assume that P is a polarization. The space Vectp(M) consists ofvector fields tangent to the fibers of P. Therefore, for f E Cp(M) wehave df E Sl ,(M). It follows that s-grad f (m) is a-orthogonal to P(M),hence belongs to P(M). So, for any fl, f2 E Cp(M) we have If,, f2} _(s-grad fl)f2 = 0.

Moreover, if (s-grad fl)f2 = 0 for all f2 E Cp(M), then s-grad f(m) EP(m) and fl is constant along the fibers, hence belongs to Cp(M). Wehave shown that Cp(M) is a maximal abelian Lie subalgebra in CO°(M).


Assume now that Cpl(M) is an abelian Lie subalgebra in C°O(M).According to formula (32) from Appendix 11.3.1 we have If,, f2} =o(s-gradf1is-gradf2). We see that skew-gradients of f E CP (M) span anisotropic subspace at every point m E M. But this subspace has dimension2 dim M, hence must be a maximal isotropic subspace in T,,, (M). 0

There is a remarkable complex analog of real polarizations.

Definition 4. A complex polarization of (M, o) is an integrable sub-bundle P of the complexified tangent bundle TCM such that each fiberP(m) is a maximal isotropic subspace in the symplectic complex vectorspace (TmM, ac(m)).

Here the integrability is defined formally by the equivalent conditions b)and c) in the Frobenius Criterion above.

The space CP (M), as before, is a subalgebra in the complexification ofC°°(M). A simple description of this subalgebra can be given in a specialcase.

Let P be an integrable complex subbundle of TCM. Then its complexconjugate P and the intersection D := P fl P are also integrable (this is aneasy exercise in application of the Frobenius Criterion). On the contrary,the subbundle E := P + P in general is not integrable.

Note that both D and E are invariant under complex conjugation, hencecan be viewed as complexifications of real subbundles Do = D fl TM andEo = E fl TM, respectively.

Proposition 5. Assume that the subbundle Eo is integrable. Then in aneighborhood of every point of M there exists a local coordinate system{u1, ... , uk; x1, , x1; y1, , y!; v1, , Vm} with the following prop-erties:

(i) Do is generated by -, 1 < i < k;

(ii) Eo is generated by -, 1 < i < k, -, 1 < j < 1, and , 1 <

j<l;(iii) P is generated by -, 1 < i < k, and IFx, + ice, 1 < j < 1. 0

The crucial case is Do = 0, Eo = TM. In this case Proposition 5is exactly the Nirenberg-Newlander theorem on integrability of an almostcomplex structure.

Let us introduce the notation zj = xj + iyj, 1 < j < 1. Then we cansay that the algebra CP (M) consists of functions which do not depend oncoordinates vi, 1 < i < k, and are holomorphic in coordinates zj, 1 < j < 1.


Or, more geometrically, the functions in question are constant alongleaves of Do and holomorphic along leaves of Eo/Do.

Remark 5. The subbundles P C T'M for which E = P +P is notintegrable are rather interesting, but until now have not been used in repre-sentation theory. In this case CP (M) is still a subalgebra in C°°(M). Thenature of this subalgebra can be illustrated in the following simple example.?

Let M = R3 with coordinates x, y, t, and let P be spanned by a singlecomplex vector field f = 8x + i0y + (y - ix)8t. In terms of the complexcoordinate z x + iy it can be rewritten as i; = 8i - izOt. Then P isspanned by = 8z + iz8t, and E = C ® C is not integrable since

J = 2iOt E.The equation £ f = 0 is well known and rather famous in analysis. The

point is that the corresponding non-homogeneous equation l: f = g has nosolution for most functions g.

Some of the solutions to the equation f = 0 have a transparent inter-pretation. Consider the domain

D = {(z, w) E C2 I Imw > Iz12}.

The boundary 8D is diffeomorphic to R3 and is naturally parametrized bycoordinates z and t = Re w. It turns out that the boundary values ofholomorphic functions in D satisfy the equation C f = 0. However, they donot exhaust all the solutions which can be non-analytic in a real sense. G

5.2. Invariant polarizations on homogeneous symplectic mani-folds.

In the representation theory of Lie groups one is interested mainly inG-invariant polarizations of homogeneous symplectic G-manifolds.

We know also that the latter are essentially coadjoint orbits. In thissituation the geometric and analytic problems can be reduced to pure alge-braic ones. Let G be a connected Lie group, and let ft c g` be a coadjointorbit of G. Choose a point F E S2 and denote by Stab(F) the stabilizer ofF in G and by stab(F) its Lie algebra.

Definition 5. We say that a subalgebra C g is subordinate to a func-tional F E g' if the following equivalent conditions are satisfied:

(i) F I [4.4]= 0;

(i') the map X F-, (F, X) is a 1-dimensional representation of 1).Note that the codimension of h in g is at least irk BF.

71n this example M is not a symplectic manifold but a so-called contact manifold. In a sense,contact manifolds are odd-dimensional analogues of symplectic manifolds.


We say that h is a real algebraic polarization of F if in addition thecondition

(ii) codima h = irk BF (i.e. h has maximal possible dimension dim 2rk9)

is satisfied.

The notion of a complex algebraic polarization is defined in thesame way: we extend F to gc by complex linearity and consider complexsubalgebras 4 C 9C that satisfy the equivalent conditions (i) or (i') and thecondition (ii).

An algebraic polarization h is called admissible if it is invariant underthe adjoint action of Stab(F). Note, that any polarization contains the Liealgebra stab(F), hence is invariant under the adjoint action of Stab°(F), theconnected component of unity in Stab(F).

The relation of these "algebraic" polarization to "geometric" ones de-fined earlier is very simple and will be explained later (see Theorem 5). Itcan happen that there is no real G-invariant polarization for a given F E g`.The most visual example is the case G = SU(2) where g has no subalgebrasof dimension 2.

However, real G-invariant polarizations always exist for nilpotent andcompletely solvable Lie algebras while complex polarization always exist forsolvable Lie algebras. It follows from a remarkable observation by MicheleVergne.

Lemma 9 (see [Ver1, Di2]). Let V be a real vector space endowed with asymplectic bilinear form B. Consider any filtration of V:

where dim Vk = k. Denote by Wk the kernel of the restriction B I vx . Then

a) The subspace W = >k Wk is maximal isotropic for B.b) If in addition V is a Lie algebra, B = BF for some F E V* and all

Vk are ideals in V, then W is a polarization for F. 0

Note that in [Di2] it is also shown that for a Lie algebra g over analgebraically closed field K the set of functionals F E g' that admit apolarization over K contains a Zariski open subset, hence is dense in g`.

Example 10. Let G = Sp(2n, K), K = R or C. The Lie algebra gconsists of matrices of the form SJ where S is a symmetric matrix with\elements from K and J = I On On" I . The dual space g' can be identified

with g using the pairing (X, Y) = tr (XY). Consider the subset f C g`given by the condition rk X = 1. This set is a single G-orbit in the caseK = C and splits into two G-orbits f2± in the case K = IR.


We show that for n >_ 2 there is no algebraic polarization for any F E Q.Indeed, it is easy to compute that rk BF = 2n for F E Q. So, the would-bepolarization h must have codimension n in g. It also must contain stab(F),hence, h/stab(F) has to be an n-dimensional Stab(F)-invariant subspace inTFQ ^_' g/stab(F) = Ken.

Consider the action of Stab(F) on TFQ ^_' Ken in more detail. Wecan write the matrix F in the form F = vvtJ for some column vectorv E K2,. Two vectors v and v' define the same functional F if v = ±v1.Therefore Stab(F) consists of matrices g satisfying gv = ±v. The connectedcomponent of the unit in Stab(F) is defined by the condition gv = v. It isthe so-called odd symplectic group Sp(2n - 1, ]R) (see Section 6.2.2).

The map v ' F = vvtJ is G-equivariant and identifies the linear actionof Stab(F) on TFQ with the standard one on Ken. According to the Witttheorem,8 this action admits only two non-trivial invariant subspaces: the1-dimensional space Kv and the (2n - 1)-dimensional space (Kv)l. Hence,an n-dimensional invariant subspace can exist only for n = 1. Q

Now we explain the relation between the notions of geometric and alge-braic polarization. As before, denote by pF the map from G onto 1 definedby pF(g) = K(g)F, and by (pF). its derivative at e which maps g onto TFcl.

Theorem 5. There is a bijection between the set of G-invariant real polar-ization P of a coadjoint orbit fl C g' and the set of admissible real algebraicpolarization lj of a given element F E Q.

Namely, to a polarization P C T11 there corresponds the algebraic polar-ization h = (pF).-1(P(F)).

Proof. We use the following general result

Proposition 6. Let M = G/K be a homogeneous manifold.a) There is a one-to-one correspondence between G-invariant subbundles

P C TM and K-invariant subspaces h C 9 containing Lie(K).b) The subbundle P is integrable if and only if the corresponding subspace

h is a subalgebra in 9.

Proof. a) Choose an initial point mo E Al and denote by p the projectionof G onto M acting by the formula p(g) = g mo. Let p. (g) be the derivativeof p at the point g E G. Define a subbundle Q C TG by

Q(g) := p.-, (g) (P(g . mo))

'The Witt theorem claims that in a vector space V with a symmetric or antisymmetricbilinear form B any partial B-isometry Ao : Vo - V can be extended to a global B-isometryA: V -V.


The following diagram is clearly commutative:

(L9).9

(P),Tg(G) Q(g)

(32)

Tm0M(gam i P (g mo)

(Here L. is a left shift by g E G and an asterisk as a lower index means thederivative map.)

It follows that Q is a left-invariant subbundle of TG. Conversely, everyleft-invariant subbundle Q C TG with the property h := Q(e) D Lie(K) canbe obtained by this procedure from a G-invariant subbundle P C TM: wejust define P(g mo) as p.(g)Q(g)

b) Being G-invariant. P is spanned by G-invariant vector fields k, X EE). By the Frobenius criterion P is integrable if the space Vectp(M) is aLie subalgebra. But the last condition is equivalent to the claim that is aLie subalgebra in g.

Let us return to the proof of Theorem 5.For a given subbundle P C TS2 we define h as in Proposition 6 (with

SZ in the role of M and F in the role of mo). We saw in Section 2.1 thatp'(e)a(F) = Bp. Therefore P(F) is maximal isotropic with respect to aiff the same is true for h with respect to BF. The remaining statements ofTheorem 5 follow from Proposition 6.

Remark 6. Let h be a real polarization of F E g`, let P be the realpolarization of ) which corresponds to h, and let H = exp h. Then the leavesof the G-invariant foliation of S2 determined by P have the form K(gH)(F).In particular, the leaf passing through F is an orbit of coadjoint action ofH. Q

In conclusion we note that Theorem 5 can be easily reformulated andproved in the complex case. It looks as follows.

Theorem 5'. There is a bijection between the set of all G-invariant complexpolarizations P of a coadjoint orbit 1 C g' and the set of all complex alge-braic polarizations 4 of a given element F E fl. As before, to a polarizationP C Tn there corresponds the subalgebra 4 = p.(F)-1(P(F)) C gc.

Chapter 2

Representationsand Orbits of theHeisenberg Group

There are several reasons to consider the Heisenberg group separately andin detail before the exposition of the general theory.

First, it is the simplest non-abelian nilpotent Lie group, actually theonly one of dimension 3. Therefore, it has many different realizations.

Second, it appears naturally in quantum mechanics. The description ofall unitary representations of the Heisenberg group is essentially equivalentto the description of all realizations of the canonical commutation relations.

Third, representation theory for the Heisenberg group enters as a "build-ing block" in the general theory, like representations of SL(2) are widely usedin the general theory of semisimple groups.

Finally, the Heisenberg Lie algebra is a contraction of many other im-portant Lie algebras. Therefore, the formulae concerning unitary repre-sentations of the Heisenberg group can be considered as limit cases of thecorresponding formulae for SL(2, R), SU(2, C), and groups of motions ofthe Euclidean and pseudo-Euclidean plane.

31

32 2. Representations and Orbits of the Heisenberg Group

1. Heisenberg Lie algebra and Heisenberg Lie group

1.1. Some realizations.The Heisenberg Lie algebra is a 3-dimensional real vector space

with basic vectors X, Y, Z, satisfying the following commutation relations:

(1) [X,Y]=Z, [X,Z]=[Y,Z]=0.It has a simple matrix realization by upper triangular 3 x 3 matrices:

0 1 0 0 0 0 0 0 1

X= 0 0 0, Y= 0 0 1, Z= 0 0 00 0 U U 0 U 0 0 0

or

(2)

0 x z

xX+yY+zZ= 0 0 y0 0 0

Sometimes it is more convenient to consider a 4-dimensional realization:

(3)

bn

0 x y 2z

xX+yY+zZ= to 0 0 ,y

0 0 0 -x0 0 0 0

In particular. this realization suggests the natural generalization. Letdenote the (2n + 1)-dimensional algebra of block-triangular matrices of

the form0 xt yr 2z0 0 0 y

(4) 0 0 0 -x0 0 0 0

where S, yp are column vectors and :i= t = (xi, ... , and y' = (yi, ... , yn)are the transposed row vectors.

We call fl,: a generalized Heisenberg Lie algebra. From the pointof view of the general theory of Lie groups, hn is the nilpotent radical of amaximal parabolic subalgebra p in the symplectic Lie algebra sp(2n1+2, R).It is also a non-trivial central extension of the abelian Lie algebra 1 2" by a1-dimensional center.

It has the natural basis {Xg, Y;, 1 < i < it, Z} with the commutationrelations:

(5)[Xi.X3]=[Y;,Y?]=[Xi,ZI =[Y, Z]=0,[Xi, Yi] = 6ijz.


Practically, all we can say about the Heisenberg Lie algebra h can beextended to the general 4,,. Moreover, in the appropriate notation the cor-responding formulae look exactly the same as (3) and (4) above.

Let H,, be the simply connected Lie group with Lie h,,. We callit a generalized Heisenberg group. For n = 1 we call it simply theHeisenberg group and denote it by H.

The group H can be realized by matrices

1 it jt 2z

(6) h(x, y", z) = exp (xiXi + yiYi) + zZ = 0 1 00 0 1 -x

3=1 0 0 0 1

We see that the exponential map defines a global coordinate system onH, the so-called exponential coordinates. Hence, H is an exponentialLie group.

Sometimes other coordinate systems are useful, e.g. one can write anelement h E H as

h(a, b, c) = exp aX exp by exp cZ, a, b, c E R.

The triple (a, b, c) form canonical coordinates relative to the orderedbasis {X, Y, Z} in h.

Exercise 1. Find the relation between exponential coordinates (x, y, z)and canonical coordinates (a, b, c) in H.

Answer: a=x, b=y, c= z- 2xy.Exercise 2. Describe all closed connected subgroups in H and deter-

mine which of them are normal.Hint. All such subgroups have the form A = exp a where a C h is a Lie

subalgebra. Normal subgroups in H correspond to ideals in h.Answer: The list of subgroups:a) The trivial subgroup {e}.b) One-dimensional subgroups {exp L} where L is a line (i.e. one-di-

mensional subspace) in h.

c) Two-dimensional subgroups exp P where P is any plane (i.e. two-dimensional subspace in h) that contains the line ]]Y Z.

d) The whole group H.The list of normal subgroups:All subgroups of dimension 0, 2, or 3 and the 1-dimensional subgroup

C = exp ]R Z, which is the center of H. 46


Exercise 3. Consider the space R2n endowed with the standard sym-plectic form or = Fni=1 (dxi A dyi). Show that polynomials of degree < 1form a Lie algebra with respect to Poisson brackets (see Appendix 11.3.2 forthe definition of Poisson brackets). Check that this Lie algebra is isomorphicto 1)n.

Exercise 4. Consider the set 4 of all smooth maps x ' 4(x) de-fined in some neighborhood of zero and subjected to the conditions 0(0) =0, 0'(0) = 1. Fix some natural number N and say that two maps 01 and02 are equivalent if 0(k) (0) _ O2(k) (0) for k = 2, 3, ..., N + 1.

1

Let GN = 4i/- be the set of equivalence classes. It is convenient towrite an equivalence class g E GN as a formal transformation

g(x) = x + aix2 + a2x3 + + aNxN+1 + o(xN+1).

a) Show that GN is an N-dimensional Lie group with global coordinatesystem (a1, ... , aN) where the multiplication law is defined by compositionof transformations.

b) Let gN = Lie(GN). Find the commutation relations in an appropriatebasis in 9N-

c) Show that the group G3 is isomorphic to the Heisenberg group H.Hint. Use the realization of Lie(G3) by equivalence classes of formal

vector fieldsd

v = (c x2 + Qxs +'Yx4 + o(x9))dX

and prove the relations

I xkd

, xtd

J

= (l - k)xk+1-1 d

IL dx dx dx

41

Exercise 5.` Show that for any real x E (0, a)

Sin sin sib (x)N times

c

N--

for some positive constants c and a. Find these constants.Hint. Use the fact that the flow corresponding to the vector field v =

x3 Z9 has the explicit form

(Dt(x)x _

1 - 2tx -x + tx3 + o(It).


It follows that for any c > 0 the inequalities

-6-E(x) S sinx < 1+c(x)

hold for x > 0 in some neighborhood of 0. Derive from it the estimation

6_((x) < sinsin sin (x) S 4 a +, (x).

N times

Answer: or = 2 , c = f . In particular, you can check on yourcalculator that in sin sin (x) 0.17 for a randomly chosen x E (0, 1). r

100 times

1.2. Universal enveloping algebra U(f)).The universal enveloping algebra U(4) is by definition (see Appendix

111.1.4) an associative algebra with unit over C which has the same genera-tors X, Y, Z as the Lie algebra h, subjected to the relations

(7) XY - YX = Z, XZ=ZX, YZ=ZY.

So, the elements of U(fj) are "non-commutative polynomials" in X, Y, Z.There are three natural bases in the vector space U(h). All three estab-

lish a vector space isomorphism between U(lj) and the ordinary polynomialalgebra C[x, y, z] S(h).

First, according to the Poincar6-Birkhoff-Witt theorem, for any basisA, B, C E h the monomials AkBlCm, k, 1, m E Z+, form a basis in U(ry).In particular, we get two isomorphisms

a : C[x, y, z] - U(b) : xkylzm 1 - XkylZm,

b : C[x, y, z] U(()) : xkylzm i . YlXkZm.

Moreover, we also have the symmetrization map(8)sym : C[x, y, z] - U(4)

xkylzm the coefficient atCYkO'Ym

in(aX + /3Y + yZ)k +'l+m

'--+k!l!m! (k+l+m)!

An equivalent formulation is

a a a(8') sym(P(x, y, z)) = P aa, aQ' ay

e.X+,3Y+yzIa-a-7-0


Exercise 6. Show that the map sym has the property

(9)

1sym(alaz...a") = 71 E sym(a9(l))sym(as(z)) ...sym(a8(n))sES

where a1, ... , an are arbitrary linear combinations of x, y, z. In particular,we have

(9') sym((ax + by + cz)') = (aX + by + cZ) ".

Hint. Use the fact that both sides of (9) are polylinear in al, ... , an.Therefore, it suffices to check it for the case when all the ai take valuesx,y,z. 4

There is a beautiful relation between these three maps. Let D be a linearoperator in G[x, y, z] given by the differential operator of infinite order

= as 92 zz 04z

2 8x 8y+ 8(8x)z(8y)z+....

Proposition 1. The following diagram is commutative:

C[x, y, z] D '' C[x,y, z] G[x, y, z]

(10) al SYm I Ib

U(h) U(h) U(4)Id Id

Proof. One of the simplest proofs can be obtained using the realization ofU(f)) by differential operators acting on C[s, t]:

X ~ dtY ' - st, Z S.

Then

k I m _ d k! _ l1 l i d k-: m+l

a(x i Z ) =(dt) ° (st) o s' i> (k)

i(l - i)!t (dt) s '

((b o D2)) xkyIZmr k! l!

b xk-iyl-izm+i=ioi!(k-i)!(l-i)! )

k-i

t.O i! (k - i)!"(l - i)!(st)I-'(dtd

) sm+i


and we see that a = b o Dz. The equalities a = sym o D and sym = b o Dcan be proved in the same way.

We see that sym is sort of a "geometric mean" between a and b suchthat we have the proportion

a: sym = sym:b = D-1.

A useful consequence of Proposition 1 is the following formula:

sym(DxD-1 P) = sym ( I x + I zay) P ) for P E C[x, y. z],

which follows from the obvious relation

\\a(xP)

= Xa(P).Using this formula, we can reduce to the symmetric form the product of

two elements of U(4) written in the symmetric form. The result is

00k

(11) sym-1(sYm(P) ' sYm(Q)) _ E P o Qk=O

where the k-th multiplication law o is defined by

k (2)k. kk a akp akQ(12) PoQ = \2(a)(-1) (Ox)k-a(ay)a

(ax)a(ay)k_..

a=O

In particular,

0 1 1aP N OP aQPoQ = PQ, PoQ =2 ax ay ay ax

1.3. The Heisenberg Lie algebra as a contraction.There is one more remarkable feature of the Heisenberg Lie algebra h:

it appears as a contraction of several other Lie algebras.The notions of deformation and contraction of Lie algebras are described

in Appendix 111.2.2. Let us consider the special class of unimodular Liealgebras. Geometrically this means that on the corresponding connectedLie group G there exists a two-sided invariant top degree differential form,hence, a two-sided invariant measure.

In terms of structure constants the unimodularity condition is given bythe equation

tr(adX)=O forallXEg or ca`k:=Eck=0.


We show in Appendix 111.2.2 that the structure constants of 3-dimen-sional unimodular Lie algebras form a 6-dimensional space 1R6 which is nat-urally identified with the space of 3 x 3 symmetric matrices B with realentries el = btk. k, l = 1, 2, 3. More precisely, to a matrix B = 11b' 11 therecorresponds the collection of structure constants

c A = ci?tbkt

The Lie algebras corresponding to matrices Bl and B2 are isomorphicif there exists a matrix Q E GL(3, R) such that

B1 = QtB2Q det Q-1 where Qt denotes the transposed matrix.

The space R6 of symmetric matrices splits under these transformations intosix GL(3. R)-orbits Ok, 1 < k < 6. The representatives Bk of these orbits,their dimensions, the corresponding unimodular Lie algebras, and their com-mutation relations are listed below:

1 0 0

1 . B1= 0 1 0

0 0 1

1 0 U

2.82= 0 -1 U ,

U 0 1

3. B30 0 0

0 1 00 0 1

0 0 04.B4= 0 -1 0

0 0 1

5.B5= 0 0 00 0 1

0 0 0

dim O1 = 6, g ^_- su(2),

[X,Y]=Z, [Y,Z]=X, [Z,X] =Y.

dim 02 = 6, g - sl(2, R),

[X,YJ=Z, [Y,Z]=X, [Z, X]=-Y.

dim 03 = 5, g ^- so(2, R) a R2

[X, Y] = Z, [Y, Z] = 0, [Z, X) = Y.

dim O4 = 5, g ^-. so(1, 1, 1R) x. 1R2,

IX,Y]=Z, [Y,Z]=0, [Z, X]=-Y.

dim05=4, g^-'h,

[X,YJ=Z, [Y, Z]= [Z,X]=0.6. B6=0, dimO6=0, g^-]R3,

[X,YJ=[Y,Z]=[Z,X]=0.


The matrix B of type 5 can be obtained as a limit of matrices of anyprevious types.

Exercise 7. Find the explicit formulae for the contraction of the Liealgebras su(2, C), sl(2, R), so(2, 1R) x 1R2, so(1, 1, 1R) x lR2 to the HeisenbergLie algebra h.

Answer: In the notation above let X, = eX, YF = fY, Z( = e2Z be thevariable basis in g. When e - 0, the variable commutation relations tendto the limit relations [X0, Yo] = Zo, [Yo, Zo] = [Xo, Zo] = 0.

We see that the Heisenberg algebra is indeed situated in the core of theset of all 3-dimensional unimodular Lie algebras. In physical applicationsthis fact is intensively used: the representation theory of these Lie algebrasand corresponding Lie groups can be constructed as a deformation of therepresentation theory of the Heisenberg group.

2. Canonical commutation relationsThe algebraic structure of the universal enveloping algebra U(h) is relativelysimple. To describe this structure it is convenient to use the formalismof free boson operators from quantum field theory. Here we explain themathematical (mostly algebraic) content of this formalism.

2.1. Creation and annihilation operators.A quantum mechanical description of a free boson particle with one

degree of freedom involves two self-adjoint unbounded operators: the coor-dinate (or position) operator q and the momentum (or impulse) operator p,which are subjected to the

Canonical commutation relations (CCR):

(13)h

Here h is a Planck constant and h := 2 is the so-called normalizedPlanck constant.

Note that in physical applications h and h are dimensional quantities.Their dimension is

[h] = [h] = length2 x mass x time-1 or energy x time.

The numerical value of h is

h 10-27. 9 ' c7nz

= 10-34. kg mz

sec sec


In ordinary scale it is a very small number, practically zero. This ex-plains why in classical mechanics p and q are considered as commuting quan-tities.

In quantum field theory the so-called free boson operators or, moreprecisely, the creation and annihilation operators a*, a are used insteadof p and q.

These operators are defined by the equations

(14) a= -(q + ip), a* = -(q - ip)

and satisfy a variant of CCR: aa* - a*a = h I.In mathematical physics a system of units where h = 1 is mostly used,

so that the commutation relation between a and a* simply becomes

(15) aa* - a*a = 1.

It turns out that many important operators can be written as polyno-mials or power series in a and a* and their properties can be studied in apurely algebraic way using only the commutation relation (15). We shallsee some examples of this below.

Let us denote by Wn the so-called Weyl algebra: the associative alge-bra with unit over C generated by n pairs of operators ak, ak satisfying therelations

(16) akaj = ajak, akaj = ajak, aka - akj.

Sometimes, instead of creation and annihilation operators ak, ak, we shalluse other generators pk = 72'(ak - ak) and qk =72. (ak + ak) with commu-

tation relations

(16') PjPk = Pkpj, qjqk = gkgj, pjgk - gkpj = -ibkj.

The algebra W can be realized as the algebra of all differential operatorswith polynomial coefficients in n variables zl,... , zn. The generators in thisrealization have the form

ak = zk (multiplication), ak = 8/Ozk (differentiation).

The space of polynomials C[zl, ..., zn] admits a special scalar productsuch that ak is adjoint to ak with respect to this product. Namely:

(P, Q) = j P(z)Q(z)e-IZI2Idnzl2

n

where Izl2 = Ekzkzk and Idnzl2 = AdynI, Zk = xk+2Yk


2.2. Two-sided ideals in U(lj).The structure of U(h) is described in

Theorem 1. a) The center Z(h) of U(f)) is generated by the central basicelement Z E f), i.e. Z(t)) = C[Z].

b) Any non-zero two-sided ideal in U(t)) has a non-zero intersection withZ(h)

c) The maximal two-sided ideals in have the form

I,, _ (Z - A 1)U(h)

and for any A 0 0 the quotient algebra is isomorphic to the Weylalgebra Wl.

Proof. a) We can use any of the maps a-1, b-1, sym-1 to identify ele-ments of U(t)) with ordinary polynomials in x, y, z. In all three cases thecommutators with basic vectors X, Y in this realization act as differentialoperators:

(17) ad X = z8/ay, ad Y = -z8/8x.

Therefore the central elements correspond to those polynomials P(x, y, z)that satisfy 8P/8x = OP/ft = 0. Hence, P = P(z) and a) is proved.

b) It is clear that starting with any non-zero polynomial P(x, y, z) andapplying the operators (17) several times we eventually come to a non-zeropolynomial in z. This proves b).

c) Now let I C U(f}) be a maximal two-sided ideal. Then its intersectionwith Z(4) is a maximal ideal in Z(fl) C[Z]. Hence,

I D Ia. But the quotient algebra U(4)/I,\ for A 36 0 is clearly isomorphicto the Weyl algebra W,. Since W, has no non-trivial ideals, I = Ia and weare done. O

2.3. H. Weyl reformulation of CCR.The physical formulation of CCR given above is short and computation-

ally convenient but not quite satisfactory from the mathematical point ofview. The point is that the product of two unbounded operators is not welldefined,' so that (13) and (15) make no sense in general.

We discuss here the correct reformulation of CCR due to H. Weyl.

'More precisely, the domain of definition for the product can be the zero subspace {0}.


According to the M. Stone theorem (see Appendix V.1.2), for any self-adjoint, not necessarily bounded, operator A in a Hilbert space H the oper-ator iA is a generator of a one-parametric group U(t) of unitary operators:

U(t) = eitA, A =i dtU(t) It=0

If the operators p and q satisfy CCR, then the formal computation givespq" = q"p+ 2h-ringn-1. It follows that for any polynomial F we have

pF(q) = F(q)p+ 2rriP(q)

Extending this relation to power series in q, we getpeitq = eitq(p + ht) or a-ityp eit4 = p + ht.

This impliese-itgF(p)eitq = F(p + ht)

and, finally,e-itge'$Peitq = eis(p+ht).

Let us introduce the notation u(s) := etBP, v(t) := eitq. Then the aboverelation takes the form

(18) u(s)v(t) = e"thv(t)u(s).

We have seen that (18) is a formal consequence of (13). Conversely, one caneasily obtain (13) by differentiating both sides of (18) with respect to s andt. So, on the formal level, (13) and (18) are equivalent.

But there is an essential difference between these two relations. Namely,the latter relation involves only bounded (actually, unitary) operators. So,there is no questions about the domain of definition.

We formulate now the Weyl form of CCR:Two self-adjoint operators p and q satisfy CCR if the corresponding 1-

parametric groups of unitary operators u(s) = ei'P and v(t) = eitq satisfythe relation (18).

Although the Weyl form (18) of CCR has the advantage of mathematicalrigor, the original, Heisenberg form (13) of CCR looks simpler and easier touse. Therefore there were many attempts to find some additional conditionson the operators p and q that together with (13) would guarantee (18). Weformulate here the most convenient one found by E. Nelson.

Nelson condition:There is a dense subspace D C H which is stable under p and q such

that the energy operator E := (p2 + q2) I v = (a*a + a'a) Ia is essentiallyself-adjoint.


Theorem 2 (see [Nel]). If the self-adjoint operators p and q satisfy CCR(13) and the Nelson condition above, then they satisfy (18). 0

Remark 1. Consider four operators X := ip, Y:= iq, Z := ih 1, T :=ah E. They span a so-called diamond Lie algebra D with commutationrelations

[T,X]=Y. [T. Y]=-X, [X,YJ=Z, [Z,DJ=0.

Note that the first three basic vectors span the ideal in D isomorphic tothe Heisenberg Lie algebra h. The Nelson condition can be reformulated inrepresentation-theoretic terms. Namely, we require that the given represen-tation of the Heisenberg algebra 4 can be extended to a representation ofthe diamond Lie algebra a D h by skew self-adjoint operators in H. V

2.4. The standard realization of CCR.The standard realization of CCR in the complex Hilbert space L2 (R, dt)

is

(19)dp=-ihd, q =t.

Here we assume that p and q have the so-called natural domain ofdefinition (see the remark below).

Remark 2. For a differential operator A = >k ak (t) d with smoothcoefficients the natural domain of definition DA consists of those functionsf E L2 (R, dt) for which there exists a function g E L2 (R, dt) such thatA f = g in the distributional sense.

It means that for any test function r0 E A(R) we have

J g(t)O(t)dt = J f (t)A*b(t)dt

where A' = > k(-1)kd o ak(t) is the formal adjoint to A. V

In our case DQ consists of those functions f E LC (R, dt) for whichthe function t f (t) belongs to L2 (R, dt). The domain Dp consists of ab-solutely continuous complex functions f E L2 (R, dt) with the derivativef E LC (1R, dt).

Exercise 8. Show that both operators p and q with the natural domainsof definition are self-adjoint.

Hint. For q it can be checked directly by the definition. For p the directproof is also possible, but it is simpler to make the Fourier transform thatsends the operator p to -hq. 46


The creation, annihilation, and energy operators for h = 1 have the form

(20) a*=-(t-dt), E=-d2t +t2.

Exercise 9.' Check that the Nelson condition is satisfied for the oper-ators above.

Hint. Take D = S(R), the Schwartz space. It is certainly stable underboth operators p and q. Check that E with the domain V is essentiallyself-adjoint using the criterion from Appendix IV.2.3. 4

We say that a bounded operator B commutes with an unbounded opera-tor A if B preserves the domain of definition DA and the equality AB = BAholds in DA.

Equivalent formulation: B commutes with all bounded functions of A(actually, it is enough to check that B commutes with the spectral functionEc(A) for all c E R).

Theorem 3. The standard realization of CCR is operator irreducible, i.e.every bounded operator A which commutes with p and q is a scalar operator.

Proof. We start with a result which is an infinite-dimensional analogue ofthe following well-known fact from linear algebra:

If a diagonal matrix T has different eigenvalues, then any matrix A thatcommutes with T is itself a diagonal matrix.

This result is formulated as follows:

Lemma 1. Any bounded operator A in L2 (iR, dt) which commutes withmultiplication by t is an operator of multiplication by some (essentiallybounded) complex function a(t).

Proof. Consider the subset Ho C LC' (R, dt) consisting of functions of theform f (t) = P(t)e_t2 for some polynomial P(t). We have

A f (t) = A P(t)e-t2 = A o P(q) a-t2 = P(q) o A e-t2 = a(t)f (t)

where a(t) = et2Ae-t2. So, A is indeed the multiplication by a(t) on thesubset Ho. Since Ho is dense2 in L2 (R, dt), for any operator B the norm ofthe restriction BIHO is equal to IIBII.

Therefore, the function a(t) must be essentially bounded and the oper-ator A acts as the multiplication by a(t) everywhere.

2For the proof of this non-trivial statement, see, e.g., [KG]. Cf. also the proof of the Stone-von Neumann uniqueness theorem in Section 2.6.


Now, we use the fact that A commutes with p. On the subset of smoothfunctions from LC' (R, dt) we have

2i [a(t)f (t)]' = p o A f= A o p f= 2 i a(t)f'(t).

It follows that a(t) has a zero generalized derivative, hence is a constantalmost everywhere.

We finish our description of the standard realization by introducing theoperator N = a`a. In the standard realization we have N =

2(p2 + q2 - 1).

The physical meaning of this operator is the number of particles: theeigenvector Vk of N corresponding to the eigenvalue k E N describes thek-particle state of the system. In particular, the vector vo corresponds tothe 0-particle state and is called a vacuum vector. It can also be definedas a unit vector satisfying av = 0.

It is easy to check that the operator N satisfies the commutation rela-tions(21)IN, a] _ -a, IN, a'] = a*, or Na = a(N - 1), Na* = a* (N + 1).

It follows from (21) that if v E 7 l is an eigenvector for N with aneigenvalue k, then av and a"v are also eigenvectors for N with eigenvaluesk T- 1. This property justifies the terminology of creating and annihilatingoperators.

In the standard realization the vacuum vector is the function vo EL2 (R, dt) satisfying the equation avo = 0, or hvo = -tvo. It is uniqueup to a phase factor:

(22)e'' L2vo(t) _

72=h . e zn

We shall continue the study of N later.

2.5. Other realizations of CCR.Although all Hilbert spaces of Hilbert dimension No are isomorphic, the

concrete realizations can be very different. Here we describe the realization7-( consisting of holomorphic functions. It has the following advantage: ele-ments of 7-( are genuine functions, not equivalence classes as in the realizationH = L2 (R, dt).

Consider the complex plane C with coordinate z = x + iy. Endow itwith a normalized measure

it = 1 e-(x2+b2)dxdy.IT


Let f be the set of all holomorphic functions f on C that belong to L2 (C, µ).We list the main properties of N in

Proposition 2. a) 7{ is closed in L2(C, µ), hence is a Hilbert space.b) The space of polynomials C[z) is dense in N.

c) The monomials pk(z) = zk,, k > 0, form an orthonormal basis in 7-l.

d) The evaluation functional FF(f) = f (c), c E C, is continuous on 7-land can be written as a scalar product:

(23) Fe(f) = (f, e.) where e,; E 7-l is given by ec(z) = e.Cz

e) The functions e, c E C, form a continuous basis in N in the fol-lowing sense: for any f E 71 we have

(24) P z) = f f (c)ec(z)df2(c) e,)e,(z)dp(c)e

Proof. a) Assume that a sequence of holomorphic functions If,,) convergesin L2(C, µ). We show that it also converges uniformly on bounded sets.Indeed, holomorphic functions have the mean value property:

f (a) = 1 f (z)dxdy for any a E C, r > 0.7rr2 fz-al<r

It follows that the value f (a) can be written as a scalar product (f, fa) forsome fa E L2(C, µ). For example, we can take

eIZ12

fa(z) = r2 Xr,a(z)

where Xr,a is the characteristic function of the disc Iz - al < r.It is clear that when a runs through a bounded set in C, the functions

fa (for a fixed r > 0) form a bounded set in L2(C, µ). Hence, f,, (a) - f (a)uniformly on bounded sets.

Since a uniform limit of holomorphic functions is holomorphic, we seethat 7-1 is closed in L2(C, IL).

b) First, we note that the {c¢k} form an orthonormal system in N. Theproof follows from the direct computation:

1 zkzl r2k 2

(Ok, 451) _ f k lie-ZZdxdy = bkl fk.

e-r dr2 = bkl.Lo


In the case where f is a holomorphic function in L2(C, p), its Taylorseries F-n>o f(n)(0) converges both pointwise and in the norm of L2(C, µ).(The latter follows from the Bessel inequality.)

The pointwise limit is obviously f (z). Consider the L2-limit. Accordingto a), it is a holomorphic function in L2(C, µ). Also, it has the same co-efficients with respect to the orthonormal system {4k}k>o as f. But thesecoefficients up to a factor are just the Taylor coefficients. Hence, the L2-limitis also equal to f.

c) From b) it follows that the system {(k} is complete, hence, is a Hilbertbasis.

d) Let us check that (23) is true for f = 4.k. Indeed,

(Ok, e,,) _ Ok, Echin

- 4k, F_ c = 4hk(c)n>0 n. n>0 n'! n>0

By linearity, the equality holds for any polynomial function f. Finally, theargument we used to prove a) and b) shows that (23) holds for any f E W.

e) Compare the coefficients of both sides with respect to the basis {Ok},k > 0. The right-hand side gives

ICf(c)(ec, cbk)d,u(c) = J f(c) R dl.L(c) = (f, 0k),

c k!

which is exactly the left-hand side. 0

The remarkable feature of f[ is that the operators of multiplication byz and differentiation with respect to z are conjugate:

(25)49

W1, f2) = (

(f1, f.) _ f f1(z)f2(z)e-Z dxdy = J e-z-z fl(z) t f2(z)) dxdy7r 71ff C C

Z

_ JC f2(z)(e-zYf1(z))Zdxdy = zfl(z)f2(z)e-z'dxdy = (zfi, f2)

7r

C

Moreover, we have the obvious relation [ a, z] = 1. Therefore, we obtaina realization of CCR in 7{ putting

49(26) a - , a* = z.z


It is called the Fock realization. The vacuum vector in this realization isvo(z) - 1 and the k-particle state is vk(z) = Zk!

'

Exercise 10. Establish an explicit correspondence between the Fockrealization and the standard one.

Answer: The desired maps A : f L2(R, dt) and A-i are given by

(Af)(t) = + tV- 2 dp., (A)(z) =46

There is one more remarkable realization of CCR. Its existence is relatedto the following fact. Put u = e2irih-'p and v = e2niq. From (23) wesee that u and v commute. It turns out that the Laurent polynomialsr_,,,, nEZ form a maximal commutative subalgebra, isomorphic tothe algebra of smooth functions on the 2-dimensional torus T2 L 82/72.

The realization in question acts on the Hilbert space r2 (L) of squareintegrable sections of a certain complex line bundle C -+ L - T2.

To describe it, we start with the space r(L) of smooth sections. Accord-ing to the algebraic definition of a line bundle (see Appendix 11.2.1), F(L)is a module of local rank 1 over the algebra A(T2). We identify A(T2) withthe algebra of all smooth double-periodic functions on JR2. The space I'(L)consists of all smooth functions .0 on R2 satisfying

(27) O(r + m, 0 + n) = 0(r, 0)e2,r:me for all m, n E Z.

It is evidently the module over the algebra of double-periodic functions.It turns out that 1,(L) can be naturally identified with the Schwartz

space S(R) of rapidly decreasing smooth functions on R.

Proposition 3. The equations

f(28) (r, 0) f (r + k)e2e and f (t) = 4)(t, 0)dOkEZ

establish the reciprocal bijections 46 +-, f between r(L) and S(R).

Proof. First, we observe that for any f E S(R) the series in (28) convergesuniformly together with all derivatives. It follows that this series defines asmooth function 4)(r, 0), which satisfies (27).

Conversely, for any 0 E I'(L) the integral in the second part of (28)defines a smooth function f on R. We want to show that this functionbelongs to S(R), i.e. that

m jtmfiti(t)I<oc for allk,IEZ+.


From (27) and the second part of (28) we obtain

Therefore

f (t + k) = f {(t, 9)e21rikodO.

0

II(27rk)mf(!)(t + k) I <_ r ¢(t, 9)Id9

0

Now, if we write tin the form t = k + {t} where k E Z and (11 E [0, 1), weobtain

It,..f(`)(t)I < (Ikl + 1)...If(')(k + {t})I < E ( ) (27r)s < x.8=0 M

Finally, the two maps (28) are reciprocal because

ft 1 if k = 0,e2nikod9

0 otherwise.

The operators q = t and p = 2i j, act on S(R). After the identificationabove they take the form

(29)h

dTp -,27ri 27ri

Define the scalar product of two smooth sections of L by

1

'(30) (01 b2) =10 f 01 (7, 9)02(T, 0)d rd6.0 0

(Note that because of (27) the integrand above is a periodic function on R2,so the integration is actually over the torus T2.)

It is not difficult to see that the operators (29) initially defined on thespace of smooth sections are essentially self-adjoint with respect to the scalarproduct (30). Moreover, they evidently satisfy CCR.

The computation of the vacuum vector in this realization leads to veryinteresting number-theoretic investigations, which we leave aside (see [LV]for details).

2.6. Uniqueness theorem.One very important fact is that the standard realization of CCR is ac-

tually the unique one. We give here three formulations and outline threedifferent proofs of this theorem.

q=T - - d o


Theorem 4 (Stone-von Neumann uniqueness theorem). Let p and q be self-adjoint operators in a Hilbert space H satisfying the canonical commutationrelation in the Heisenberg form (13) and also the Nelson condition.

Then H is isomorphic to a direct sum of several copies of L2(R, dt)where the operators p and q act by the standard realization (19).

In particular, any irreducible realization of CCR is equivalent to thestandard one.

Proof. The idea is to investigate in detail the spectrum of the self-adjointoperator N = a* a.

Proposition 4. a) In any realization of CCR there exists a vacuum vector.h) For an irreducible realization the vacuum vector is unique (up to a

scalar factor).c) All irreducible realizations of CCR are equivalent to each other. There

exists an orthonormal basis {Vk}k>o such that the operators a, N, a' aregiven by

(31) avk=v"i Vk-1, avk=

Proof of Proposition 4. a) Observe that N is a non-negative operator:

(Nv, v) = IavI2 > 0.

So, the spectrum of N is contained in the right half-line.Let E,,, c E R, be the spectral function for N. From (15) we conclude

that Ea = aEc+1.3 Let co > 0 be the minimal point of the spectrum ofN. Then for any e E (0, 1) the projector is non-zero, while aEEt,+e _Ec{,+E-la = 0. Therefore, all vectors from the subspace H,,0+f := Eco+EHare annihilated by a. Hence, they are vacuum vectors (eigenvectors for Nwith zero eigenvalue).

We see that actually co = 0 and the zero eigenspace for N is differentfrom {0}. Thuus, any realization of CCR has a vacuum vector.

b) and c). Choose an arbitrary vacuum vector vo of unit length and put

(32)(a.)k

vk :=kl

vo

3Here and below we use the fact that all vectors in the range of E. belong to the domainof definition of the operators a and a'. It follows easily from the study of the graph of N (seeAppendix IV.2.3).


Then the last two equalities (31) will be satisfied. Let us prove the first one.Indeed,

avk = J=[a, (a*)k]vo = kf(a*)k-1vo

= VIC 7Jk-1.

The vectors vk are orthogonal since they belong to different eigenspacesof N. From the equalities

1Vk12 = (vk, k 2a*vk-1) = (k zavk, Vk-1) = jVk_112

we see that IVkI = 1 for all k > 0. Therefore, {vk} forms an orthonormalsystem.

Suppose now that the realization in question is irreducible. Accordingto (31), the Hilbert subspace spanned by {vk}k>o is invariant under bothoperators a and a*. Then it must coincide with the whole space.

The uniqueness of the vacuum vector in H now follows immediatelyfrom (31) : for any vector v = Ek>o ckvk we have av = Fk> 1 ck f vk_ 1. So,av=0iffck=0for k>1,i.e. v=covo. 0

We return to the proof of Theorem 4. Consider an arbitrary realizationof CCR in a Hilbert space H. Let Ho be the subspace of vacuum vectors.Let {vo°i }aEA be an orthonormal basis in Ho and put vk°1 = ° kk voal. Let

H(a) be the Hilbert space spanned by {vka)}k>O. We leave to the reader tocheck that H = ®QEAH(a) (direct sum in the category of Hilbert spaces)and in each Heal we have the standard realization of CCR. 0

Note that in the course of the demonstration we obtained another proofof the irreducibility of the standard realization. Indeed, we have seen thatin the standard realization we have a unique vacuum vector (22).

The explicit formula for the functions vk can be obtain using the relationL2 1 d t2a`=e2o---oe2.

T2 dt

Namely,

(a*)k e z 1 d k _,2 Hk(t) _e2vk =

VPvo =

P12-d/ e =

k1e 2

where Hk are the so-called Hermite polynomials. We recover the well-known fact that functions vk, called Hermite functions, form an orthonor-mal basis in L2(R, t).

The second formulation and proof of the uniqueness theorem uses theWeyl form of CCR directly.


Theorem 4'. Let H be a Hilbert space, and let u(s) = else, v(t) = e`t9 bethe two 1-parametric groups of unitary operators in H that satisfy the CCRrelation in the Weyl form (18). Assume also that the space H is irreduciblewith respect to these groups.

Then there is an isomorphism a : H -+ L2(IR, dx) such that for all sand t the following diagrams are commutative:

H H H V(t)H

al la aI la

L2(IR, dx)U(s)

L2(IR, dx), L2(IR, dx) L2(IR, dx)

where

(33) (U(s) f) (x) = f (x + hs), (V (t) f)(x) = e:txf (x)

Proof. We shall use the spectral theorem for the self-adjoint operator q =-iv'(0), the generator of the group {v(s)} (see Appendix IV.2.4). Accordingto this theorem, there exists a family (p1, p2i ..., per) of pairwise disjointBorel measures on R such that H is isomorphic to the direct sum of one copyof L2(]R, p1), two copies of L2(R, 112), ... , and a countable set of copies ofL2(R, per) where q acts as multiplication by x. Moreover, these measuresare defined uniquely up to equivalence.

The relation (18) implies that u(s)qu(s)-1 = q+hs 1. So, the operators qand q + a 1 are unitarily equivalent for all a E R. But the spectral measures{pk} for the operator q + a 1 are just the spectral measures {pk} for qshifted by a. It follows that each measure pk is equivalent to the shiftedmeasure. Such measures are called quasi-invariant under the shifts. Wenow use the following fact from measure theory.

Lemma 2. Any non-zero Borel measurep on IR, which is quasi-invariantunder all shifts, is equivalent to the Lebesgue measure A = IdxJ.

Proof of Lemma 2. Consider the plane R2 = R x R with the productmeasure A x it. For any Borel subset A C R denote by XA the subset of theplane defined by

XA={(x,y)ER2 I y-xEA}.We can compute the (A x p)-measure of XA in two different ways usingFubini's theorem:

1. (A x p)(XA) = fxA A x it = fR A(y - A)dp(y) = \(A)p(R).

p(x + A)dx =0

2. (.1 x p)(XA) = f p x A = f Iif p(A) = 0,

xA R > 0 if p(A) > 0.


We see that the conditions µ(A) = 0 and A(A) = 0 are equivalent.Hence, the measure p is equivalent to A. O

We return to the proof of Theorem 4. The measures pk are pairwisedisjoint and at the same time equivalent to A, if non-zero. We conclude thatall /2k vanish except for some jz which is equivalent to A.

Hence, we can assume that H is a direct sum of n copies of L2(IR, dx)where q acts as the multiplication by x. The space H can be interpretedin two ways: either as a tensor product W ® L2(R, dx) where W is an n-dimensional Hilbert space, or as the space L2(IR, W, dx) of W-valued func-tions on R. We shall use both interpretations.

Thus, we have reduced the operators v(t) to the canonical form

(34) v(t) = 1® V(t)

where the V (t) are given in (33).

The next step is reducing the operator p to the form 27ri , or, what isthe same, reducing u(s) to the form 10 U(s). To this end we observe thatu(s) and 10 U(s) interact in the same way with v(t):

u(s)v(t)u(s)-1 = eit"v(t) = (1 ® U(s)) v(t) (1 (& U(s))-1.

Indeed, both equalities follows from (18) if we recall that v(t) = 1 ® V(t).Therefore, the operator u(s)-1(1(& U(s)) commutes with v(t) for all t.

Now, we use the following general fact.

Lemma 3. Any bounded operator in L2(IR, W, dx) that commutes with mul-tiplication bye", t E IR, is itself a multiplication by an operator-valued func-tion A(x) with values in End W.

Proof of Lemma 3. For k = 1 the statement is similar to Lemma 1 inSection 2.4 and can be proved in the same way. Namely, we consider thesubspace Ho C L2(IR, dx) that consists of functions of the form f (x) =

T(x)e_x2

where T(x) is a trigonometric polynomial (a linear combinationof functions eitx for different t's). This subspace is dense in L2(IR, dx) andan operator A that commutes with multiplication by all eiex, t E R, on thesubspace Ho coincides with multiplication by the function a(x) =

ex2Ae_x2

For general k we proceed as follows. Let w1, ..., w,, be any orthonormalbasis in W. Introduce the functions

ai9 (x) =ex2 (A(e-x2

®wj ), Wi)W, 1 < i, 3 < k.

It is straightforward that on the subspace Ho ® W the operator A coincideswith the multiplication by the matrix-function A(x) := JIa,j(x)JI. Since


Ho ® W is dense in H, the norm of A(x) is almost everywhere bounded byIIAII. Hence, the multiplication by A(x) defines a bounded operator on H.This operator coincides with A on Ho 0 W, hence everywhere.

Applying Lemma 3 to the operators u(s)-1(1(9 U(s)), we reduce u(s) tothe special form

(35) u(s) = A(s, x) ® U(s)

where for all s E IR the matrix A(s, x) defines a unitary operator in W foralmost all x E R.

Since u(s) as well as U(s) form 1-parametric groups, we get the relations

A(sj, x)A(s2i x + s1) = A(sl + 82i x), A(-s, x + s) = A(s, x)-1,

which hold for any s1, 52i s for almost all x.Let us put B(x, y) := A(y - x, x), 81 = y - x, s2 = z - y. Then

A(s, x) = B(x, x + s) and the above relations take the simple form

(36) B(x, y)B(y, z) = B(x, z), B(x, y) = B(y, x)-1

for almost all (x, y, z) E R3 (respectively, for almost all (x, y) E 1R2)

Remark 3. If we knew that (36) is true not only almost everywhere,but everywhere, this equation would be easy to solve. Indeed, put C(x) =B(0, x) = B*(x, 0). Then (36) for z = 0 implies that B(x, y) = C(x-1)C(y).Conversely, for any unitary-valued operator function C(x) the expressionB(x, y) = C(x)-1C(y) satisfies (36). Q

Lemma 4. Any measurable unitary-valued operator function B(x, y), whichsatisfies (36) for almost all x, y, z, has the form

(37) B(x, y) = C(x)C(y)-1 for almost all (x, y) E R2

where C(x) is a measurable unitary-valued operator function on R.

From the lemma we get u(s) = A(s, x) O U(s) = C(x)C(x + s)-1 ®U(s)and the transformation f (x) C(x) f (x) sends u(s) to the desired form10 U(s) and preserves the form of the operators v(t) = 1 ® V(s).

Hence, our realization is indeed a direct sum of n copies of the standardone.

Proof of Lemma 4. To make the exposition easier, we consider at firstthe case dim W = 1.


Then B(x, y) will be simply a complex-valued function with absolutevalue 1 almost everywhere. We shall denote it b(x, y). Our goal is tofind a complex-valued function c(x) of absolute value 1 such that b(x, y) _c(x)-lc(y) almost everywhere.

Let a, ,3, -y be test functions from A(R).

fIthe notation

fa(y) := f a(x)b(x, y)dx, ca(x)b(x, z)ry(z)dxdz.

Let us multiply the basic relation b(x, y)b(y, z) = b(x, z) by a(x) ,3(y)-y(z) and integrate over Iit3. Using the relation b(y, z) = b(z, y), the resultcan be written as

J fa(y)f. (y)/3(y)dy = ca,7 f J3(y)dy

It follows that fa(y) fy(y) = ca,.y as a distribution on ]ft, or

(38) fa(y) f-,(y) = ca,-, almost everywhere.

From (38) we conclude that ca,a = I fa(y)12 > 0 and fa(y) = ..y) ifGr,ry 96 0.

Hence, all functions f(x), a E A(IR), are proportional to some fixedfunction with absolute value 1, which we denote by c(x) . More precisely,we have

f,,, (x) = ca c(x), where IcQI2 = ca,a and cac.y = C.'-Y.

Put (a) = fR a(x)c(x)dx. Then from equalities

fa(x)b(x , z)y(z)dxdz =J

fa(z)y(z)dz = ja(x)f.)(x)dxI

we get ca,ry = ca (y) = (a) c?, which implies (a) = ca. It follows thatb(x, z) = c(x)c(y) as desired.

The general case dim W > 1 is technically more difficult, but the schemeis the same. The operator function B(x, y) satisfies the relations

(39) B(x, y)B(y, z) = B(x, z), B(x, z) = B'(z, x) = B-1(z, x).

Let a, -y be test vector-functions on P with values in W. We introduce thevector-valued functions Fr(y) and the numbers ca,ry by

F.y(y) = j B(y, z)y(z)dz, can = f2 (B(x, y)a(y), y(x))u.dxdy.


Then from the relations (39) we get for any 0 E A(IR):

f (Fa(y), F7(y))yy$(y)dy = ca,7 f 3(y)dy.R R

This implies that (Fa(y), F. (y))jt, = Ca,., for almost all y E R.Geometrically, the last relation means that the configuration of vectors

{FQ(y),a E A(IR)} has the same shape for almost all y E R. Therefore,there exists a collection of vectors {va, a E A(R)} and a unitary operatorC(y) such that Fa(y) = C(y)va for almost all y E JR and all a E .A(R). Itfollows that B(y, z) = C(y)C-1(z) almost everywhere.

The third proof is based on the Inducibility Criterion from AppendixV.2.1. Namely, consider the representation (ir, H) of the Heisenberg groupdefined by the formula

(40) ir(a, b, c) = e2! v(b)u(a).

It is easy to check, using the Weyl form of CCR, that (40) is indeed a unitaryrepresentation of the Heisenberg group (see Section 3.1).

Repeating the first part of the previous proof, we can assume that Tr actsin L2(R, W, dx) by the formula

(41) (7r(a, b, c) f)(x) = e`(Rc+bx)A(a, x) f (x + ha).

We also define the *-representation (II, H) of A(R) by

(42) (11 (0)f)(x) = O(x)f(x)

From (41) and (42) we conclude that the representation (40) of the Heisen-berg group is compatible with the representation (II, H) of A(R) if we con-sider R as a homogeneous manifold where the Heisenberg group acts by therule

(43) (a, b, c) x = x + ha.According to the Inducibility Criterion, (ir, H) is induced from some repre-sentation (p, W) of the stabilizer of some point xo E R. If (7r, H) is irre-ducible, so is (p, W). But the group Stab(xo) is abelian, so its irreduciblerepresentations are 1-dimensional and have the form (0, b, c) ' ehc+ab.

The induced representation, constructed along the standard proceduredescribed in Appendix V.2.2, looks like

(44) (7r(a, b, c) f)(x) = e`(hc+b(x+.\)) f(x + ha).

Exercise 11. Show that representations (44) for any two values of \ E 1Rare equivalent.

Hint. Use the shift operator to intertwine both unirreps.It remains to observe that for A = 0 the formula (44) coincides with the

standard representation.


3. Representation theory for the Heisenberg groupHere we collect the main facts about the unirreps of H using the results ofthe previous sections.

3.1. The unitary dual H.We recall that by the unitary dual G we mean the set of equivalence

classes of all unirreps for the given topological group G. This set has anatural topology (see Appendix V.2 and Chapter 3) and also can be viewedas a "non-commutative manifold", which we discuss later.

There is a straightforward connection between CCR in the Weyl formand unitary representations of the Heisenberg group. Indeed, let us writethe general element 9a,b,c of H in the form

1 a c9a,b,c = exp cZ exp by exp aX = 0 1 b

0 0 1

The 1-parametric subgroups of H satisfy

exp aX exp by = exp abZ exp by exp aX.

Therefore, if u(s), v(t) satisfy (23), we can define for any A E R theunitary representation Ira of H by the formula

1a(9a.b,c) = e27r:acv(Ab)u ()

This representation is irreducible, if the representation {u(s), v(t)} of CCRhas that property. In particular, for the standard representation of CCR wehave

(45) [7ra(9a,b,c)f] (x) = e27ria(c+bt) f(x + a).

Conversely, if it is any unitary irreducible representation of H, then theelements of the center are represented by scalar operators. Hence, we have

(46) 7r(expcZ) = e21rAc . 1 for some A E R.

If A 71- 0, we can define the irreducible representation of CCR in the Weylform by

(47) u(s) = ir(exp shX), v(t) = ir(exp to-'Y).

It is clear that the two correspondences are reciprocal. So, due to theuniqueness theorem, for any real number A 54 0 there exists exactly one


unirrep it of the Heisenberg group satisfying (46), namely the representationIra given by (45).

The remaining irreducible representations of H are trivial on the centerC = exp R R. Z. Hence, they are actually representations of the abelian groupH/C and must be 1-dimensional. The general form of such representationsis

(48) 21ri(aju+bv)7rµ,v (ga,bx) = e

The final result can be formulated as

Theorem 5. The unitary dual H for the Heisenberg group H splits intotwo parts:

a) the 1-parametric family of equivalence classes of infinite-dimensionalunirreps ira, A # 0, and

b) the 2-parametric family of 1-dimensional representations 7r,,,,.

Note that the equivalence class of Ira can be realized as a unirrep in thespace 71 of holomorphic functions on C. Here the representation operatorsare given by

(49) (7r,\ (a, b, c)F)(z) = eaz+RF(z - 5)

where a - b "A, f3 = - 2 + iA(2 + c).We leave to the reader to write explicitly the third realization of the

same equivalence class in the space of sections of a line bundle over the2-torus.

Thus, H_= ]i8\{0} U R2 as a set. It is reasonable to expect that thetopology of H is the natural one on each of the pieces R\{0} and 1R2. Amuch more interesting question is: what is the closure of ]I8\{0} in H? Or,in other words, what is the limit4 lima_o Tra?

If we simply put A = 0 in (45), we obtain a highly reducible represen-tation which actually is a continuous sum of 7rµ,o, It E R. However, it doesnot yet mean that the limit contains only representations with v = 0.

For example, if we make a Fourier transform in (45) and only after thatput A = 0, we get another reducible representation, namely, the continuoussum of iro,v, v E R. And if we make an appropriate transform in (49) andthen put A = 0, we get the continuous sum of all 1-dimensional imirreps.

It turns out that the last answer is correct: the limit of Ira when A - 0is the set of all 1-dimensional unirreps. We omit the (rather easy) proof ofit. A more general fact can be found in Chapter 3.

4Since the topological space H is not Hausdorff, the limit of a sequence can be more thanone point.


3.2. The generalized characters of H.The notion of a generalized character is explained in Appendix V.1.4.For 1-dimensional representations ir,,,, the generalized characters coin-

cide with multiplicative characters and with representations themselves.For the unirreps Ira generalized characters are non-regular generalized

functions which can be computed explicitly. To do this, we pick a ¢ E .A(H).The operator

7ra(¢) :=J

¢(a, b, c)7ra(g(a, b, c))dadbdcH

is an integral operator in L2(]R, dx) with the kernel

y - x, b. c)e2idbdc.K,(x, y) =fR2

¢(

This kernel is a rapidly decreasing function on 1k2. Therefore, the operatorir,\(¢) is of trace class and its trace is given by the formula

tr7ra(¢) = fR Kc (x, x)dx = f ¢(0, b, c)e21ria(bx+c)dbdcdxa1 2riAcdC'Jo,

(50) XA(a, b, c) = -6(a)6(b)e2niac

Introduce the Fourier transform

¢(x, y, z) := f ¢(a, b. c)e2a:(ax+by+cz)dxdydz.a

In terms of ¢ the trace of 7r,\(O) can be written very simply:

(51) tr 7r,\(¢) = f ¢(x, y, A) dx A dy

We have used above the canonical coordinates on the group H. However,the final formula is also valid in exponential coordinates. These coordinatesidentify the Lie group H with its Lie algebra b. Therefore, the Fouriertransform ¢ lives naturally in the dual space lj' with coordinates x, y, z.

We see that the generalized character is the Fourier transform of themeasure dxAdy, on the hyperplane z = A.


3.3. The infinitesimal characters of H.The infinitesimal character I, of a unirrep it of H is defined in Appendix

V.1.3 as a homomorphism of the center Z(l)) of the enveloping algebra U(h)into the complex field C. given by

ir(A) = I, (A) 1 for all A E Z(h)-

As we have seen in Section 2.2, the center of U(h) is just C[Z]. So, theinfinitesimal character I,r of a unirrep 7r is defined by one number 1r(Z).

From the formulae (45) and (48) we immediately obtain

(52) IR,, (Z) = 21riA, I-"N.,,(Z) = 0.

Thus, the infinite-dimensional unirreps are determined by their infinitesimalcharacters up to equivalence, while all 1-dimensional unirreps have the sameinfinitesimal character.

Later we shall see that this situation is typical: the infinitesimal char-acters form a coordinate system on the set of "generic" representations anddo not separate the "degenerate" ones.

3.4. The tensor product of unirreps.Using the explicit formulae (45) and (48) it is easy to decompose the

product of two unirreps into irreducible components. We consider here thecase of two infinite-dimensional unirreps 7ra,, aa2. Their tensor productacts in the Hilbert space L2(R, Idxj) ® L2(R, Idyl) L2(]R2, Idx A dyl) bythe formula

((7ra, ® 'r,\2) (a, b, c) F) (x, y) =e21ri(A1(br+c)+a2(by+c)) F.(x + a, y + a).

Denote by L9 the line given by the equation x - y = s and by H8 the Hilbertspace of square-integrable functions on L8. We see that the representationabove is a continuous sump of representations if8 acting by the same formulain the spaces H.

We choose t = \IT+-\2" as a parameter on the line L8. Then the repre-a,+a2sentation 1rr8 takes the form

(*8(a, b. c)¢) (t) = e27r,(a,+A2)(bt+c))0(t + a).

It follows that all irr8 are equivalent to 7r.\,+a2, provided that Al + A2 # 0.Thus.

(53) irA1 (9 7r,\2 = Oo - 1r 1 +a2.

§4. Coadjoint orbits of the Heisenberg group 61

This result is in accordance with Rule 5 of the User's Guide (see the Intro-duction) and with the count of functional dimensions. Indeed, the functionaldimension of Ira is 1, so the product irai 0 7r,\2 has the functional dimension2, as well as the right-hand side of (53). Note, in this connection, the generalformula

71

® 7r.k = OCn-1 Ira,k=1

n

provided 0#A:=E,\,.k=1

Exercise 12. Prove the relations

(54) Ire, ® 2r_a = f 7ra,vI dp A dvi, a ®n ,, _ tea,

mµ1,41 ® 7rA2,112, = 7rµ1+µ2,1/1+ii2

46

4. Coadjoint orbits of the Heisenberg group

4.1. Description of coadjoint orbits.We shall use the matrix realization (2) for the Lie algebra h. The dual

space g' is identified with the space of lower-triangular matrices of the form

F= x * *

Z y *

Here the stars remind us that we actually consider the quotient space ofMat3(R) by the subspace ga- of upper-triangular matrices (including thediagonal).

The group H consists of upper-triangular matrices

1 a cg= 0 1 b

0 0 1

The coadjoint action is(55)

1 a c * * * 1 -a -c + abK(g)F = p(gFg-1) = 0 1 b x** 0 1 -b

0 0 1 z y* 0 0 1

* * *x+bz * *

z y-az *

or

K (9a.b.c) (x, y, z) = (x + bz, y - az, z).


in agreement with the general statement

(58) s-grad fx = K. (X) where fx(F) = (F, X).

The canonical Poisson structure on fl' in coordinates x, y, z is given by

-{fl,f2}=z Oh af2_- 19h M(ax dy dy 8x

We see that the symplectic leaves are the planes z = const # 0 and thepoints (x, y, 0) of the plane z = 0, i.e. precisely the coadjoint orbits.

4.3. Projections of coadjoint orbits.Let A C H be a closed subgroup, and let a = Lie (A) be its Lie algebra.

According to the ideology of the orbit method, the functors IndA and ResHare related to the canonical projection pa : g' -p a'. So, we consider herethe geometry of these projections for different subalgebras a.

First let dim a = 1. There are two different cases.1. a = c = lR Z is the central subalgebra. Looking at Figure 1. the

projection pa is just the horizontal projection to the vertical axis. Therefore,for any orbit Sl c g' the image p(Q) is just one point:

pa(IZA) = a, Pa(1l ,) = 0.

2. a = R R. (oX + /3Y + -yZ) is a non-central subalgebra. Then, in thesame picture, pa is a non-horizontal projection to some non-vertical line. Theimage of 1l, is the whole line, and the image of Q,,,,, is the point aµ + 13v.

Now consider a 2-dimensional subalgebra a. It can be written as Flwhere F E g* has the property (F, Z) = 0.

Looking again at Figure 1, we can view pa as the projection along thehorizontal line R F to an orthogonal 2-dimensional plane. Therefore, p a (1& . )

is a line in a*, while pa(fl,,,,,) is a point.

5. Orbits and representationsIn this section we compare the set O(H) of coadjoint orbits with the set Hof equivalence classes of unirreps. According to Theorems 4 and 5, both arethe union of two pieces: the real line with the origin deleted and the realplane.

It suggests the following one-to-one correspondence between orbits and(equivalence classes of) unirreps:

S2a e==* 7ra, SZ,,,,, . zrµ,,,.


We show below that this correspondence allows us to formulate the mainresults of the representation theory of H in simple geometric terms. More-over, these formulations can be extended to wide classes of Lie groups, aswe show in subsequent chapters.

5.1. Restriction-induction principle and construction of unirreps.We start with the following general statement which is an essential part

of the ideology of the orbit method.Restriction-induction principle:If H C G is a closed subgroup and p : g* is the natural projection

(the restriction to h of a linear functional on 9), then the restriction andinduction functors can be naturally described in terms of the projection p.

In the next chapter we make this statement more precise and write it inthe form

a) ResHrrn = J m(1l, w)irwdw,WCp(I)

b) IndH-7r,,, _

S)C K(G)p- ' (w)

where

7rn E G is the unirrep of G associated to an orbit Sl E O(G);

7rW E H is the unirrep of H associated to an orbit w E O(H);Resyan is the restriction of irn on H;IndH7r,,, is the representation of G induced by 1r,,,;

m(Q, w) is the multiplicity factor, which here takes only values 0, 1, oo;

integrals denote the continuous sum of representations (see AppendixIV.2.5) with respect to appropriate measures dw and dIl on the sets oforbits.

Right now we prefer to use the restriction-induction principle in a some-what fuzzy form in combination with common sense.

We start with the simplest case of an abelian Lie group, G = lR'. TheLie algebra Lie (G) is the vector space R' with the zero commutator. Allcoadjoint orbits are single points in the dual vector space (n)*.

Compare it with the set of unirreps for 1R". Recall that for any abeliangroup A all unirreps are 1-dimensional and are just multiplicative characters.

Let A be a locally compact abelian topological group. The set of allcontinuous multiplicative characters of A is denoted by A. It is endowedwith the group structure (ordinary multiplication of functions) and with


the topology of uniform convergence on compact sets. It turns out thatA is also an abelian locally compact topological group which is called thePontryagin dual of A.

Pontryagin duality principle. a) The canonical map -0: A A, givenby

(41(a))(X) = X(a),

is an isomorphism of topological groups.

b) For any invariant measure da on A there exists a dual invariantmeasure dX on A such that the direct and inverse Fourier transforms

F : f (a) .,.. f (X) =f f (a)X(a)da and F : 0(X) ^"' 'Y(X) _:f O(X)X(a)dXA A

are reciprocal unitary isomorphisms between L2(A, da) and L2(A, dX). U

In particular, the Pontryagin dual Rn to the abelian group R" is itselfisomorphic to Rn and can be identified with the dual vector space (W')'.We choose the identification which associates to A E (R")* the characterX,\ (x) = e2,,i(a, x). The advantage of this choice is that the standard Lebesguemeasures dx and dA are dual to each other (there are no additional factorsin the direct and inverse Fourier transforms).

Now we consider the one-point coadjoint orbits for any simply connectedLie group G and relate them to 1-dimensional representations of G. Let {F}be a one-point orbit in g`, and let a{F} be the corresponding representationof G.

The restriction of this representation to a 1-parametric subgroup exp IRX must correspond to the restriction of F to the subalgebra R X. Using theidentification A d& Xa chosen above, we conclude that the representation7r{F} must have the form

(59) n{F}(exp X) = e2ai(F,x)

Remark 4. Formula (59) indeed defines a 1-dimensional unirrep of G.The point is that F is a fixed point for the coadjoint action. Therefore,stab(F) = g, which is equivalent to the condition F 1

[9,91=0. Hence, F

defines a linear functional F on the abelian Lie algebra 9 := 9/[9, g].

Let G be the simply connected Lie group corresponding to the Lie al-gebra g (as sets both G and 'g can be identified with Rd, d = dim 'g). Theinitial group G factors through the Lie group G = G/[G, G] and (59) actuallymeans

a{F}(exp X) = {F} (p(exp X)) = e2,ri(F,'.(x))


where p : C - G/[G,GI and p. : g g/ [g, g] are the canonical projections.The last expression is clearly a character of G. G

Coming back to the Heisenberg group, we see that a one-point orbit SZ,,,must correspond exactly to the 1-dimensional unirrep 7rµ,,,.

Now consider a 2-dimensional orbit fl ,\. We want to write the associatedunirrep in the form IndAp where A is a closed subgroup of H and p is a1-dimensional unirrep of A.

Let F E a' be the functional on a associated to p. Then, according tothe restriction-induction principle, IndA p splits into unirreps 7rS1 for whichQ intersects p-1(F). There are two cases when p-1(F) is contained in asingle orbit 1l\:

a) a is the central subalgebra c = R R. Z and

(60) (F, Z) = A;

b) a is any 2-dimensional subalgebra5 and F E a` satisfies (60).In case a) the induced representation has the functional dimension 2 (i.e.

is naturally realized in the space of functions of two variables). Hence, it istoo big to be irreducible. We shall see later that it splits into a countableset of unirreps equivalent to Ira.

Consider the case b). Let us choose a = R - Y ®R Z and define F E a'by (F, /3Y + -Z) = A -y. Then the inducing representation of A has the form

p(exp ()3Y + yZ)) =

Let us derive the explicit formula for IndA p. We identify the homo-geneous manifold M = H/A with R and define the section s : M -+ Hby

s(x) = exp xX.

The master equation (see Appendices V.2.1 and V.2.2) takes the form

exp (xX) ga,b.c = exp ((3Y + yZ) . exp (yX)

or1 x 0 1 a c. 1 0 y 1 y 0

0 1 0 0 1 b = 0 1 0 0 1 0

0 0 1 0 0 1 0 0 1 0 0 1

with given x, a, b, c and unknown ,3, y, y. The solution is

,Q=b, y=c+bx, y=x+a.SRecall that a is necessarily abelian and contains c (see Exercise 2 in Section 1.1).


Therefore, the induced representation IndAp is given by

(IndAP(9a,b,c)f) (x) = e2aia(r+bx) f(x + a),

and coincides with Ira given by (45).

Now we shall show that the equivalence class of the induced represen-tation IndAp does not change if we choose another subalgebra a' C t) oranother unirrep p' associated with a different linear functional F' E a'*(subjected as before to the condition (F', Z) = A).

Actually, these two possibilities are related.

Lemma 5. Let a' = g a g-1 and F' = K(g)F E a'*. Let p' be the1-dimensional unirrep of A' = exp a' given by p'(exp X) = e2"i(F', X) . Then

IndA P' - IndAp.

Proof. Let M = H/A and M' = H/A'. We identify M and M' with R asbefore and choose in the second case the section s' : M' H by

s'(x) = g exp xX g-1.

Then the master equation in the second case can be obtained from the initialmaster equation via conjugation by g. Therefore, the induced representationIndAp' is related to IndAp by the formula

IndA,P'(9' 9u.b.c 9-1) = IndAP(9a.b,c

0

Lemma 5 implies that often it is enough to vary either a or F. Inparticular, in our case we can assume that (F, X) = (F, Y) = 0, (F, Z) = Aand a is an arbitrary 2-dimensional subalgebra of l which contains the centerg. (Note that this subalgebra is actually an ideal in tl, hence does not varyunder inner automorphisms.)

There are two different cases:

1. a does not contain X; then we can identify G/A with JR using thesection s(x) = exp xX.

2. a = R X ® J Z; then we can identify G/A with JR using the sections(x) = exp xY.

We denote the induced representations of H in these two cases by irl and1r2, respectively. It is easy to compute the corresponding representations of


the Lie algebra f . The result is:X Y Z

Ori). d3-X

iAx+µ iA

(7r2). - iAx + v d3i is

We see that both 7r1 and 7r2 are equivalent to Ira. In the first case theintertwiner is the shift operator T 1,, : f (x) '-, f (x + µ/A). In the secondcase it is the shift operator composed with the Fourier transform.

5.2. Other rules of the User's Guide.The results of the previous section can be formulated as follows: the

correspondence

(61) Q,\ - lra, QA,V 4-4 7rµ,,

is forced by the ideology of the orbit method with respect to the restriction-induction functor. It turns out that conversely, if we fix the correspondence(61), then all rules of the User's Guide will be correct. We leave this to thereader to check using the results described above in this chapter.

6. PolarizationsThe general results about invariant polarizations look especially transparentin the case of the coadjoint orbits of the Heisenberg groups. We give herethe complete description of real and complex invariant polarizations.

6.1. Real polarizations.Consider in more detail the symplectic geometry of coadjoint orbits. We

leave aside the trivial case of one-point orbits and consider a 2-dimensionalorbit 12a, A E R\{0}. It is a 2-dimensional plane which is a homogeneoussymplectic manifold with respect to the group K(H) acting by translations.

We are interested in the real polarizations of Sta that are invariant undertranslation. It is clear that every such polarization is just a splitting of SZainto the union of parallel lines ax + by = const.

In accordance with the general theory (see Chapter 1), these lines are justthe orbits of the subgroup K(A) C K(H) where A C H is a 2-dimensionalsubgroup of H. We observe that in this case all 2-dimensional subalgebrasa C h are abelian, hence are subordinate to any functional.

The functions on 11A which are constant along the leaves of the polar-ization have the form F(x, y) = f (ax + by) and form an abelian subalgebrain C°°(SZA) with respect to the Poisson bracket.


6.2. Complex polarizations.Let P be a complex polarization of Q,\, which is translation invariant.

Then P is generated by a constant vector field v = 8x + T8, where T is anynon-real complex number.

The functions satisfying the equation vF = 0 can be described quiteexplicitly. Namely, we introduce on f2a a complex coordinate w so that thefield v takes the form v = aiff,. Solving the system

wt+Tw, =0, we+Tw'y=1

we obtain w :=. In particular, for T = i we have w =z

(x + iy).Therefore, the solutions to the equation vF = 0 are exactly the holomor-

phic functions of w. They form a maximal abelian subalgebra in C°°(f A)with respect to the Poisson bracket.

6.3. Discrete polarizations.We have seen that the real polarizations of h are related to maximal

abelian connected subgroups in H. The complex polarizations are relatedto analogous subgroups in the complexification He = exp 11c.

There is one more class of groups which are abelian modulo the kernelof a given unirrep Ira of H. Namely, in exponential coordinates g(a, b, c) _exp(aX + by + cZ), such a group is given by the condition:

aEo.Z, bE,3.Z, cER where

This polarization was used in Section 2.5 for constructing the third real-ization of CCR. It has an important analog in the representation theory ofp-adic groups (see [LVJ).

Chapter 3

The Orbit Method forNilpotent Lie Groups

The class of nilpotent Lie groups is the ideal situation where the orbitmethod works perfectly. It allows us to get simple and visual answers toall the important questions of representation theory. You can find the cor-responding "User's Guide" in the Introduction to this book.

The nilpotent case can also serve as a model for more general and so-phisticated theories considered in subsequent chapters.

All results of this chapter were obtained in [Kill with one exception:the fact that the bijection G O(G) is a homeomorphism was provedlater in [Br].

1. Generalities on nilpotent Lie groupsHere we list the basic properties of nilpotent Lie groups and Lie algebrasthat we use in this lecture. The proofs can be found in [Bou], [J], or derivedfrom Appendix III.1 and Appendix 111.2.

Definition 1. A Lie algebra g is called a nilpotent Lie algebra if itpossesses the properties listed in the proposition below.

Proposition 1. The following properties of a Lie algebra g are equivalent:a) There exists a sequence of ideals in g

(1) {0} = 9o C 91 C ... C gn-1 C 9n = 9

such that

(2) [9, 9k] C 9k-1, 1 < k < n.

71

72 3. The Orbit Method for Nilpotent Lie Groups

b) The some as a) with the additional property dim gk = k.c) For any X E g the operator ad X is nilpotent: (ad X )N = 0 for N big

enough.

d) g has a matrix realization by strictly upper triangular matrices X =IIXi,,II, i.e. such that Xij = 0 for i > j.

Definition 2. A connected Lie group G is called a nilpotent Lie groupif its Lie algebra g is nilpotent.

Proposition 2. The following properties of a connected Lie group are equiv-alent:

a) G is a nilpotent Lie group.b) There exists a sequence of connected normal subgroups

(3) {e}=

such that Gk+1/Gk is in the center of G/Gk.c) Same as b) with the additional property dim Gk = k, 1 < k < n.d) G has a matrix realization by upper triangular matrices of the form

g = Ilgi; II satisfying1 for i = j,

gig 1 0 fori > j.0

The minimal n for which the sequence (1) (respectively (3)) exists iscalled the nilpotency class of g (resp. G). The abelian Lie groups havenilpotency class 1 while the generalized Heisenberg groups Hn have nilpo-tency class 2.

We also use the following features of nilpotent Lie groups.

Proposition 3. Let G be a connected and simply connected nilpotent Liegroup. Then

a) the exponential map exp: g -+ G is a diffeomorphism that establishesa bijection between subalgebras lj C g and closed connected subgroups H C G;

b) in exponential coordinates the group law is given by polynomial func-tions of degrees not exceeding the nilpotency class;

c) G is unimodular and a two-sided invariant measure dg is just theLebesgue measure dnx = Idxl A ... A dxnl in exponential coordinates;

d) for any F E g` we have Stab(F) = exp stab(F). Hence, Stab(F) isconnected and the coadjoint orbit SiF ^_- G/Stab(F) is simply connected.


2. Comments on the User's GuideHere we explain in more detail the practical instructions given in the User'sGuide and supply some additional information.

2.1. The unitary dual.By definition, the unitary dual G to a topological group G is the

collection of all equivalence classes of unirreps of G. It is a topologicalspace with the topology defined in Section 4.5. For a compact group Gthis topology is discrete. For discrete G the space d is quasi-compact (i.e.compact but not necessarily Hausdorff).

For an abelian group G the set a consists of all continuous homomor-phisms X : G -+ T', called multiplicative characters.' If, moreover, G islocally compact, the set a has itself a structure of a locally compact group.It is called the Pontryagin dual of G (cf. Chapter 2, Section 5.1).

For non-abelian groups the set G is no longer a group, but some dualitybetween G and G still exists. For example, there is a generalized Fouriertransform from L2(G) to L2(G) (see Appendix V.1.5). The attempt todevelop this duality and put G and G on equal rights has led ultimately tothe discovery of quantum groups (see [Dr] for historical comments).

2.2. The construction of unirreps.A big part of the unirreps for all kinds of groups can be constructed

using the induction procedure. For nilpotent Lie groups all unirreps can beconstructed by applying the induction functor to 1-dimensional unirreps, i.e.multiplicative characters, of some connected subgroups H C G.

Lemma 1. Any multiplicative character of a connected Lie subgroup H C Ghas the form

(4) PF,H(exp X) = e2xi(F, x)

where F is a linear functional on g = Lie (G) with the property F1 (4, 41 = 0.

Proof. First note that any character p of a Lie group H is a smooth functionon H (see Remark 4 in Appendix V.1.2). By Theorem 2 in Appendix 111.1.3we conclude that p is given by the equation (4) with F E h'. Since p : g' -h' is surjective, we can replace it with an F E g'.

The character p has the value 1 at all points of the commutator subgroup[H, H]. Therefore, the functional F must vanish on the commutator [t4, h]of h. 0

'The term "character" in representation theory is overloaded. Do not confuse the notion ofa multiplicative character with notions of ordinary, generalized, distributional, and infinitesimalcharacters of Lie groups defined in Appendix V.1.3-4.


According to Rule 2 of the User's Guide, all unirreps of a nilpotent groupG are among the representations Ind

HPF,H We shall prove in Section 3

below that the representation 1TF,H = Ind y pF,H is irreducible if f) is asubalgebra of maximal dimension among the subalgebras subordinated toF. Actually, we have dim dim G+dim H

2

Moreover, the equivalence class of 7rF,H depends only on the coadjointorbit Sl C g' that contains F.

2.3. Restriction-induction functors.The statements of Rules 3 and 4 of the User's Guide can be formulated

a bit more precisely. Let in denote the unirrep of G corresponding to theorbit Il C g', and let p", denote the unirrep of H corresponding to the orbitwCl)'.

Denote by Sn the set of H-orbits in p(1l) C h` and by S' the set of G-orbits which have non-empty intersection with p -1(w) C g'. For nilpotentLie groups the sets Sn and S°' are finite unions of smooth manifolds, so theycarry a natural equivalence class of measures defined by differential formsof top degree. We denote these measures by dw and dit respectively.

The decomposition formulae look like

(5)

and

ResH7fn=Jwcp(n)

(6) Ind ypW = J m(SZ, w) 1r,, dSl.( f))DW

By the Frobenius Duality Principle, the multiplicity function m(1l, w) is thesame in (5) and (6). For nilpotent Lie groups it takes only values 0, 1, andoo. It is convenient to write it in the form m(SZ, w) = ook(11,"') where ook isinterpreted as 0, 1, or oo when k is negative, zero, or positive respectively.

According to the ideology of the orbit method, the integer k must bedefined by geometry of the triple (Sl, w, p). Recall that fI and w are sym-plectic manifolds and p : g --i fj is a Poisson map, so that I'p fl (Sl x w) is acoisotropic submanifold in the symplectic manifold (fl x w, an - a",). Theprecise answer is given below in Section 3.3.

2.4. Generalized characters.For a nilpotent Lie group G the generalized characters are defined for

any unirrep it as linear functionals on the space M(G) of all smooth rapidlydecaying measures on G. In exponential coordinates the elements of M(G)look like p = p(x)d"x where p E S(R"), the Schwartz space. If we fix a


smooth measure on G, e.g., the Haar measure dg, then generalized charactersbecome the tempered distributions on G.

They are in general non-regular distributions. But the general theory ofdistribution guarantees that they can be written as generalized derivatives(of a certain order) of regular distributions. The order in question dependsof the geometry of the orbit. To find the explicit form of this dependence isan open problem.

2.5. Infinitesimal characters.In the paper [Dill Dixmier proved that for a nilpotent Lie group G

the algebra (Pol(9*))G of G-invariant polynomials on g' is always a poly-nomial algebra in a finite number of independent homogeneous generatorsP 1 ,.. . , Pk. We can also assume that they are real-valued polynomials. So,the center Z(g) of U(g) is a polynomial algebra with homogeneous Hermitiangenerators Aj = (27ri)deg' sym(P3), I< j < k.

For any unirrep it of G its infinitesimal character I,t is defined by thereal numbers cl = I,r(Aj). If irn is the unirrep corresponding to a coadjointorbit fl, then Rule 7 of the User's Guide claims that cj = Pj(f ). (Theright-hand side makes sense because Pj is constant along t)

Note that both the infinitesimal character I,r, : Z(g) C and theevaluation map evn : (Pol(g*))G -, C are algebra homomorphisms. Itfollows that for any nilpotent Lie algebra g the map sym : (Pol(g'))G -iZ(g) is an algebra homomorphism. This property is not true for general Liegroups.

2.6. Functional dimension.We say that a unirrep (7r, V) of a Lie group G has the functional

dimension n if the Hilbert space V is naturally realized by functions of nvariables. Here the word "naturally" means that the space V°° consists ofsmooth functions on which elements of U(g) act as differential operators.

For nilpotent groups we can say more. The space VI can always beidentified with the Schwartz space S(R") so that the image of U(g) is thealgebra W" of all differential operators with polynomial coefficients (seeSection 4.3).

The correctness of the definition of the functional dimension is based onthe following fact.

Theorem 1 (see [GK]). The algebra, W" has GK-dimension 2n, hence W,and W"' are not isomorphic for n 0 n'.


We recall that the GK-dimension of a non-commutative algebra A isdefined as follows. Let a = {al, ..., ak} be any finite subset in A. Let [c]Ndenote the subspace in A spanned by all monomials in al, ..., ak of degree< N.

Then we put

(7) dim GK A= sup limlog dim [a] N

a N-oo log N

This definition generalizes the notions of Krull dimension for commutativerings and of transcendence degree for quotient fields.

2.7. Plancherel measure.Let G be a unimodular topological group with bi-invariant measure dg.

In the space L2(G, dg) the natural unitary representation of the group G x Gis defined. Namely,

(8) (n(91, 92)f)(9) = f(91_'992)-

This representation is nothing but IndGxC 1 where G is considered as thediagonal subgroup in G x G. Therefore, according to Rules 3 and 8 of theUser's Guide, it decomposes into unirreps of the form 7ra,_n ^_- 7rn x in.

Let Vn be the representation space for urn. The representation spaceVn,_n for 7rn x 7rs2 is V1® Vi, i.e. the space of all Hilbert-Schmidt operatorsin V12. The representation it acts as

7r(91, 92)A = 7r (91) . A . 7r0(92)-1

The explicit isomorphism between L2(G, dg) and the continuous sum ofHilbert spaces V0 0VV. Il E O(G), plays the role of the non-commutativeFourier transform. It looks as follows:

(9)

f '-' 711(f) _ f(9)nn(9)d9, f(9) =J

trC' d(C}

Ilirsa(f)II2du(1l)IIIIIL2(C,dg)10(c)

The measure p entering these formulae is called the Plancherel measureon the unitary dual G, which here is identified with the set O(G) of coadjointorbits.

Note that this Fourier transform keeps some features of the classicalFourier transform on abelian groups. For example, left and right shifts go


under this transform to left and right multiplication by an operator-valuedfunction:

xsl(L91R92f) = irn(g1)-I7rsi(f)7rc (g2).

Another interpretation of the Plancherel measure can be obtained if we putg = e in the second equation in (9). We get

(10) 5(g) = J(G)

Xn(g) dp(1)

o, the Plancherel measure gives the explicit decomposition of the delta-S

function concentrated at the unit into characters of unirreps.After the Fourier transform in canonical coordinates this relation ac-

quires a geometrically transparent form. Namely, it gives a decompositionof the Lebesgue measure on g' into canonical measures on coadjoint orbits.

3. Worked-out examplesIn this section we show how to apply the orbit method to a concrete nilpotentLie group. The simplest example is that of the Heisenberg group. It wasalready treated in the previous chapter and will be mentioned again.

Here we choose the next, more complicated and more typical case.Among 4-dimensional nilpotent Lie algebras there exists a unique Lie al-gebra g that cannot be split into a direct sum of ideals.2 It has a ba-sis {X1, X2, X31 X4} with the commutation relations (we list only non-zerocommutators):

(11) [X1,X2] = X3, [X1,X3]= X4.

This Lie algebra admits the following upper-triangular matrix realization:

0 xl 0 x4

4 to'0 x1 x3X=>x X;=

0 0 0 x2t-10 0 0 0

The corresponding matrix Lie group is denoted by G. A general elementg E G in exponential parametrization looks as follows:

g(x'

(- )' x4 3 xl 2221 x l 2

+22 + 6

1 3 xt22x2, x3, x4) = exp X= 0 1 x x+ 2

U 0 1 2

0 0 0 1

2Representations of this Lie algebra are used in the theory of an anharmonic oscillator (see[KI]) and were first described in [Dil].


3.1. The unitary dual.The first question of representation theory for a given group G is the

description of its unitary dual G, i.e. of the collection of equivalence classesof unirreps.

There is a natural non-Hausdorff topology in G which can be defined inthree different but equivalent ways (see, e.g., [Di3] or [Ki2J). We explainthe most convenient definition later in Section 4.5.

We shall see below that for a nilpotent Lie group G the set e) is alwaysa finite union of smooth manifolds that are glued together in an appropriateway.

The exponential coordinate system, being very useful in theory, is oftennot the best for computations. In our case we get simpler expressions if weuse the so-called canonical coordinates of the second kind and put

g'(a', a2, a3, a'1) := exp a4X4 - exp a3X3 exp a2X2 exp al X1

(1 a1 (a')2 a4

0 1 al a3

0 0 1 a2

0 0 0 1

The element F E g' with coordinates {X1, X2, X3, X4} can be written asa lower triangular matrix of the form

0 0 0 04_F X1 0 0 0

with (F X) = tr(FX) _ >xaX;0 0 0 0

, .

i=1X4 X3 X2 0

These matrices form a subspace V C Mat4(R) and the projection pMat4(IR) -+ g' parallel to g-i has the form

0 0 0 0

p(IIA=,II)=1A21+A32 0 0 0

0 0 0 0A43 0A41 A42

Note that the subspace V is not the only possible choice of a subspace oflower triangular matrices that is transversal to gi. We could, for instance,take as V the set of matrices of the form

0 0 0 0 0 0 0 0

0 0 0 0 1X1 0 0 0or 2

0 X1 0 0 0 2X1 0 0X4 X3 X2 0 X4 X3 X2 0


We leave it to the reader to compare the resulting formulae for the coadjointaction with those shown below.

Exercise 1. Explicitly compute the coadjoint action of G on g' in theabove coordinates using formula (4) from Chapter 1.

Answer:(12)

K(g(x', 22, 23, 24))(X1, X2, X3, X4)

+1 +22X3+ (23 - 2122

2)X4,X2-21X3+

(2

2

)2X4,X3-x1X4,X4

K(9 (al, a2, a3, a4)) (X 1, X2, X3, X4)al 2

I.( ) X4, X3-a1X4,X4=(Xl+a2X3+(a3-ala2)X4,X2-a1X3+ 2

For future use we describe here all polynomial invariants of the coadjointaction. One of them is obvious: P(F) = X4. This is immediately seen from(12) but can also be explained more conceptually. Namely, the basic elementX4 E g belongs to the center of g. Hence, the corresponding coordinate X4on g' is unchanged under the coadjoint action of G.

Note that P cannot be the only invariant, because the generic orbit can-not have codimension 1. (Recall that coadjoint orbits are even-dimensionaland dim g = 4.) To find another invariant, we follow the scheme describedin Chapter 1 (see I.3.1).

Let us consider the plane S given by two linear equations Xl = X3 =0. It is clear from equation (5) that almost all orbits meet this plane.3An easy computation shows that the orbit St passing through the point(X1, X2, X3, X4) with X4 # 0 intersects the plane S in the single pointwith coordinates (0, X2 - 24, 0, Xq). It follows that the quantity R(F) _

X2X2 - is a rational invariant. So, the second polynomial invariant isQ(F) = 2R(F)P(F) = 2X2X4 - X.

Lemma 2. Let Pol(g')C be the algebra of all G-invariant polynomial func-tions on 9'. Then

Pol(gS)G S(9)G = C[P, Q] with P = X4, Q = 2X2X4 - X.

Proof. The first isomorphism holds for any Lie algebra (see Appendix III).To prove the second equality we consider the common level set S,,,, of thetwo invariants P, Q given by

(13) Sp,q = IF E g` I P(F) = p, Q(F) = q}.

3E.g., all orbits with X4 # 0 have this property.


Exercise 2. Check that the set Sp,qfor p 54 0, is a single orbit S1p,q passing through the point (0, 2L, 0, p);

for p = 0, q < 0, splits into two orbits 1 q passing through the points(0, 0, fv/---q, 0);

for p = 0, q > 0, is empty;for p = q = 0, splits into 0-dimensional orbits Sto'o`2 = {(cl, c2, 0, 0)},

which are fixed points of the coadjoint action. aNow, let 4' E be any invariant polynomial on g'. Then it takes

a constant value on each coadjoint orbit and, in particular, on every set Sp,qwith p#0.

On the other hand, the restriction of 4' on the plane S is a polynomial0 in coordinates X2 and X4. Compare 4' with the function 0(-2& P). Theycoincide on Sn {X4 54 0} and are both G-invariant. Therefore, they coincideon g' n {P 0 0}, hence everywhere. Thus, 4' has the form 4' = A P' forsome polynomial A and some integer N. But 4), being polynomial on g, isregular on the hyperplane P = 0. It follows that 4) E C[P, Q]. 0

The final description of the topological space O(G) looks as follows.Take a real plane l,q, delete the line p = 0, and glue to the remaining set

a) two points instead of each point of the deleted ray p = 0, q < 0,b) a whole 2-plane instead of the deleted origin.The topology of O(G) is the standard quotient topology: a set of orbits

is open if the union of all orbits from this set is open in g'. In particular,the limit4 of the sequence {SlE,,,c} where en -+ 0 is

a) two points S2o for c < 0;

b) the whole plane of zero-dimensional orbits Q `c' for c = 0;

c) no limit for c > 0.According to Rule 1 of the User's Guide, there is a homeomorphism, i.e.

a bicontinuous bijection, between the set G of all unirreps (considered up toequivalence) and the set O(G) of coadjoint orbits.

In this example the K(G)-invariant polynomials separate the genericorbits 0p,q but do not separate the special orbits S2o

qand the degenerate

orbits clc 0`2

3.2. Construction of unirreps.According to Rule 2 of the User's Guide, the representation 7rn is induced

from a 1-dimensional unirrep pF,H of an appropriate subgroup H C G.

'Recall that since the space in question is non-Hausdorff, the limit is non-unique.


To get the explicit formula we have to make the following steps:

(i) pick any point F E Il;(ii) find a subalgebra h of maximal dimension which is subordinate to

F. i.e. such that BF Ia= 0. or F Ilh.hl= 0;

(iii) take a subgroup H = exp h and define the 1-dimensional unirrepPF,H of H by the formula

(14) PF.H(expX) = e21r:(F,x);

(iv) choose a section s : X = H\G -+ G and solve the master equation

(15) s(x) g = h(x, g) s(x g);

(v) compute the measure p, on X and write the final formula

(16) (i11 (9)f) (x) = PF,H (h (x. 9))f (x - g)

for the realization of lrn in the space L2(X, E,,).

Remark 1. In our case (i.e. when G is a connected and simply con-nected nilpotent Lie group G and H is a closed connected subgroup) the ho-mogeneous space X = H\G can be identified with lRk, k = dim G - dim H,as follows.

Choose any k-dimensional subspace p c g that is transversal to h =Lie(H). Then any point of G can be uniquely written as g = p, p E p.So, any coset x = Hg has a unique representative of the form exp p, p E p,hence, can be identified with p E P.

If we define the section s : X -+ G by s(p) = exp p, one can check thatthe measure µ, is just the Lebesgue measure on p.

Here we make the explicit computations for all unirreps of the 4-dimen-sional Lie group G discussed in the previous section. We keep the notationfrom there.

Let us start with zero-dimensional orbits. Here the form BF is identicallyzero, since rk BF = dim e = 0. Hence, h = g, H = G, and Trn = PF H.

Thus, the 1-dimensional representation iris of G associated with thesingle-point orbit SZ = {F} is given by

(17) 7ro(exp X) = e"' (F,x)


In our case, we associate the representation

(18)CI .C2 27ri (cj a 7 +c2a2 )

70,o (9(al, a2, a3, a4)) = e

to the orbit 5200X2 = {(cl, c2, 0, 0)}.

All remaining orbits are 2-dimensional. They correspond to infinite-dimensional representations which have functional dimension one (seeRule 9 of the User's Guide).

The latter statement, roughly speaking, means that the representation inquestion is naturally realized in a space of functions of one variable. Indeed,each of these representations, according to Rule 2 of the User's Guide, canbe obtained by the induction procedure from a 1-dimensional representationof a subgroup H of codimension 1. Therefore, it acts in the space of sectionsof a line bundle L over a 1-dimensional manifold. Such a section locally isgiven by a function of one variable.

The precise definition of the notion of functional dimension was discussedin Section 2.6 above (see also the Comments on the User's Guide).

So, we have to find a 3-dimensional subalgebra 4 C g that is subordinateto F. The general procedure is described in Lemma 9 of Chapter 1, Section5.2. In fact, for our group G we can choose the same subalgebra l) for allremaining unirreps.

Namely, take as h the linear span of X2, X3, X4. The point is that t),being abelian, is subordinate to any functional F E g' and has the rightdimension.

Remark 2. Note that the observed phenomenon is not a general rule.For other nilpotent groups it may happen that we need to use infinitelymany different subalgebras in the role of h to construct all the unirreps.

Consider for example the universal nilpotent Lie algebra g of nilpotencyclass 2 with 3 generators. This 6-dimensional Lie algebra g = Lie(G) has abasis {Xi, Y4}1<i,j<3 with commutation relations

[Xi, Xj] = EijkYR, [Xi, Y'] = [1", Yk] = 0.

The generic orbit in g' is a 2-dimensional plane defined by the equations

and ciXi=c.

The kernel stab(F) of the form BF is spanned by all YJ and X = czXi. Therole of h can be played by any subalgebra of codimension 1 that containsstab(F). It remains to observe that a finite number of 2-planes cannot coverthe whole I[Y3. C)


To construct the induced representation IndN

PF,H, we apply the stan-dard procedure described briefly in the beginning of this section and in fulldetail in Appendix V.2.

In our special case it is convenient to identify the homogeneous spaceM = H\G with IR and define the section s : ]R -i G by s(x) = exp xXl.Then the measure p, will be the standard Lebesgue measure ldxl on R.

We have to solve the master equation (15), which in our case takes theform

1 2 0 1 l a2 4x a a0 1 x 0 0 1 al a3

0 0 1 0 0 0 1 a2

0 0 0 1 0 0 0 1

1 0 0 h4 l y 2 0_ 0 1 0 h3 0 1 y 0

0 0 1 h2 0 0 1 00 0 0 1 0 0 0 1

with given ai and x and unknown hi and y. The solution is

y = x + al, h2 =a 2 , h3 =a 3 +a 2X, h4 = a4 +a3x + Ia2x2.

The next step is to write the explicit formula for PF, HFor generic orbits Stp,q with p # 0 we take the representative F =

(0, p, 0, p) while for the remaining orbits Stog, q < 0, we take the repre-sentative F = (0, 0, ±.,I--q, 0).

We obtain:21ri(a2 2p+a4p)

27ri(F,log h) = e for f2p,g,PF,H(h) = e

e±27ria3 Vr--q for f2}0,9

So, we come to the following final results:

(19) a2, a3, a4))f)(x) =e27ri(p(a4+a3x+Za2x2)+°a2)f(x+a')

and

(20) g (g'(al, a2, a3, a4)) f) (x) = e±2hi*,,r---q(a3+a2x) f(x + a').Here the notation g' reminds us that we use the canonical coordinates ofthe second kind.

From (19) and (20) we easily derive formulas for the representations ofU(h). The results are collected in Table 1. From these results we concludethat the space H°° of smooth vectors for the representations (19) and (20)coincides with the Schwartz space S(RI), where U(4) acts as the algebra ofdifferential operators with polynomial coefficients. Later, in Section 4.3, weprove that this is a common feature of unirreps for all nilpotent Lie groups.


Table 1. The images of basic elements of g in all unirreps of G.

7rP,4f

7rO,qCI ,C2

"0,0

X1d d

2iricl

X2

TX

7ripx2 + Zq

dx

f 27rixP 27ric2

X3P

27ripx 27ri 0

X4 27rip 0 0

Note that the commutation relations (1) are obviously satisfied.

3.3. Restriction functor.Rule 3 of the User's Guide for simply connected nilpotent Lie groups can

be formulated in a more precise form than was stated in the Introduction.We mentioned in the Comments on the User's Guide that not only thespectrum of Res N7rfl but also the multiplicity function can be described interms of orbits. It turns out that for nilpotent Lie groups it takes only threevalues: 0, 1, 00.

To formulate the result. we introduce some notation. Let 1 be a coad-joint orbit in g', and let w C p(1t) be a coadjoint orbit in lj'. Denote by rthe graph of the projection p : g* h'. It is a subset in g' x l}` consistingof all pairs (F, p(F)), F E g'. Define the number k(Q, w) by the formula

(21) k(SZ, w) = dim (r fl (S2 x w)) - 2 (dim Q + dim w).

The more precise form of Rule r3 has the form

(22) Res y 7rn = J m(fl, w) 7r,,, dw0(11)

where

(23) M(9, w) = k(Sl, v))

Here we interpret m = ook as 0, 1, or oo according to whether k is negative,zero, or positive. In particular, m(SZ,w) = 0 if w ¢ p(SZ).

We continue to explore the group G from Sections 2.1 and 2.2. In thissection we consider the restriction of the generic representation 7rp,q of G todifferent subgroups.


Consider first the abelian subalgebra a spanned by X2, X3, X4. Therepresentation lrp,q after restriction to the subgroup A = exp a must splitinto 1-dimensional unirreps of A. These unirreps have the form

ua,µ,V ( exp(a2X2 + a3X3 + a4X4)) = e21ri(a2A+a3µ+a4V).

Formula (19) shows that ResA is a continuous sum of a 1-parametric familyof unirreps u,\,M,V with

2

(24) A=p2 +2 , p=px, v=p, xER.p

Thus,

ResH lrp,q = J!P

where w2 = {ff! + p, px, p}. We see that m(fl, w..,) = 1 for all x E R.On the other hand, the projection of the orbit 11p,q C g* to a* is the

parabola given in coordinates A = X2, It = X3, v = X4 by equations v =p, 2Av - p2 = q.

But (24) is exactly the parametric equation for this parabola. Moreover,in our case k(Ilp,q, wx) =

z(2 + 0) - 1 = 0 for all x E R. So, Rule 3 in the

precise form (22), (23) gives the correct answer.

Consider now the subalgebra 4 C g generated by X1, X3, X4, which isisomorphic to the Heisenberg Lie algebra. Let H = exp h be the correspond-ing subgroup. We denote by h(a. b, c) the element of the group H given byexp aX l exp bX3 exp cX4.

The representations and coadjoint orbits for H were considered in detailin the previous chapter. In particular, we showed that H has a series ofinfinite-dimensional unirreps which depend on one real parameter A and actin L2(R, dx) by the formulas

(xa(h(a, b, c))f)(x) = e2;riA(bx+c)f(x+a).

The projection of g' to 1" is the natural map from R4 with coordinatesX1, X2, X3, X4 to R3 with coordinates X1, X3, X4 (the projection parallelto the X3-axis). The orbit S2p,q given by equations (13) is projected to theplane X4 = p. The number k(flp,q, wp) is equal to

2(2 + 2) - 2 = 0.

On the other hand, from (19) we see that the restriction Res H irp,q isexactly Ira with A = p.

So, again Rule 3 gives the right answer.

5This formula follows also from Rule 2 of the User's Guide.


Exercise 3. Derive from the refined Rule 3 the fact that ResG 7r0a is

equivalent to ira with A = f / . Then check it using formula (20).It would be interesting to investigate in more detail the properties of the

number k(ft, w). The following lemma is a step in this direction.

Lemma 3. Assume that p(Il) splits into an s-dimensional family of coad-joint orbits ws C f)', x E X, of the same dimension 2m and that p-1(p(1l)) =11 + 1)1 splits into a t-dimensional family of coadjoint orbits Q. C g', y E Y,of the same dimension 2n and the same projection to If. Then for allx E X, y E Y we have

(25) k(fly,wx)=n-s-m=m+r-t-nwhere r = dim hl = dim g - dim tj.

Note that these relations agree with the "naive" count of variables. Forinstance, if lrn acts in the space of functions of n variables, then the restrictedrepresentation acts in the same space. On the other hand, an s-dimensionalfamily of unirreps with the functional dimension m needs only s + m vari-ables. So, we have n - s - m extra variables and the multiplicity must be

,,n-s-m

3.4. Induction functor.Here again we formulate a more precise result than Rule 4 of the User's

Guide. Namely,

(26) Ind Ham", = J m(Sl, w)irn d 1fCp- 1 (K(G) w)

where the multiplicity function m(1l, w) is the same as in (22).A simplest variant of an induced representation of a Lie group G is

a representation 7r = IndG 1 in the natural L2-space on a homogeneousmanifold M = H\G.

Our group G has the matrix realization with canonical coordinates al,a2, a3, a4 described in Section 2.1. Therefore, it acts on the vector space IIt4with coordinates {x', x2, x3, x4}:

1 1 2 4 xl l 2 + n2Lx3 + a4x41 +a a x a x0 1 a1 a3 I I x2 = x2 + alx3 + a3x40 0 1 a2 x3 x3 +a 2 X 4

0 0 0 1 x4 xa

This linear action preserves the last coordinate x4 (because the matrices inquestion are strictly upper triangular with units on the main diagonal).


In geometric language this means that G preserves an affine hyperplanex4 = c, hence acts by affine transformation of ]R3:

12(Xi x1 + a1x2 + -x3 +a4c

/(a',a2, a

3,a

4).

x2 = x2 + alx3 + a3c

x3 x3 + a2c

Note that this is a left action.Since the affine transformations in question are unimodular, we get a

unitary representation 1r, of G in L2 (R3, d3x):

(lrc WW, a2, a3, a4)-1) f) (xi, x2, x3)

.= f (x1 + alx2 + (2x3 + a4c, x2 + alx3 + a3c, x3 +a2c)

(The inverse element (g')-1 appears in the formula because we deal with aleft action of G.)

The question is, what is the spectrum of this representation? Or, inother words, how does it decompose into irreducible components?

We consider our hyperplane X4 = c for c # 0 as a left homogeneousspace X = G/B where the subgroup B is the stabilizer of some point x0 ofthe hyperplane.

Choosing xo = 0, the origin, we get B = exp b = Therefore,7r, is just the induced representation Ind41 where 1 denotes the trivial 1-dimensional unirrep of the subgroup B. In particular, ir,, belongs to thesame equivalence class for all c 36 0.

According to Rule 4, to describe the spectrum of 7r we have to considerthe projection p: g' -+ b', take the G-saturation of p-1(0), and decomposeit to G-orbits. In our case p is just the coordinate projection onto theX1-axis and p-1(0) is the hyperplane X1 = 0. The G-saturation of thishyperplane contains all 2-dimensional orbits f lp,q and SZog

and also a 1-parametric family of 0-dimensional orbits 000,0 . Since in decompositionproblems one can neglect the sets of measure 0, we can restrict ourselves tothe representations lrp,q corresponding to generic orbits. The final answerlooks as follows:

(27) V = J1rp,dL(P,)

where p is any measure equivalent to the Lebesgue measure on R2p,q.

Remark 3. The space L2(R3,d3x) has functional dimension 3 whilethe generic unirreps have functional dimension 1. So, it is natural that thedecomposition involves a 2-parametric family of unirreps.


The decomposition (27) can be interpreted as a simultaneous diago-nalization, or spectral decomposition, of the two Laplace operators Di =a(A1), i = 1, 2 (see the next section).

Now we check this answer and give the explicit formula for the abstractdecomposition (27). For any function f E L2(1R3, d3x) we introduce thefamily of functions ?ip,q E L2(lR, dt):

(28) op,g(t) = f (p, pt, p2t2 + q)2p

where f is the Fourier transform of f.

Proposition 4. a) The correspondence f i-- V) is invertible:

j1,p,q(t)e_21+123)(29) f (xl, x2, x3) =

and unitary:

(30) IfIL2(R3,d3x) = if Iop,g1L2(R,di)dp

2 dq

b) When f is transformed by the operator er(g), g E G, the corresponding1,ip,q is transformed by lrp,q(9).

Note that in the concrete decomposition formula (30) the abstract mea-sure dp(p, q) from (27) takes the concrete form l dpndg

2

3.5. Decomposition of a tensor product of two unirreps.Here, as an example, we compute the spectrum of the tensor product

lrp,q ®iro r. According to Rule 5 of the User's Guide we have to consider thearithmetic sum of f p,q and Sto,,.

The generic points of these orbits have the coordinates (X, , Y, p)and (x, y, fvr--q, 0), respectively. It follows that the arithmetic sum is thehyperplane X4 = p. This hyperplane is the union of all orbits Stp,q withgiven p # 0. So, the answer is: the spectrum consists of all representationsTrp,q with fixed p, and 7rp,q ®7o,, is a multiple of f 7rp,gdµ(q).

The more delicate question about multiplicities can be answered by thecount of functional dimensions: for the tensor product 7rp,q ® 7r6±,, the func-tional dimensions add to 1+1= 2. On the other hand, the continuous sum,or integral, of all 7rp,q with a fixed value of p also has the functional dimension2. This suggests the equality

(31) lrp,q ®'ro r = f irp,gdp'(q)


The abstract formula (31) has a concrete form (compare Proposition 4above). It can be obtained directly if we look at the tensor product of therepresentations given by (19) and (20). We use the fact that the Hilberttensor product of L2(R, ldxl) and L2(IR, idyl) is naturally isomorphic toL2(R2, IdxAdyl).

So, our representation has the form

(32) ((,q ®?ro r) (9 (al, a2, a3, a4)) f) (x, y)

= e2nilp(a4+a3x+Za2x2)+Zpa2f,,' (a3+a2i!)) f(x + al y + a1).

Let us split the plane with coordinates x, y into the family of parallellines y = x + t. Then the space L2 (R2, dxdy) splits into the continuousdirect sum, or direct integral, of smaller Hilbert spaces. Namely, for f EL2(1R2, l dx A dyl) define the family of functions i&t(x) = f (x, x + t).

It is clear that

(33) If l i2(Re2, dxdy) = j l V't 12 (R, dx)dt

and that z(it transforms according to 7rp,q, for some q' when f transformsaccording to (32).

So, (33) shows that the representation (32) splits into a direct continuoussum of representations of type irp,q, with fixed p and various q'.

Exercise 4. Find the exact value q' corresponding to the line y = x + t.Answer: q' = q + 2tpr - 2r2. 46

Exercise 5. Decompose the tensor product of two generic unirreps intounirreps.

Answer: 7rp,, q' ® lrpn,, q11 =f 00 7tp'+p", q dq

+30o (7r0, r ®ir0, r) drf o

ifp'+p"54 0,

ifp'+p"=0. 4

3.6. Generalized characters.Here we shall compute the generalized character Xp,q of the unirrep lrp,q.

According to Rule 6 of the User's Guide we have to compute the integral

(34) Xp,q(exp X) = e2ni(F;x)vp,q

as a generalized function on G in the exponential coordinate X = log g.We start with the calculation of the canonical form on f p,q which we de-

note by ap,q. From (12) we get the following expression for the infinitesimalcoadjoint action of g on g`:

K.(X1) = X30--X8, Kr(X2) = -X381, K*(X3) = -X4C81i K.(X4) = 0.


The orbit Slp,q is given by equations (13). We choose x := X1 andy := X3 as global coordinates on the orbit and obtain the parametric repre-sentation of fp,q:

q+y2Xl=x, X2= 2p , X3=y, X4=p.

In terms of coordinates x, y the g-action on SZp,q takes the form

K.(Xl) =poi, K.(X2) = -y,9x, K.(X3) = -p8x, K.(X4) = 0.

Definition 1 from Chapter 1 in this case implies

(35) Cp,q =dx A dy

p

We are now in a position to compute the integral (34). For

0 al 0 a4 0 0 0 00 0 a1 a3 _ x 0 0 0

10 0 0 a2F to 0 0 0

0 0 0 0 p y 2p 0it can be written as

JJ1dxAdyl

pUsing the well-known relations

JR

e2aioxd, = 5(a)

we get the final formula

and e7ri bx2 dx =1 + i sgnb

IbI

1 + i sgn (pat) rri(2pa4- o3A+vp )(36) Xp,q(9(a,, a2, a3, a4)) = 1

e 2 b(al)Ipa21 2

Remark 4. It is worthwhile to mention that the same result can beobtained in the pure formal way. Namely, we can consider the representationoperator (19) as an integral operator with distributional kernel and computeits trace by integrating this kernel along the diagonals

This principle works in many other cases, in particular, for all unirrepsof nilpotent Lie groups and for the representations of principal series ofSL(2, R). It would be useful to have a general theorem of this kind whichis an infinite-dimensional analogue of the Lefschetz fixed point formula. G

GCornparing the two approaches we have to keep in mind the equality g(a1,a2,a3,a4) _

exp(E°_2a,X,) exp(aiXi).


3.7. Infinitesimal characters.Consider the same group G as a basic example. We know from Lemma

2 that the algebra Pol(g*)G is generated by two polynomials P(F) = X4and Q(F) = 2X2X4 - X.

According to the general rule, we define for any element A E Z(g) thepolynomial PA E related to A by the formula

A = sym(pA(27riX1, ..., 27riX,a)).

Since in our case generators X;, 2 < i < 4, commute, we can omit sym inthis expression and obtain the following basic elements in Z(g):

=1 1 -2X X3 2 4)Al

21riX4,

A2 =47x2

(X2

In Section 3.2 we computed the images of these elements for all unirreps(see Table 1). The results are given in the table:

C1.r.2±P.9 U,q 7rU,O

Al p 0 0

A2 q q 0Compare this table with the values of the invariant polynomials P = PA,

and Q = PA2 on the coadjoint orbits:

f1m f l q Slo o

P P 0 0

Q q q 0

We see that Rule 7 works perfectly.

Note also that the infinitesimal characters separate generic represen-tations 7rp,q but do not separate 7roq and 7r0-,q or 7ro o"2 for different pairs(c1,c2)

3.8. Functional dimension.The notion of functional dimension often allows us to predict the general

form of the answer in many decomposition problems. Here we illustrate iton the example of the regular representation of G. The space L2(G, dg) hasfunctional dimension 4, while the generic unirreps have functional dimension1 and depend on two parameters.

We conclude that these unirreps must enter with multiplicity o01. Thisis an analogue of the following well-known fact for compact groups: anyunirrep occurs in the regular representation with multiplicity equal to itsdimension.


Now let us consider the space L2(G, dg) as the representation space forIndGxc

Exercise 6. a) Show that the generic unirreps entering in the decom-position of Ind G"c 1 have the form 7rp,q x Irp q.

b) Prove that 7rp.q ir_p,q.

Hint. Use Rules 4 and 8 of the User's Guide. 46

Thus, this time the irreducible components have functional dimension2 and, as before, depend on two parameters. Therefore, we can expect themultiplicity to be finite (it is actually 1 in this case).

This is also the analogue of a well-known fact: for a compact group Gall the representations 7r x 7r*, it E G, of G x G occur with multiplicity 1 inL2(G, dg).

3.9. Plancherel measure.Let us now compute the Plancherel measure p on G. We shall use the

values p, q as local coordinates on the part of G which consists of genericrepresentations. The remaining part has measure zero and can be neglected.

By definition of the Plancherel measure, we have the equality

(37) a(g) =JR2

Xp,9(g)dp(p, q)

The direct computation of p using formula (36) for the generalized characteris rather complicated. But Rule 10 of the User's Guide gives the answerimmediately:

(38) p =IIdpAdgl.

Indeed, from the parametrization above we see that for any integrable func-tion d on 9* we have

ig -

4X f ( q+y2 )dxAdQAdYAdP©(X1,X2,X3,X4)d =JR4 I\x, 2p ,?!,p

2p

=f Idpndgl2 2 S2n.v

O(Fi.g) - ap.q

Applying this formula to mb = f, the Fourier transform of f, we get

f (O) _ RZ tr 1rp,q(f) I dp2 dqj

which is equivalent to (37) with p given by (38).


3.10. Other examples.We highly recommend that the reader (or the lecturer who will use this

book) explore independently other examples of nilpotent groups. We listbelow some of the most interesting and instructive examples.

1. The group Gn of all upper triangular real matrices of order n withunit diagonal. The corresponding Lie algebra gn consists of strictly uppertriangular real matrices (those with zero diagonal). The classification ofcoadjoint orbits in this case is still unknown and is related to deep combi-natorial problems (see [Kilo] for details).

2. Let 9n,k be a universal nilpotent Lie algebra of nilpotency class kwith n generators. By definition, it is a quotient of a free Lie algebra withn generators by its k-th derivative spanned by all commutators of lengthk + 1. The corresponding group Gn,k was used by Brown [Br] to prove thehomeomorphism between G and O(G) (see Section 3.1).

3. Let On be the Lie algebra with basis X1, ... , Xn and commutationrelations - U -i)Xi+i ifi+j <n,

[Xi X.i]0 otherwise.

The corresponding Lie group Vn can be realized as a group of (equivalenceclasses of) transformations 0 of a neighborhood of the origin in the real linethat have the form:

n

O(X)=X. 1 + E akxk + O(xn+l ).k=1

The group law is the composition of transformations. Note that the 1-parametric subgroups corresponding to basic vectors in on can be explicitlycalculated:

exp (tXk) : x'-+x

= x + txk+1 + 0(t).k l - ktxk

The structure of Z(vn) is known. When n is odd, the center has onlyone generator - the central element Xn E V.

When n = 2m is even, there exists another generator An, = sym (Pm)where P,n E Polvn (t ) is a polynomial of degree m and weight nt(2rm - 1).(Here we agree that Xk has weight k.) For small m it can be found usingthe general scheme of Chapter 1. It turns out that for polynomials Pm thereexists a nice generating function.7

7This result is due to my former student A. Mihailovs.


Namely, let us introduce polynomials Qm(y1, , ym) by the formula

1 + E Qmtm = ji + > yktkm>1 V k>1

Then the polynomial Pm can be written as

Pm(Xm, ... , X2m) = X2-,,, ' Q,-X2.m-1 ' ...' XX2m)

4. Generalizing example 1, we can consider a unipotent radical N of aparabolic subgroup P in a semisimple group G. There are a few generalresults in this case but the detailed analysis of representations has neverbeen done.

A nice particular case is the group GIn

of upper triangular matrices thatare either symmetric or antisymmetric with respect to the second diagonal.The study of their orbits and representations along the scheme of [Ki10] isa very interesting and difficult problem.

5. Among two-step nilpotent groups (i.e. groups of type IRm a R') themost interesting are the so-called groups of Heisenberg type. They aredefined as follows.8

Let A be one of the following: the complex field C, the skew field H ofquaternions, or the non-associative division algebra 0 of Cayley numbers(octonions).

Denote by Ao the subspace of pure imaginary elements of A. In thespace gk,1(A) := Ak ® A' ® A0 we define a Lie algebra structure by

[(X', Y', Z'), (X", Y", Z")] = (X'+X", Y'+Y", Z'+Z"+3`(X'X"+Y"Y'))

iii k i " 1 " iwhere X X = E;_1 X;X; , Y Y = YjY,, and 3 denotes theimaginary part of an element of A.

The characteristic property of the Lie algebras gk,i(A) is that all coad-joint orbits are either points, or linear affine manifolds of the same dimension(k + 1) dim A. The case A = C corresponds to the generalized Heisenbergalgebra bk+l.

The corresponding Lie groups Gk,l(A) were used in [GW] and [GWW]to construct non-diffeomorphic compact Riemannian manifolds with thesame spectrum of the Laplace-Beltrami operator.

8This construction is taken from A. Kaplan, Trans. Amer. Math. Soc. 258 (1980), no. 1,147-153.

§4. Proofs 95

4. ProofsThe main merit of the Orbit Method is that it provides the right geomet-ric formulation of representation-theoretic facts. When this formulation isfound, the proofs of the results mentioned in the User's Guide become rathernatural and simple, although sometimes we have to use some deep facts fromfunctional analysis and Lie theory.

For nilpotent groups most of the proofs use induction on the dimensionof the group.

4.1. Nilpotent groups with 1-dimensional center.Nilpotent Lie groups have a very nice property (which they share with

exponential groups, see below): the exponential map is one-to-one. Thismap establishes the bijection between Lie subalgebras in g = Lie(G) andconnected Lie subgroups in G, and between ideals in g and connected normalsubgroups in G.

Lemma 4. Let g be a nilpotent Lie algebra. Then any subalgebra go c g ofcodimension 1 is an ideal.

Proof. From the definition of a nilpotent Lie algebra it follows that theadjoint action of g is given by nilpotent operators.' If X E go. the operatorad X is nilpotent on g and preserves go. Hence, it defines a nilpotent op-erator in the quotient space g/go. But this space is 1-dimensional and thecorresponding operator must be zero. Hence, [X, YJ E go for any Y E g andgo is an ideal.

Another property of a nilpotent Lie algebra g which follows from thedefinition is that g has a non-zero center S. In the induction procedure belowthe important role is played by nilpotent Lie algebras with 1-dimensionalcenter. Here we describe the structure of such algebras.

Lemma 5. Let g be a nilpotent Lie algebra with a 1-dimensional centera. Then there exists a basis {X, Y, Z, Wl, ..., Wk} in g with the followingproperties:

b=R'Z, [X,Y] =Z, [W:,Y]=o, 1<i<k.

Proof. Let Z be any non-zero element of S. The algebra g = 9/3 is nilpo-tent, hence it has a non-zero center j. Choose Y E g so that Y = Y mod 8is a non-zero element of j.

91n fact, this property is characteristic. By Engel's theorem, if ad X is nilpotent for anyX E g, then 9 is a nilpotent Lie algebra.


Since Y 0 0, Y is a non-central element of g. Hence, there exists X E gsuch that [X, Y] 0 0. But Y is central in 'g, hence Y is central modulo S.We conclude that [X, Y] E S. So, replacing X by cX if necessary we canassume that [X, YJ = Z.

Now, let go denote the centralizer of Y in g. Again using the fact that Yis central modulo 3, we see that go has codimension 1 in g. Let { W1, ... , Wk)be elements of go which together with Y and Z form a basis in go. Then allequations above are satisfied. 0

In the rest of this section we fix the following notation:Go C G - the Lie subgroup corresponding to the Lie subalgebra go C g;

it is the centralizer of an element Y under the adjoint action.A C Go - the normal abelian 2-dimensional subgroup in G correspond-

ing to the Lie subalgebra a spanned by Y and Z.C = exp 3 - the central subgroup of G.

Theorem 2. Let (7r, H) be a unirrep of G. Then one of the following mustoccur:

1) the representation Tr is trivial on C, that is, 7r(c) = 1H for all c E C;or

2) the representation it has the form Indcop where p is a unirrep of Go,non-trivial on C.

Proof. We start by introducing the structure of a G-manifold M on thereal plane. Namely, we put M = A, the Pontryagin dual to the abeliannormal subgroup A = exp a C G.

The group G acts on A by inner automorphisms, hence it acts on M = A.We shall describe this action in more detail.

Introduce the coordinates (y, z) in the vector space a with respect to thebasis (Y, Z) and transfer these coordinates to A via the exponential map.So, the point exp(yY + zZ) E A has coordinates (y, z).

The dual group A consists by definition of multiplicative characters ofA that have the form

x, (y, z) = e27r:(µy+az)

So, M = A is a 2-dimensional plane with coordinates (p, A).

Lemma 6. Let g E G have the form g = go exp tX, go E Go, t E R. Thenthe action of g on M is

(39) K(g) : (µ, A) t-. (µ - tA, A).

§4. Proofs 97

Proof. First, we note that the subgroup Go acts trivially on A, hence on M.It remains to compute the action of the 1-parametric subgroup exp tX, x E JRon A and on A. We have

Ad(exptX)(yY + zZ) = exp(ad tX) (yY + zZ) = yY + (z + ty)Z.

The coadjoint action K(g) on A is the dual one, hence given by (39).

Lemma 7. The action of G on M is tame.

Proof. The G-orbits in M are lines A = const # 0 and points (p, 0).Consider the following family of G-invariant sets:

the stripes Sa,b : a < A < b and the intervals Ia,b : A = 0, a < u < bwith rational a and b.This family is countable and separates the orbits. Therefore, the action

is tame.

The next step is the construction of a representation 11 of M compatiblewith a given unitary representation (jr, H). Let f E A(M). Denote by fthe function on A given by

1(y, z) =IM

f (µ, A) (y, z) dp A dA.

Then we put

11(f) = r(1) = f Y(Y' z)7r(exp(yY + zZ)) dy A dz.A

The equalities

11(1112) = ir(fif2) = ir(fh * f2) = r(f1)r(f2) =11(11)11(12),

11(7) = r(7) = r(7)V = r(1)'

show that (11, H) is a representation of M. The compatibility conditionfollows from the definition of the action of G on M.

Suppose now that r is irreducible. According to Theorem 10 in Appen-dix V.2.4, the representation II of M is in fact the representation of someG-orbit ci C M. If this orbit is a point (µ, 0) E M, r is trivial on C. If f2 isa line A = const # 0, r is induced from a stabilizer of some point (µ, A) E Q.This stabilizer actually does not depend on µ and is exactly Go.

The real impact of Theorem 2 is that in both cases of the alternativethe study of a unirrep r of G can be reduced to the study of some unirrepof a smaller group: either the quotient group G/C or the subgroup Go C G.


4.2. The main induction procedure.In this section we give the proofs of most of the results mentioned in the

"User's Guide". We shall do it by induction in the dimension of G. It isa sometimes tedious but straightforward way. Of course, more conceptualproofs would be nicer but as of now they are known only for some of theresults.

Base of induction. When dim G = 1, we have G R and all statementscan be easily verified. Note, however, that in the course of this verificationwe have to use the following classical but quite non-trivial fact.

Proposition 5. The Pontryagin dual to the group R is isomorphic to Ritself.

In other words, all unirreps of IR are 1-dimensional and have the form(40) 7r,\(x) = e2'", X E R.

For the sake of the interested reader we recall the short proof of thisfact.

We know that any family of commuting unitary operators in a Hilbertspace can be simultaneously diagonalized (see Appendix IV.2.4). Therefore,a unirrep of any abelian group must be 1-dimensional.

Thus, we have to solve the following functional equation that expressesthe multiplicativity property

X(t + s) = X(t) . x(s).There are two ways to do it.

First, we can use Proposition 3 from Appendix V.1.2, which claims thatall characters are smooth functions. From the functional equation above wederive the differential equation X'(t) = a X(t) with a = X'(0). The generalsolution has the form X(t) = cent. Since JXi = 1 and X(0) = 1, the constanta is pure imaginary and c = 1. Hence, X(t) has the form (40).

Second, if we do not know a priori that X is smooth, we can consider itas a generalized function and use the fact that all generalized solutions tothe above equation are in fact ordinary smooth functions. Technically it ismore convenient to deal with log X instead of X itself. 0

Now we are in a position to prove the main results by induction on thedimension of the nilpotent Lie group G in question.

Induction Theorem. Assume that Rules 1-10 are true for all connectedand simply connected nilpotent Lie groups of dimension < n. Then they arevalid also for a connected and simply connected nilpotent Lie group G ofdimension n.

§4. Proofs 99

Proof. Let g = Lie(G), and let b denote the center of g. First of all weseparate two cases:

1. dim 3 > 1;

II.dim3=1.=In the first case the group G has no faithful unirreps. Indeed, if dims

k > 1 and Z1, ... , Zk is a basis in 3, then for any unirrep it of G theoperators 7r (exp Ek1 x;Zj) are scalar, hence have the form

k

7r exp x3 Zj = e2ai E; a, x,

Let 3o = IF_, xjZj I >j Ajx., = 0}. Then Co = exp 3o is in the kernel of it.Therefore, we can consider it as the composition it o p where p : G -+

G_ = G/Co is the projection on the quotient group and ii is some unirrep ofG.

The next step is verification of Rules 1-10 for the group G under theassumption that they are valid for G. For some of them, namely for Rules2-4 and 6-9, this verification is almost evident.

Consider, for example, Rule 2. We know by assumption that this rule istrue for the group G. So, any unirrep r of G has the form

9r = 7r fl = IndHpp,i,

for some orbit Il C g'' and a point E i2.

It remains to verify that for H := p-1(H) we have

7r:= iF o p =

where 1 = p'(SI) and F E p*(1l). (Here we denote by p' the canonicalinjection of g' into g' dual to the projection p : g -+ `g.) This verification isan easy exercise on the definition of an induced representation and we leaveit to the interested (or sceptical) reader.

The same scheme works for the other rules except Rules 1, 5, and 10.Rule 5 for G actually follows from Rule 3 for G x G since the tensor

product can be written in terms of a direct product:

7r1 ® 7r2 = ResGXG(7rl x 7r2).

For direct products all the rules follow immediately from the same rules forthe factors.


So we have to check Rules 1 and 10. Their common feature is that theserules deal not with an individual representation but with the total set G.Let Z be the center of G.

To prove that G = O(G) as a set, it is enough

(i) to split the set G of all unirreps according to their restrictions to Z;(ii) to split the set O(G) of coadjoint orbits according to their projection

to 3*; and

(iii) establish bijections between corresponding parts.This is also an easy exercise and can be left for the reader. But to prove

that the final bijection is a homeomorphism is a more difficult problem. Weshall discuss it later.

Finally, Rule 10 follows from Rule 6 which gives the explicit formula forthe generalized character. Indeed, if we consider characters as distributions,the Plancherel formula gives the decomposition of the 5-function supportedat e E G into distributional characters of unirreps according to (37) (seealso (10) in Section 2.7). After the Fourier transform in the exponentialcoordinates and using Rule 6 this formula becomes

(41) f, f (F)dF = 10(G) (d()fn f (F)voln(F) I .

This gives exactly Rule 10 for G.

Consider now case II. Here we have the alternative of Theorem 2. Fora representation of the first kind, Rules 2-6 and 8-9 are verified exactly asin case I. For representations of the second kind, this is also true but foranother reason. As an illustration we give here the proof of Rule 6.

We keep the notation of Theorem 2. Let 7r = Ind G p be a unirrep of G.Let p act in a Hilbert space V. Then 7r can be realized in 1l = L2(R, V, dx)by the formula

(42) (7r (go exp tX) f)(x) = p(exp xXgo(exp xX)-1) f (x + t).

In the next section we shall prove the existence of the generalized characterand show that it can be computed by the standard formula using the distri-butional kernel of 7r(g). Here we perform the corresponding computation.

From (42) we obtain the formula for the distributional kernel of it interms of the distributional kernel of p:

K.,,(go exp(tX) I x, y) = Kp(exp xXgo(exp xX)-1)6(x + t - y).

§4. Proofs 101

Figure 1

It follows that

(43) X, (goexp tX) = 5(t) jXp(exP xXgo(exp xX)1)dx,

which is an exact analog of the Frobenius formula for the character of in-duced representation of a finite group (see Appendix V.2.1).

Now we use Rule 6 for the group Go and write

ai(Fo,Xo)voloo(Fo)Xp(exp Xo) = foo e2

This together with (43) gives us the formula

X,r(exp Xo exp tX) = 5(t) J1 I dx r 2.i(K(..p.X)Fo.Xo).aolno(Fo))

(44) R \ J/s2o

= 5(t) j (dx J I

R nm

where we use the notation S2x := K(exp xX)Sto and denote by Fr a genericelement of Q .

On the other hand, Rule 6 for 7r looks like

(45) fe2X0+t'0vo1ci(F).

But the orbit 1 is a cylinder with UZER SZx as a base and R X as a directrix(see Figure 1).

The generic point of S2 has the form F = Fx+sFi where FF E f , s E R,(F1,X)=1, and(F1,Xo)=O forXoEgo.


Therefore the integral over Sl in (45) takes the form

X.(exp(Xo + tX)) = J dxds Je2n:(F=+sFl,xo+ex)voln=(Fx)

R,

c /_ (dxdsJ (Fo) J

R2\ cz

_ 5(t) [ (dx I e2ai(Fz, xo)vol (F) I .S2Z x /R S2z

The last expression coincides with (44) and proves Rule 6 for G.

We turn now to Rule 7. The difficulty with this rule is that the centerZ(g) of U(g) changes when we pass from g to a subalgebra go or to a quotientalgebra g. Fortunately, this change can be controlled.

It is convenient to identify Z(g) with Pol(g')c, Z(go) with Pol(go)co,and Z(g') with Pol(g)c.

Consider the case when we pass from g to a quotient algebra g/cwhere c is a subalgebra of S. We denote by p the projection of g onto g (andalso the projection of G to G). Then the dual map p` identifies g' with thesubspace cl C g`'. Let Sl be a G-orbit in this subset.

We denote by p' : S(g) Pol(g') -- S(g) ^_' Pol(g'') the projectionof algebras that corresponds to the projection p of sets. For any elementA E Z(g) ^_' Pol(g')c we have p(A) E Z(-g). Hence, Rule 7 is valid for p(A):under the unirrep Tsz of d it goes to the scalar PP(A)(S1). But this meansthat under the unirrep Tn := to o p the element A goes to PA (Q).

Suppose now that the second case of the alternative of Theorem 2 holds.From the commutation relations of Lemma 5 it follows that G-invariantpolynomials on g' do not depend on the variable X, hence can be consideredas polynomials on go. Moreover, Z(g) = Z(go)e"p

Let 7r be a unirrep of G induced by a unirrep p of Go. We assumethat p corresponds to some orbit f 0 C go, and we denote by S2 the orbitcorresponding to ir.

Formula (42) shows that any A E Z(g) goes under 7r to a scalar operatorwith the same eigenvalue as p(A). But the former coincide by Rule 7 withPA(12o). We can consider PA E Pol(go) as a polynomial on g* which doesnot depend on the X-coordinate. Moreover, this polynomial is invariantunder the action of exp R X. Hence, it takes the same value on Sl as on Tlo(cf. Figure 1 above). 0

§ 4. Proofs 103

4.3. The image of U(g) and the functional dimension.The main result of this section is

Theorem 3. Let G be a connected nilpotent Lie group, and let 7r be aunirrep of C corresponding to a 2n-dimensional orbit ft C g'. Then 7r canbe realized in the Hilbert space H = L2(W', d"x) so that

a) H°O coincides with the Schwartz space S(R");b) the image of U(g) under 7r coincides with the algebra W" of all dif-

ferential operators with polynomial coefficients.

Proof. If the representation 7r is not locally faithful (i.e. has a kernel ofpositive dimension), the statement of the theorem reduces to the analogousstatement for a group of lower dimension. So, the crucial case is that of arepresentation (7r, H) of a Lie group G with 1-dimensional center for whichwe have the second alternative of Theorem 2. We keep the notation of thistheorem and its proof. Without loss of generality we can assume, using (38),that the inducing representation p has the property p. (Y) = 0.

Then from (42) we conclude that the images of basic elements of 9 under7r, are:

7r. (X) _ -, 7r.(Y) = 27riAx, 7r.(Z) = 27riA,

7r.(Wj) = p.(Ad(exp xX)Wj).

Since the action of ad X in g is nilpotent, the elements

Wj(x) := Ad(exp xX )Wj = exp (ad xX) W3

can be written as linear combinations of Y, Z, Wl, ..., Wk with coefficientswhich are polynomials in x.

Note that the constant term of Wa(x) is W3(0) = Wj.Now we use the induction hypothesis and choose the realization of V in

the form of L2 (R', dy) so that the operators p.(Wj) generate the algebra ofall differential operators with polynomial coefficients in variables yl, ... , y,".

We see that in the chosen realization the image of U(g) is containedin the algebra of differential operators with polynomial coefficients in vari-ables x, yl, ..., Moreover, this image contains the operators x and 8y.It follows that together with p.(W3) the image contains p.(W3). Indeed,the constant term of any polynomial P in x can be expressed as a linearcombination of P, xB P, ..., (x8y)deg Pp.

We conclude that 7r(U(g)) contains all differential operators with poly-nomial coefficients.


We proved the second statement of the theorem. The first follows fromit because for any generalized function on ]R" the conditions

D4 E L2(R'z, d"x) for any D E A,, and 0 E S(1R")

are equivalent. 0

4.4. The existence of generalized characters.In this section we prove the existence of generalized characters for all

unirreps ir and show that they can be computed directly in terms of distri-butional kernels of the operators ir(g), g E G.

We deduce this from Theorem 3.

For our goal we need an explicit example of a differential operator D withpolynomial coefficients in L2(R' , d"x) for which D-i exists and belongs tothe trace class.

We start with n = 1 and consider the operator H = - + x2. Wealready considered this operator in Chapter 2. Here we recall briefly thefacts needed.

Lemma 8. The spectrum of H is simple and consists of all odd positiveintegers 1, 3, 5, ....

Proof. Recall the definition of the annihilation and creation operators:

a=x+dx a*=x- d .

In Chapter 2 we derived the relations

(46) H = a*a + l = aa* - 1, Ha = a(H - 2), Ha` = a* (H + 2).

From the last two relations in (46) we conclude that if a function fis an eigenfunction for H with an eigenvalue A, then a f and a* f are alsoeigenfunctions with eigenvalues A - 2 and A + 2, respectively (only if theyare non-zero).

Frther, the equations

of =0, a'f =02 12

have general solutions f = ce- 2 and f = ce 2 , respectively.It follows that the creation operator a` has the zero kernel in L2(R, dx)

and the annihilation operator a has the 1-dimensional kernel spanned by

fo(x) = e 2

§4. Proofs 105

From the first relation in (46) we see that H fo = fo. Hence, if we putfn (a*)nfo, then H fn = (2n + 1) fn.

To complete the calculation of the spectrum, it remains to show that thelinear span V of fo, , fl, ... is dense in L2 (R, dx), so that H has the pointspectrum described in the lemma. This was done in Chapter 2. 0

We see that the operator H has a bounded compact inverse H-1 thatbelongs to the Hilbert-Schmidt class C(L2(R, dx)). Therefore H-2 is oftrace class.

In the space L2(Rn, dnx) we define the operator Hn byn

k=1

The eigenfunctions and eigenvalues of Hn have the formn

/n

lPk(xl, ... , xn) = f Ok, (xi), \k = (2ki + 1)i=1 i=1

where k = (k1i ..., kn) is a multi-index and the 0k are eigenfunctions of H.Then an elementary estimation shows that Hn n-1 is of trace class.Now we are ready to prove

Theorem 4. Let G be a connected nilpotent Lie group, and let (7r, H) bea unirrep of G. Define the Schwartz space S(G) using the diffeomorphismexp. Rn g - G.

Then for any 0 E S(G) the operator 7r(0) is of trace class in H. (Inother words, the character Xn is a tempered distribution on G.)

Proof. One way to prove the theorem is to show, using Theorem 3, thatH can be realized as L2(Rn, dnx) so that 7r(0) is an integral operator withkernel in S(R2n). We leave this to the reader.

Another way is more direct. Choose an element A E U(g) such that 7r.sends it to the operator B with the inverse B-1 of trace class (e.g., to theoperator Hn+1 considered above). Then for 0 E S(G) we can write

7r(O) = B-1B7r(0) = B-17r.(A)7r(O) = B-17r(AO)In the last term the first factor is an operator of trace class while the secondis a bounded operator. So, 7r(0) is of trace class. Moreover, the trace normof 7r(0) can be estimated:

IIir(0)II1 = Imlax Itr (B-1ir(AO)C)I(47)

< trB-1 II7r(AO)IIH <- trB-1 IIAOIIL1(c)0


4.5. Homeomorphism of G and O(G). .

Let d denote the set of all equivalence classes of unitary (not necessarilyirreducible) representations of a topological group G. For brevity we usuallyuse the same notation for representations and their equivalence classes.

Recall the definition of the topology on G.

Definition 3. A neighborhood of a given representation (ir, H) is deter-mined by the following data:

(i) a compact subset K C G;(ii) a positive number e;

(iii) a finite collection of vectors X = {xl, ... , x, } in H.The neighborhood UK,E,x(ir) of (ir, H) consists of all unitary represen-

tations (p, V) such that there exists a finite family Y = {yl, ..., yn} ofvectors in V satisfying

(48) I(x(g)xi, x3)H - (p(g)yi, yi)vl < e for any g E K.

The set G of unirreps is a subset of G and inherits the topology from there.

Recall one more notion from the general representation theory. We saythat a unirrep 7r is weakly contained in a unitary representation p if thepoint 7r E G C G is contained in the closure of the point p E G.

Exercise 7. The 1-dimensional representation ira(t) = e21riAt of R isweakly contained in the regular representation p of lii; acting by translationsin L2(R, dx):

(p(t)f) (x) = f (x + t).

Hint. Make the Fourier transform and consider a 8-like sequence in thedual space L2(1R', dA).

Note that Ira is not a subrepresentation of p.The main problem we discuss in this section splits into two parts:

(i) prove that the map O(G) -+ G is continuous;(ii) show that the inverse map is continuous.The first part is easier and goes as follows. Suppose that a sequence

of orbits {Stn} goes to a limit St. By definition of the quotient topology inO(G) this means that there exists a sequence of functionals {Fn} such thatFFESlnand limFF=FESl.

n-.ooLet Ijn be a subalgebra of maximal dimension subordinated to F, and let

Hn = exp 1)n be the corresponding subgroup of G. Passing to a subsequence

§4. Proofs 107

if necessary, we can assume that all hn have the same codimension 2r andhave a limit h in the Grassmannian Gtr (g). It is clear that his subordinate toF. But it can happen that it does not have the maximal possible dimension(since rk BF can be less than rk BF = 2r).

Using Lemma 9 from Section 5.2 of Chapter 1 we can construct a sub-algebra 4 -of maximal dimension subordinate to F so that h C h, henceH=exp4 exp b = H.

Denote by rr (resp. 7r) the induced representation IndH pF, H (resp. theunirrep 7rsn = IndH p,, x)

We need to show that ii is contained in the limit10

lim irn = lim Indy,pF,,H, .

Lemma 9. We have lim rro,, in d contains 7rn.n-»oc

Sketch of the proof. Since the sequence hn tends to h and all h,, havethe same dimension, we can identify Xn = H,,\G with the standard space]Rn, so that the actions of G on Rn arising from the identification Rn ^_' Xnhave a limit. It is clear that this limit corresponds to the identificationIItn X = H\G. The statement of the lemma follows from the explicitformula for the induced representation given by Rule 2 of the User's Guide.

In the case rkBF =2r=rkBF wehaveH=H, ir=rrandweobtainthe desired relation lim irst = xn

Assume now that rk BF < 2r, hence H is a proper subgroup in H. Thenrr is a reducible representation.

Lemma 10. The representation if is weakly contained in ir, hence is con-tained in lim irn .n-oo

Sketch of the proof. First, we observe that pF H is weakly contained in

IndH pF, jr. This can be proved exactly as in the statement of Exercise 7.Next we use the following general fact, proved by G. M. J. Fell.

Theorem 5 (see [Fe]). The induction functor IndH defines a continuousmap from h to G.

"Note that because d is not necessarily Hausdorff, the limit of a sequence need not be apoint, but is some closed subset in C.


Consider now the continuity of the inverse map: G - O(G). To give anidea of how to prove this continuity, we describe the approach invented byI. Brown in [Br). It is based on two simple observations:

1. Any nilpotent Lie algebra can be considered as a quotient of gn,k(see example 2 in Section 3.10) for appropriate n and k. Consequently, thetopological space G is a subspace of Gn,k with inherited topology.

2. The Lie algebra gn,k has a huge group of automorphisms induced byautomorphisms of the free Lie algebra.

The first observation reduces the general problem to the case g = gn,k,while the second allows us to prove the theorem for gn,k by induction on k.

Chapter 4

Solvable Lie Groups

1. Exponential Lie groupsThe next class of Lie groups where the orbit method works well is the classof exponential Lie groups. Most of the prescriptions of our User's Guide(namely, Rules 1, 3, 4, 5, 8, 9) are still valid in this more general situation.

These results were obtained mainly by the bench school (see [BCD]):the validity of Rules 3, 4, and 5 was first proved in [Bu]. As for the topo-logical isomorphism of G and O(G), this was established only recently in[LL].

Rules 2, 6, 7, and 10 need modifications, which we discuss below.

1.1. Generalities.A Lie group G is called exponential if the exponential map exp : g G

is a diffeomorphism. So, the exponential coordinates give a single chartcovering the whole group. In particular, all exponential groups are connectedand simply connected.

A Lie algebra g is called exponential if the corresponding connectedand simply connected Lie group is exponential.

Example 1. Let E(2) denote the Lie group of rigid motions of theEuclidean plane. We denote by G the connected component of this groupand by G its universal covering. All three groups E(2), G, and G have thesame Lie algebra g = e(2).

The group G consists of rotations and translations. It has a convenientmatrix realization by complex matrices of the form

T 1L'

grow= 0 1 , T,wEC, ITI=1.

109

110 4. Solvable Lie Groups

Hence, the Lie algebra g can be realized by matrices

it zAt.. = 0 0 '

It admits the natural basis

tEIR, z=x+iyEC.

T= (0 0)' X= (0 Q)' Y= (0 0)with the commutation relations

IT, X] = Y, IT, YJ = -X, [X, Y] = 0.

The Lie group G can be realized by 3 x 3 matrices

feit 0

9:.f» = 0 1

0 0

W

t , tER, wEC.1

We haveC eit

eit

expc(tT + xX + yY) = 0

0

e z s(t)(x + iy) ) sin(t/2)1 , s(t) = t/2

0 e.zs(t)(x+iy)1 t0 1

We see that elements 9,,0 E G have infinitely many preimages, while ele-ments 92.R,,,w, n E Z\{0}, w 54 0, are not covered at all by the exponentialmap. Therefore, G and G are not exponential Lie groups and g is not anexponential Lie algebra. Q

Actually, the example above is a crucial one, as the following criterionshows.

Proposition 1. Let G be a connected and simply connected Lie group. ThenG is exponential if its Lie algebra 9 satisfies the following equivalent condi-tions:

a) The operators ad X, X E g, have no pure imaginary non-zero eigen-values.

b) The Lie algebra g has no subalgebra isomorphic to e(2).

From Proposition 1 it follows that the class of exponential Lie algebrasis hereditary: any subquotient (i.e. quotient algebra of a subalgebra) of anexponential Lie algebra is also exponential.

We note also that the class of exponential Lie algebras is situated strictlybetween nilpotent and solvable Lie algebras.


1.2. Pukanszky condition.An important correction is needed to extend Rule 2 of the User's Guide

to exponential Lie groups. To formulate it we introduce

Definition 1. Let us say that a subalgebra h C g. subordinate to F E g*,satisfies the Pukanszky condition if

(1) P 1(p(F)) C lF,

i.e. the fiber over p(F) in g lies entirely in a single G-orbit. Note that thisfiber can be written asp ' (p(F)) = F + hl

The modified Rule 2 differs from the initial one by the additionalrequirement:

The Lie subalgebra h must satisfy the Pukan8zky condition.

In fact, the necessity of the condition follows immediately from Rule 4 of theUser's Guide. According to this rule, the spectrum of the induced represen-tation IndHUF.H consists of representations zrn, for which i intersects theset p' (Sl). So, the spectrum is reduced to a single point if the Pukanszkycondition is satisfied.

Counting the functional dimension by Rule 9 of the User's Guide, wecan expect that the multiplicity of the spectrum is finite (in fact, it is equalto 1) when the following equality takes place:

(2) codim,4 = 2 dim SZF,

i.e. when h is a maximal isotropic subspace of g for the bilinear form BF.This is indeed true.

Theorem 1 (Pukanszky). Conditions (1) and (2) together are necessaryand sufficient for the irreducibility of the representation

(3) 7M. = Ind H UF,ff.

Proof. The proof follows essentially the same scheme that we used in theprevious chapter to prove Rule 2 for nilpotent Lie groups. However it ismuch more involved. So, we omit it and refer to [P1], [BCD], or [Bu] forthe details. 0

Theorem 1 holds, in particular, for nilpotent Lie groups. We have not in-cluded the Pukanszky condition in the initial Rule 2 because of the followingfact.


Proposition 2. For nilpotent Lie groups the Pukanszky condition (1) issatisfied automatically.

The statement follows from two lemmas.

Lemma 1. Let g be any Lie algebra, and let lj C g be any Lie subalgebrathat is subordinate to F and has codimension 1 dim HF in g. Then the affinemanifold F + 111 has an open intersection with HF (i.e. the local version ofthe Pukanszky condition is always satisfied).

Proof. Let H = exp b. Consider the H-orbit of F in g`. Let us check thatthis orbit is contained in F + hl. For any X, Y E lj and h = exp(X) wehave

(K(h)F, Y) = (F, Ad(h)-1Y) = (F, ead(-X)Y) = (F, Y)

since ead(-X )Y E Y + [(), ll] and F I[h.hl= 0. So, F and K(h)F have thesame values on h, hence, the difference is in ll.

Now we compare dimensions of K(H)F and 41. The first dimension isequal to

dim H - dim (Stab(F) fl H) > dim H - dim Stab(F) = 2 dim Q.

The second is dim g - dim h = z dim H.

We see that dim K(H)F > dim hl. But K(H)F is contained in F+f)1.Hence, both sets have the same dimension. It follows that the intersectionQF fl (F + f)1) contains a neighborhood of F in F + f)1.

The same argument can be applied to any point F1 E F + f)-L. Hence,the set OF fl (F + I)1) is open in F + 4-L. 0

Lemma 2. Let G be a connected unipotent' Lie subgroup in GL(n, 1[Y).Then all G-orbits in 1Etn are affine algebraic subvarieties in R' (i.e. they aredefined by a system of polynomial equations). In particular, they are closedin the ordinary topology of lRn.

Proof. Induction by n (see details in [Kill or [Dill). 0

Let us return to the proof of Proposition 2. According to Lemma 2, fornilpotent Lie groups all coadjoint orbits are closed. Hence, K(G)Ffl(F+f)1)is closed in (F + f)1).

'A group G C CL(n, R) is called unipotent if it consists of matrices with all eigenvaluesequal to 1. (Equivalent property: G is conjugate to a subgroup of N, the group of triangularmatrices with 1's on the main diagonal.)


But according to Lemma 1, it is also open. So, it must coincide with(F + hl).

For general exponential groups, Lemma 2 is no longer true (see for in-stance Example 2 below) and we have to include the Pukanszky conditionin the formulation of Rule 2.

1.3. Restriction-induction functors.Rules 3 and 4 of the User's Guide remain true and in some cases admit

a more precise formulation. Namely, assume that an orbit Q C g' intersectsthe preimage p-1 (w) along a set with several connected components. Thenthe multiplicity m(12, w) depends on the number of components (see [BCD]for details).

1.4. Generalized characters.Rule 6 of the User's Guide needs corrections for the two reasons discussed

below.

First, for non-nilpotent exponential groups the generalized characters ofunirreps are not necessarily well defined as distributions. Namely, it couldhappen that the operator 7r(O) does not belong to the trace class even for

E A(G).The explanation of this phenomenon by the orbit method is very simple.

The coadjoint orbits are no longer closed submanifolds in g'. As a conse-quence, the canonical volume form associated with the symplectic structureon an orbit can have a singularity at the boundary and the integral formulaof Rule 6 does not define a distribution on G.

So, we have to restrict the domain of definition of the generalized char-acter by imposing additional conditions on the test functions 0 E A(G). Inthe simplest case (see Example 2 below) the additional condition is verynatural: the Fourier transform

(4) (F) = f(exP X) e2"'(F, x)dX

must vanish on the boundary of the orbit.In general, a more severe restriction is needed to compensate for the

singularity of the canonical measure at the boundary. Note that for closedorbits these additional restrictions are unnecessary.

Second, for nilpotent groups the Fourier transform '--+ given by (4)defines the unitary transformation from L2(G, dg) to L2 (g*, dF) where dg isthe Haar measure on G and dF is the Lebesgue measure on g', both suitablynormalized. In terms of this transform Rule 6 looks like

r(5) tr Tc(4) _ (F) QI where r = 12rfu


This can be briefly formulated as follows:

The Fourier transform of Xn is the canonical measure on the coadjointorbit Q.

In the case of non-nilpotent groups we have to take into account the

more complicated relation between invariant measures on G and on g'.

For non-unimodular groups the densities of the right and left Haar mea-sures have the following form in canonical coordinates:

x _ -aaxd,.(exp X) = det e X 1) dX, di(exp X) = det (1

_

ad X ) dX.(adad

For unimodular groups both expressions can be written as q(X)dXwhere

q(X) = det( sinh(ad X/2)

ad X/2

This is an analytic function on g whose zeros coincide with singular points ofthe exponential map. For exponential groups, q(X) is everywhere positiveand so the function

smh(ad X/2) z

(6) p(X) = q(X) = det ad X/2 P(o) = 1,

is well defined and analytic.We define the modified Fourier transform by

(4') O(F) = re2,,i(Fx)O(exp X)p(X)dX.J9

For unimodular exponential groups the Fourier transform is a unitarybijection of L2(G, dg) to L2(g*, dF) and we shall use it instead of (4) inexpression (5).

Also, we can replace the differential 2r-form r by the non-homogeneousform e° = 1+a+ 2 +... and agree that the integral of the non-homogeneousform over a k-dimensional manifold depends only on the homogeneous com-ponent of degree k.

Then the modified Rule 6 acquires the elegant form

(7) tr Tn(expX) = 1

n2,ri(F,X)+o

P(X)

Since p(X) = 1 for nilpotent groups, the modified Rule 6 coincides inthis case with the original one.


It turns out that the modified Rule 6 is valid for a wide class of non-nilpotent groups provided that test functions are subjected to additionalrestrictions.

Remark 1. Consider the function j of one complex variable z definedby

(8)j2(z) _

sin:k z2

,

2

j(0) = 1.

It is a holomorphic function in the disc I z I < 21r with the Taylor decompo-sition

z zj(z) = 1 +

48 + 23 440 +and it has a two-valued analytic continuation to the whole complex planeexcept at the branching points zn = 27rin, n E Z\{0}.

The function p(X) that we used in the modified Fourier transform canbe expressed in terms of j as follows. Let Al, ..., a be eigenvalues of theoperator ad X. Then

n 2d v d X 2 2 7 d X 4tr (a (tr (a ) ) tr (a ))(9) p(X) = fl i(Ak) = 1 +

AR + A n.Q S71Mk=1

Maxim Kontsevich observed that the well-known identity

r(x)r(1 - x) =

implies the equality

xsin irx

j-2(Z)=r(1+2z)r

Since ad X is a real operator, its eigenvalues are either real or split intocomplex conjugate pairs. Taking into account the property r(z) = r(z), weconclude that

p(X)=r17-1(Ak)=Ifir(1+Z iII.k=1 k=1 JJ

This strongly suggests that we introduce the function

1

k=1 detII(,Nxx)\ 2iri


where II(z) = t(1 + z), and use it instead of p(X) in the definition of themodified Fourier transform.

The advantage of this replacement is that p is an entire function,hence k(X) is well defined on g and even on its complexification. Q

Example 2. Let G = Aff+(1,R) be the group of orientation preservingaffine transformations of the real line.2 The group G is a semidirect productof the subgroup A of dilations and the normal subgroup B of translations.It has a matrix realization by 2 x 2 matrices of the form

9(3,b =(00

1) , a,bEIR, a>0.

The elements of the Lie algebra g and of the dual space g' are realizedby real 2 x 2 matrices of the form

X-\0 0/' F- \y 0/and the coadjoint action is

K(a, b) : (x, y) '-' (x + a-'by, a-'y).

We see that g' splits into two 2-dimensional open orbits Sgt = {(x, y)I f y >0} and a family of 0-dimensional orbits S2x = {(x,0)}.

Exercise 1. Compute the exponential map for the group G.

Hint. For a 34 0 use the identity I0 0) = Q . (0 c' 0 0) .

Q-1 for an

appropriate invertible matrix Q.

Answer: exp ( )=( ei

a) =( ea 2j1(a)Q

Now we compute the modified Fourier transform (4') for the group G.We start with a function 0 E A(G). Its modified Fourier transform is afunction io = on g* given by

7'(x, y) = J e2 eo,e- 1 a/2

2 ( a sink (a/2)dad.

The unirreps of G corresponding to the open orbits Sgt are given by asimple formula. We omit the standard computations according to Rule 2and write down the final result.

2 We refer to (LL) where this simple but instructive case is treated in detail (see also [BCD)and (K12J).

§ 1. Exponential Lie groups 117

Namely, consider the space L2 (R X, L`) , the natural Hilbert space asso-ciated with the smooth manifold R" = R\{0} (see Appendix V.2.2). Theunitary action of G on this space has the form

(1r(9a,b) f)(x) = e21ribx f(ax).

This representation is manifestly reducible: the functions concentrated onpositive and negative parts of R' form two invariant subspaces. Thesesubspaces are irreducible and the corresponding unitary representations areexactly at.

Let us compute the generalized characters of these representations. Fora test measure ¢(a, b) dadb with 0 E A(G) we have

(r(0)f) (x) = J O(a, f(ax) daa b

aER"bER

0 (y b) e2-ib f(y) dydb/ .

YR

x y

So, the kernel of the operator 7r±(b) is

K (x, y) = f 0 (x' b) ±y > 0, ±x > 0.R

For the trace of 7r±((b) we get the expression

J 2,rtnxdbdx.trir±(0) = Kt(x, x)x = 0(1, b)eXX xR l' X

In terms of the Fourier transform = w the last expression can be writtenas

tr7r±(ti) = j :;(x,y)dxdy

= f b(F)a(F)ty z t

in perfect accordance with Rule 6.

1.5. Infinitesimal characters.The initial formulation of Rule 7 of the User's Guide also must be cor-

rected. Recall that in Section 2.5 of Chapter 3 we defined the bijectionA +--+ PA between Z(g) and Pol(g`)G using the symmetrization map sym.For a homogeneous element A E Z(g) we have

A = (27ri)deg PASym(PA)


The point is that this map is the isomorphism of linear spaces, but in generalnot an algebra homomorphism.

It forces us to modify Rule 7 since both the maps P' P(1) and A HI,r(A) are algebra homomorphisms. We shall have an opportunity to speakmore about this in Section 3.6 in Chapter 5 and Section 6.2 in Chapter 6and here only give the correct formula.

Namely, the modified Rule 7 looks exactly as before:

1"n (A) = PA (Q),

but with a differently defined correspondence A PAew

To define this new correspondence we consider the Taylor decompositionof the function p(X) defined by (6) and (9). The formal Fourier transformassociates to this power series a formal differential operator J of infiniteorder with constant coefficients on g'. This operator acts on Pol(g*) and isinvertible.

If {X;} and {F3} are any dual bases in g and g', respectively, then J isobtained from p(X) by substituting eF, instead of X;.

For A E U(g) we now define PA"' E Pol(9*) so that

(10) A = (27ri)de$ p"sym(JPAe") or PAew = J-yAod.

Note that for nilpotent g the definition of PAnew coincides with that of PAdbecause p(X) - 1 and J = 1 in this case.

2. General solvable Lie groups

2.1. Tame and wild Lie groups.There is a new phenomenon that occurs when we consider non-exponen-

tial solvable Lie groups. Some of these groups are wild, i.e. do not belongto type I in the sense of von Neumann.

This means that the representation theory for these groups has severalunpleasant features. See Appendix IV.2.6 for details.

It is interesting that there are two different reasons why a solvable Liegroup may be wild. Both have a simple interpretation in the orbit picture.

The first reason is that the space O(G) can violate the separation axiomTo. This is because coadjoint orbits are not necessarily closed, even locally.The simplest example of such an occurrence was discovered by F.I. Mautnerin the 1950's and was rediscovered later many times. We describe it here.


Figure 1. Coadjoint orbits for GQ.

Example 3. Consider the family of 5-dimensional Lie groups G° thatdepend on a real parameter a and have the following matrix realization:

eit 0 a(11) G°3g(t,a,b)= 0 et°t b , wheretER, a,bEC.

0 0 1

It turns out that G° is tame if and only if a is a rational number.fit 0 a

The Lie algebra g° consists of matrices 0 iat b , and the elements0 0 0

of the dual space g' can be realized by matrices

iT 0 0

F.,w,T = 1 0 0z w 0

The coadjoint action in the coordinates z, w, r has the form

K(g(t, a, b)) (z, w, T) = (e-'tz, a-1QLw, T + Im (az + abw)).

A picture of the coadjoint orbits for irrational a is shown in Figure 1.We see that the topological space O(G°) does not satisfy the axiom To.Exercise 2.* Consider two families of representations of Ga: the family

Ua,b, a, b E C, acting in L2(R, dr) by the formula

(Ua,6(t, z, w)O)(r) = e27ri Re(e''az+e'o'bw)0(,r + t),

and the family r1, r2 E IR+, s E R, acting in L2(T2, d91d92) by theformula

(Vr1,r2,s(t, z, 00)(011 92)= e2 1ri Re(ts+e"r1 z+e'I'r2w), ,(91 + t mod 1, 92 + at mod 1).


a) Show that there are two different decompositions of the right regularrepresentation p of G into irreducible components:

(12)

and

p =J

U,,,bdadadbd6CxC

,.2,4drndr2ds.(13) p =LXR

Vr,

b) Show that no Ua,b is equivalent to any 6;.1,,.2,4.

Hint. See [Ki2], Part III, §19. 40The second reason is that the canonical form a on some orbits can be

non-exact. This innocent looking circumstance implies that after applyingthe Mackey Inducibility Criterion (see Appendix V.2.4 or [Ki2], §13) we runinto representations of some non-abelian discrete groups that are usuallywild.

Example 4. The simplest group for which this situation occurs hasdimension seven and can be realized by block-diagonal 6 x 6 matrices withdiagonal 3 x 3 blocks of the form

fets 0 z 1 s r(14) 0 Pit w and 0 1 t , s, r, t E 1R, z, w E C.

0 0 1 0 0 1

We give here the description of the orbits and representations of thisgroup following [Ki2J, §19.

Let us introduce the seven real coordinates x, y, u, v, s, t, r on g so thatz = x+iy, w = u+iv. We denote by X, Y. U, V, S, T, R the correspondingbasic vectors in g which also serve as dual coordinates in 9'. The non-zerocommutators are

[S,X]=Y, [S,YJ=-X, [T, U] = V. [T, V]=-U, [S,T]=R.

Let G be a connected and simply connected Lie group with Lie(G) = g.Exercise 3. Show that the coadjoint orbits of maximal dimension in

g' are diffeomorphic to T2 x R2 and the canonical symplectic form or is notexact.

Hint. Check that for non-zero r1, r2i r3 the equations

X2+Y2=ri, U2+V2=r2, R=r3


define a coadjoint orbit with canonical 2-form

v= doAdS+dhindT+r3 doAdb

where the angle coordinates 4 and ip are defined by

X = r1 cos 0, Y = rl sin 0, U = r2 COS V, V = r2 sin .

4The fact that G is wild can be established in the following way. There

is an abelian normal subgroup A C G whose Lie algebra a is spanned byX, Y, U, V. The Pontrjagin dual group A consists of characters

X(exp(xX + yY + uU + vV)) = e2"i(°x+ey+«.+dv)

The splitting of A into G-orbits is tame: orbits are given by the equations

a2 + b2 = r2 c2 + d2 = r22.1+

So we can apply the Inducibility Criterion in Appendix V.2.4 to thissituation. The conclusion is that a unirrep it of G has the form

7r,.p = IndH' p

where X E A, HX is the stabilizer of the point X in G, and p is a unirrep ofHX such that p(a) = X(a) 1 for a E A C HX.

The representation in question remains in the same equivalence classwhen we replace the pair (X, p) by g (X+ p) = (X o A(g-1), p o A(g-1))where A(g) : x H gxg-1. So, unirreps of G are actually labelled by thefollowing data: a G-orbit Sl C A and a unirrep p of the stabilizer HX ofsome point X E 11 (subjected to the condition PIA = X 1).

Call X non-degenerate if a2 + b2 0 0 and c2 + d2 34 0. Let Ao be the setof all non-degenerate characters. The stabilizer H. of X E A0 in G actuallydoes not depend on X. It is the semidirect product

HX = HZ x A where HZ := exp (Z S + Z . T + ]R R).

Let Go be the open part of G that consists of representations whoserestrictions to A split into non-degenerate characters. It follows from abovethat Go is in fact homeomorphic to (Ao)C x HZ. We see that G and HZ areof the same type: either both are tame or both are wild. So, we have toshow that H7 is a wild group.


The group HZ is a subgroup of the Heisenberg group that consists ofmatrices

1 m rh(m, n, r) = 0 1 n withm,nEZ, rER.

0 0 1

It has a normal abelian subgroup B defined by the condition n = 0. Ageneric character X of B has the form

X9,c(h(m, 0, r)) = e2ai(mO+rc), 0 E ]R/7d, c E R.

The group H acts on B and this action factors through H/B since B actstrivially on B. An easy computation shows that the generator of H/B ^_' Zdefines the transformation

(0, c) i- (0 + cmod 1, c).

We see that the partition of B into H-orbits is wild: for any irrational c theset 0 + Z . c mod 1 is dense in lR/7L.

Consider the representations 7re,c =IndBXo, of H.Exercise 4. a) Show that 7re,c and 7re1,c' are equivalent if c' = c and

0' = 0 + kcmod 1 for some k E Z.b) Show that for c irrational all the points ire,,, E H belong to any

neighborhood of ae,c.

Hint. Write the explicit formula for representations and try to find theintertwining operators between them. 60

Remark 2. It is natural for wild groups of the first kind to extend thenotion of coadjoint orbit and consider ergodic G-invariant measures on g`as virtual coadjoint orbits.3

The notion of a virtual subgroup was suggested by Mackey. The idea touse it in the orbit method was proposed in [K2). Soon after that I learnedthat this idea had already been realized by Pukanszky.

In particular, the analogue of the integral formula for generalized charac-ters was obtained in [P2). The left-hand side of this formula is the relativetrace of the operator 7r(g) in the sense of von Neumann and the right-handside is the integral over a virtual orbit. C7

3One can prove that when O(G) satisfies the separation axiom To all such measures areproportional to canonical measures on orbits.


2.2. Tame solvable Lie groups.Now we come back to the tame solvable Lie groups. The remarkable fact

is that the User's Guide still works in this case after appropriate amendmentsto all rules except 7 and 9.

Almost all the results described in this section are due to Louis Auslanderand Bertram Kostant [AK]. We shall present these results here togetherwith some simplifications and complements suggested by I.M. Shchepoch-kina [Sh].4

We start with the following simple criterion.

Theorem 2 (Auslander-Kostant). A connected and simply connected solv-able Lie group G is tame (that is, of type I) if the following two conditionsare fulfilled:

1. The space 0(G) satisfies the separation axiom To.2. The canonical form or is exact on each orbit. O

From now on we assume that G satisfies these conditions.Even for tame solvable groups the correspondence between coadjoint

orbits and representations need not be one-to-one. To describe this corre-spondence we have to modify the space g'.

The point is that, unlike the case of exponential Lie groups, a coadjointorbit Sl can be topologically non-trivial; in particular, the Betti numbersb1(St) and b2(1) can be non-zero.

A simple topological consideration (see Appendices 1.2.3 and 111.4.2)shows that the fundamental group 7r1(1) is isomorphic to Stab(F)/Stab°(F).

We define a rigged momentum for a Lie group G as a pair (F, X)where F E 9* and X is a unitary 1-dimensional representation of Stab(F)such that

(15) X.(e) = 21riF IStab(F)

Recall that such a X exists only if the orbit SZF is integral (see Proposition2 in Section 1.2.4). For the type I solvable Lie groups this condition is alwayssatisfied because a is exact, hence all orbits are integral.

The set of all rigged momenta will be denoted by g*igg. The groupG acts naturally on 9*igg and this action commutes with the projectionH : 9*igg - 9* given by II(F. X) = F.

The G-orbits in g*igg will be called rigged coadjoint orbits and theset of all such orbits will be denoted by O,.igg(G).

4Unfortunately, the complete text of her Ph.D. thesis (Moscow, 1980) was never publishedand so is still inaccessible to the mathematical community.


It is worthwhile to mention that for tame solvable Lie groups the pro-jection H is onto and the fiber over a point F E g' is a torus of dimensionequal to the first Betti number of SZF. So, the correspondence between usualand rigged orbits is one-to-many.

The modified Rule 1 is given by

Theorem 3 (Auslander-Kostant). For any connected and simply connectedsolvable Lie group G of type I there is a natural bijection between the set Gof unirreps and the space 0,t99(G) of rigged coadjoint orbits. 0

We refer to the original paper [AK] for the detailed proof and describehere only the construction of a unirrep TO associated with a rigged orbit Sl.

It turns out that it requires a new procedure of holomorphic inductionand we outline the main idea behind this notion. We shall speak again aboutit in Chapter 5.

From now on we assume that 1 E O,.,gg and (F, X) E Q.

We start with the case when a real polarization h for F exists and satisfiesthe Pukanszky condition. Then the usual induction procedure of Rule 2 ofthe User's Guide is applicable with a minor modification.

Namely, let H° = exp h, and let H be the group generated by H°and Stab(F). (Note that Stab°(F) is contained in H°.) We define the1-dimensional unirrep UFX,H of H by

(16) UF H(g) =xe21r; (F X) for g = exp X, X E h,X(g) for g E Stab(F).

The correctness of this definition is ensured by (15).Then we define the desired unirrep 7ro by the formula

(17) 7rn = IndH UFX,H,

which can be considered as the first amendment to Rule 2.This procedure is not sufficient for the construction of unirreps for all

rigged orbits. The reason is that for some F E g' there is no real polariza-tion, i.e. no subalgebra h C g that is subordinate to the functional F andhas the required codimension 2rkBF.

On the other hand, a complex polarization, i.e. a complex subalgebrap C gc with these properties, always exists. This follows from the proceduredescribed in Section 5.2 of Chapter 1. We keep the notation of this section.

The definition of the induction procedure can be further modified sothat a complex polarization p is used instead of a real one. Namely, let p bea complex polarization for F E g'.


Following the procedure described in Section 5.1 of Chapter 1, we intro-duce the real subspaces e C g and a C g such that their complexificationsare p + p and p fl p, respectively.

Note that a is always a subalgebra in 9 that contains stab(F) _Lie(Stab(F)). Let D be the corresponding connected subgroup in G andput H = D Stab(F). It is clear that H° = D.

We call the polarization p admissible if e C g is also a subalgebra anddenote by E the corresponding subgroup in G.

Recall that for any X E g there corresponds a right-invariant vectorfield X on G. We extend the map X " X to a complex-linear map from gcto Vectc(G) and use the same notation for the extended map.

Consider the space L(G, F, p) of smooth functions on G satisfying theconditions

(18)(X -27ri(F, X))f =0 for all X E p,

f (hg) = X(h) f (g) for all h E H.

Note that for X E g the real field k is the infinitesimal left shift along thesubgroup exp (RX). So, the first condition in (18) is just the infinitesimalform of the equation f (exp X g) = e2ni (F; X) f(g). We observe also that thesecond condition in (18) for h E H° follows from the first one due to (15).

From the geometric point of view elements of L(G, F, p) can be consid-ered as smooth sections of a certain line bundle L over the right homoge-neous space M = H\G with an additional condition: they are holomorphicalong some complex submanifolds isomorphic to H\E (cf. Section 5.1 ofChapter 1). Moreover, L is a G-bundle and the action of Stab(F) on the1-dimensional fiber of L over F is given by the character X.

We denote by L°(G, F, p) the subspace of those sections whose supportshave compact projections on E\G. One can introduce in Lo(G, F, P) a G-invariant inner product (see the details in the worked-out example below).The completion of Lo(G, F, p) with respect to this inner product is a Hilbertspace 7i where G acts by unitary operators.

It turns out that under a suitable positivity condition on p the resultingrepresentation is irreducible and depends only on the rigged orbit 1 thatcontains (F, X). We denote it by 7ro and consider the described constructionas the second amendment to Rule 2.

The most important particular case of this construction is when p is aso-called Ki hler polarization. This means that

1. P+p=gc.2. p fl 0 = be where 4 = stab(F).


3. The Hermitian form h(X, Y) := 2tBF(X, Y) is positive definite on9c/P.

In this case E = G and p defines a G-invariant complex structure onthe manifold M = H\G. Moreover, M as a homogeneous manifold is iso-morphic to a coadjoint orbit S C g', hence possesses a canonical volumeform. One can introduce a G-invariant Hermitian structure on L so thatthe representation 7rn acts on the space of square integrable holomorphicsections of L.

The practical application of this procedure is explained in detail in thenext section with a typical example.

3. Example: The diamond Lie algebra gWe illustrate the general theory described above using a concrete example.

3.1. The coadjoint orbits for g.Let g be the so-called diamond Lie algebra g with basis T, X, Y, Z

and non-zero commutation relations

(19) [T,X)=Y, [T, Y] =-X, [X,Y)=Z.

It admits the matrix realization as a Lie subalgebra of sp(4, R) with a genericelement

0 a b 2c

(20) S=OT+aX+bY+cZ= 0 0 -9 b

0 9 0 -a0 0 0 0

To describe the corresponding simply connected Lie group G, we considerthe group E(2) of orientation preserving rigid motions of the Euclidean planeiit2. As a smooth manifold, E(2) is isomorphic to S' x 1R2. It turns out thatour group G can be viewed as a universal covering of a central extension ofE(2) (see below).

Let us transfer the coordinates (0, a, b, c) in g to the group G by

(21) g(9, a, b, c) = exp(0T + cZ) exp(aX + bY).

Warning. The group G admits a global coordinate system and is dif-feomorphic to R4 but it is not exponential.

Exercise 5. a) Show that the center C of G consists of elementsg(0, a, b, c) satisfying the conditions: a = b = 0, 0 E 2ir7G, c E R.


b) Check that the adjoint group G/C is isomorphic to the motion groupE(2).

Hint. Use the explicit formula for the coadjoint action given below. 4The dual space g' can be conveniently realized as the set of matrices

1(22) F(t, x, y, z) =

2

0 0 0 0X 0 t 0Y -t 0 0

Z y -x 0

so that (F, S) = tr (FS) = 9t + ax + by + cz.The explicit form of the coadjoint action is:

K(g(O, a, b, c)) F(t, x, y, z) = F t - bx + ay -a2

+b2

2 z, x + bz, y - az, z ;

C /K(g(9, 0, 0, 0)) F(t, x, y, z) = F(t, x cos 9 - y sin 9, x sin 9 + y cos 9, z).

The invariant polynomials on g' are z and x2 + y2 + 2tz. Hence, thegeneric orbits are 2-dimensional paraboloids

(23a)x2 + y2

z=CI #0, t=c2-2c1

The hyperplane z = 0 splits into 2-dimensional cylinders

(23b) x2 +Y 2 = r2, r > 0,

and 0-dimensional orbits (fixed points)

(23c) fto = (0, 0, 0, c), c E R.

The quotient topology in O(G) is the ordinary topology on each piece (23a),(23b), (23c):

1 C,C} R" xR, {S2, } = R>o, IM I R.

As cl - 0 and 2c1c2 r2, the paraboloid Qc,,c, tends to the cylinder Q,;when cl --+ 0 and cic2 -+ 0, the limit is the set of all floc.

So, to obtain the topological space O(G) one should take the real planeR2 with coordinates (a = c1, b = 2c1c2), delete the ray a = 0, b < 0, andpaste in a line R instead of the deleted origin. To get the space O,igg(G),we have in addition to replace each point of the open ray a = 0, b > 0 by acircle.


3.2. Representations corresponding to generic orbits.All generic coadjoint orbits are simply connected. Hence, there is no

difference between orbits and rigged orbits and, according to Rule 1 of theUser's Guide, to any generic orbit there corresponds exactly one (upto equivalence) unirrep 7rc,,C,. We describe here how to construct theseunirreps.

It is not difficult to show that for cl # 0 the functional F has no realpolarization. Indeed, the only 3-dimensional subalgebra of g is spanned byX, Y. Z, hence is not subordinate to F.

However, there exist two 3-dimensional complex subalgebras p± C gcthat are subordinate to F. To simplify the computations, let us choose thespecial point F E 52, ,2 with coordinates (0, 0, c1, c2). Then we can put

pf

Indeed, since [p f, p fJ = C. (X ±iY), we see that FI[Dt, Dt[

= 0. Hence, bothp± are complex polarizations of F. Later on we concentrate on p+, whichwill be denoted simply by p, and only mention the necessary modificationsfor p_.

In this case we have p + g, p fl p = C T ®C Z. The 1-dimensionalcomplex space gc/p is spanned by the vector = (X - iY) mod p and theHermitian form h is given by cl. So, p is Kahler if cl > 0.

Let D = exp The manifold M = D\G is an ordinary planeR2 with coordinates (u, v). The right action of C on M factors through G/Cand produces exactly the transformation group E(2):

(u, v) exp(9 T) = (u cos 9 + v sin 0, -u sin 9 + v cos 9) (rotations);

(u, v) exp(aX + by + cZ) = (u + a, v + b) (translations).

Exercise 6. a) Establish a G-equivariant isomorphism a between themanifold M and the coadjoint orbit 1L,1,C2 with cl # 0.

b) Compute the image a'(o) of the canonical symplectic form o, underthis isomorphism.

Hint. a) Take into account that SZc,,c2 is a left G-manifold while Mis a right G-manifold. So, the equivariance of the map a : M -+ fZ,,,c2 isexpressed by the equality g a(u, v) = a((u, v) g-1).

b) Use the fact that, on linear functions on g*, Poisson brackets definethe structure of a Lie algebra isomorphic to g.

Answer: a) a(u, v) = F(-clv, c:u, cl, c2 - e1 u22v2)

b) a*(a) = cl du A dv.


According to the general theory, every complex polarization p defines aG-invariant complex structure on M. In our case this structure is given bythe global complex coordinate u! = u + iv.

Proposition 3. The space L(G, F. p) given by the system (18) coincidesin our case with the space of holomorphic sections of a (topologically trivial)holomorphic line bundle over M.

Proof. In coordinates (21) the right invariant vector fields on G are givenby

b cos 0

2

+ a sin 6c7,.;

Z = 0r;

2acos9 a, T_8e

Thus, the second condition in (18) (the "real part" of it) takes the form

do= 21ric, f, (9B = 27ric2f.

This condition is equivalent to the equality

(24) f (0, a, b, c) = e2"' (ctc+c26) f(0. a, b, 0).

Let us define the section s: M G: (u, v) '--+ g(0, u. v, 0).

The equality above implies that a solution f as a function on G is com-pletely determined by its restriction on s(M), which is actually a function0= fosonlll;2:

(25) P(u, v) := f (0, u, v, 0).

Taking (24) and (25) into account, the first condition in (18) (the '-com-plex part") can be written in the form

eie (d + i8,, + 7rc1(u + iv)) *p = 0 or 2L+ 7rclw¢ = 0.

The general solution to this equation has the form

1(26) exp (_irciIwI2) ,where i is a holomorphic function of the complex variable w.

Let us define a holomorphic line bundle L over M so that the function

ho(w) = exp (_rciIwI2)


is by definition a holomorphic section of L. Since this section is nowherevanishing, all other holomorphic sections have the form 46 = 0 ¢o with'holomorphic. 0

The action of G in the space L(G, F, p) is computed, as usual, by themaster equation

s(u, v) g(9, a, b, c) = exp(rT + yZ) s(u', v')

with given u, v, 9, a, b, c and unknown r, y, u', V. The solution is

(au + bv) sin 9 + (bu - av) cos 9r=9, )=c+ 2

U' = ucos9+vsin0+a, v' _ -using+vcos9+b.

Hence, the action has the form(27)

(7f (g(0, a, b, 0))0) (u, v) =e,ricl (bu-av) O(u + a, v + b),

(7r(g(9, 0, 0, c) )O) (u, v) = e21ri(clc+c'e) q5(u cos 9 + v sin 9, -u sin 9 + v cos 9).

The corresponding representation of the Lie algebra g in the space of func-tions 0 is

7r.(X) = au - 7ricly;7r. (Y) = a, + 7riclu;

7r.(Zi) = 27ricl ;

7r. (T) = v au - ua + 27ric2.

In terms of the holomorphic functions 7I' = 0/00, it can be rewritten as

7r. (X) = aw -7rclw;7r.(Y) = i8u, +7riclw;

7r.(Z) = 27ricl;

7r. (T) = -iw au, + 27ric2.

The group representation operators in terms of 7/1 are

(7r (g(0, a, b, c))Vj) (w) =e27riclc+7rc1(w(a-ib)-1(a2+b2)) 't'(w + a + ib),

(27')(7r(g(9, 0, 0, 0))7!i)(w) = e21ric2O . 'ry(e-iew).

Now consider the representation space f more carefully. By definition, itis a completion of L(G, F, p) with respect to a norm related to a G-invariantinner product. The natural choice for this inner product is

(VII, 02) = JM = fkl(w)2(w)e_12clduAdv.


It is clear that f is non-zero exactly when cl > 0, i.e. when p is a Kahlerpolarization.

As an orthonormal basis in 11 we can take the holomorphic functions

(ircj w)'

n!n>0.

Exercise 7. Introduce a Hermitian structure on L so that the repre-sentation space f is isomorphic to the space of all square-integrable holo-morphic sections of the bundle L.

Answer. For a holomorphic section V) define the norm of its value at wby

IkL'(w)1i2

4

3.3. Representations corresponding to cylindrical orbits.Here we consider the orbits Str, r > 0, which lie in the hyperplane Z = 0.

According to Rule 3 of the User's Guide, these orbits correspond to unirrepsof G that are trivial on the subgroup exp(R Z) of G, so, actually, to unirrepsof the quotient group E(2).

These orbits have a non-trivial fundamental group because the stabilizerStab(F) of a point F(t, x, y, 0) E 52, has the form

Therefore, al(Str) ?' Stab(F)/Stab°(F) = Z. We see that a 1-dimensionalfamily of rigged orbits fIr(r) corresponds to an orbit 1r, labelled by r EZ^_'R/Z.

More precisely, a rigged orbit 1r(r) E Orj99 consists of pairs (F, Xr,r)where F E 52, and the character Xr,r E H is given by

(28) Xr,r(exp (21rn T + a (xX + yY) + c Z)) = e2ai (nr+ar)

Exercise 8. Show that for any F E St, there exists a unique realpolarization h = R X ® R Y ® R Z and two complex polarizations pt =

Hint. Compute the action of Stab(F) in TFStr (resp. in TF fZ) andcheck that it has a unique Stab(F)-invariant 1-dimensional subspace R . 8t(resp. three Stab(F)-invariant 1-dimensional subspaces C C. (8 ± and

4The modified Rule 2 attaches the unirrep 7rnr(r) = IndGIXn,, to Ilr(r).


Exercise 9. Derive the following explicit realization of the representa-tion 7r51r(r) in L2([0, 27r), da):

(29)(7ti1,(r)(.9(0, a, b, c)) f)(a) = e27rir (a eosa+b sin (t)f (a),

(lr zr(r)(g(d, 0, 0, 0))f)(a) = e27rinrf(01)

where n E Z and 01 E [0, 27r) are defined by a + 0 = 01 + 27rn.Hint. Use the master equation.The corresponding representation 7r.:_ (urn, (T)). of g has the form

7r,(T) = Oe,. 7r.(X) = 27rircosa, 7r.(Y) = 27rirsina, 7r.(Z) = 0.

Remark 3. At first glance 7r, seems to be independent of the parameterr. The point is that the operator A = iOa with the domain A(0, 27r) CL2([0. 27r), da) is symmetric but not essentially self-adjoint. It has severalself-adjoint extensions Ar labelled just by the parameter r E R/Z.

Namely, the domain of Ar consists of all continuous functions f on [0, 27r]that have a square-integrable generalized derivative f and satisfy the bound-ary condition f (27r) = e2'rir f(0). 0

The same modified Rule 2 attaches the unirrep Trr.r to Slr, which isholomorphically induced from a subalgebra p+. We leave it to the readerto write the explicit formula for irr,, acting in the space of holomorphicfunctions on C and to check the equivalence Of Frr.r and 7r,,,.

4. Amendments to other rules

4.1. Rules 3-5.Here we discuss briefly the amendments to Rules 3, 4, and 5 sug-

gested by Shchepochkina.Let H be a closed subgroup of G. We say that a rigged orbit Sl' E

Origg(H) lies under a rigged orbit Q E O(G) (or, equivalently, 1 lies overfl') if there exists a rigged moment (F, X) E Sl and a moment (F', X') E fi'such that the following conditions are satisfied:

(30) a) p(F) = F', b) X = X on H fl Stab(F).

We also define the sum of rigged orbits Q1 and f12 as the set of all (F, X)for which there exist (Fi, Xi) E Sli. i = 1.2, such that

(31) F = F1 + F2 and X = X1X2 on Stab(Fi) fl Stab(F2).

§4. Amendments to other rules 133

Theorem 4 (Shchepochkina). Let G be a connected and simply connectedtame solvable Lie group, and let H be a closed tame subgroup. Then

1. The spectrum of IndHpst, consists of those an for which Q lies overci,

Q.2. The spectrum of ResHan consists of those pp, for which 11' lies under

3. The spectrum of an, ® 7rn2 consists of those an for which St is con-tained in ci 1 + i2 .

The proof, as before, is based on the induction on the dimension of Gand uses the following result which is interesting on its own right.

Let G be a connected and simply connected solvable Lie group, H itsclosed subgroup, and K the maximal connected normal subgroup of G thatis contained in H. Then the quotient group G/K is called a basic group.

Lemma 3 (Shchepochkina). Any basic group coincides with the simply con-nected covering of one of the following groups:

1. The only non-commutative 2-dimensional Lie group G2 = Aff (1, IR).

2. The one-parametric family of 3-dimensional groups G3(y) in Aff (1, C)acting on C by translations and by multiplications by e''t, 'y E C\1R.

3. The 4-dimensional group G4 =Aff(1,C). O

Exercise 10. (We use the notation of Section 3.) Find the spectrum ofthe following representations:

a) Resyac,,c2 where H = exp(R X ®iR Y (D 1RY. Z).b) Indco1 where CO = and 1 denotes the trivial representation.

e) Ind

d) ari(ri) (9 ar2(T2)Hint. Use the modified Rules 3-5.Answer: a) The irreducible representation with h = c1.b) The continuous sum of all ar(T), r > 0, r E IR/7L.c) The continuous sum of all a,,, ,2 with clc2 > 0-

d) The continuous sum of all ar(T) with Irl - r21 < r < rl + r2 andT=T1+T2. 4


4.2. Rules 6, 7, and 10.The amendment to Rule 6 looks as follows. Consider an orbit SZ E

O(G) and let {Il }XE;,(n) be the set of all rigged orbits in fI-1(1l). Then

2ni(F, X)+atr T ex X d en, (n)

nx ( P ) X X =P(X) Jn

In other words, the Fourier transform of the canonical measure on the orbitSZ is equal to the average of all characters of the unirreps corresponding tothe rigged orbits from II-'(Q).

It would be interesting to find the formula for the character of an indi-vidual unirrep TTz. The example in Section 3.3 could be instructive.

The modified Rule 8 needs the following natural refinement: for arigged orbit S1 passing through the point (F, X) E g*igg, we denote by -SZthe rigged orbit passing through the point (-F, X). (Note that Stab(F) _Stab(-F).)

The modified Rule 10 for a unimodular group G follows from the aboveamendment to Rule 6. Namely, the Plancherel measure u on G t--- O'i9g(G)is concentrated on generic rigged orbits; it induces the normalized Haarmeasure on each fiber of the projection II : Origg(G) O(G) and II(p) isthe conditional measure on O(G) corresponding to the Lebesgue measureon g'.

Example 5. Consider the group k(2), the simply connected covering ofE(2). It is isomorphic to the quotient group G/C° where G is the diamondgroup from Section 3 and CO = exp(R Z) is the connected component of

the center in G. So, the space E(2) is just a subspace of G which consistsof rigged cylindrical and one-point orbits.

Exercise 11. Compute the canonical measure on the orbit SZr.Answer: or = xdg -Y dx A dt. 46

By comparing this formula with the Lebesgue measure dx A dy A dt onthe momentum space, we conclude that the Plancherel measure on the setof rigged orbits SZr(T) is given by

µ = rdr A dr.

0Remark 4. If we consider the group M(2) itself, then the dual space

M(2) will be a subset of M(2). Namely, only representations 7r,(0) and1-dimensional representations corresponding to integer one-point orbits willbe single-valued on the group. O

Chapter 5

Compact Lie Groups

The case of compact Lie groups is the oldest and the most developed part ofthe representation theory of Lie groups. The simplest compact Lie group isa circle S', also known as the 1-dimensional torus T = R/Z. The represen-tation theory of this group has already been used (under the name Fourieranalysis) for two centuries.

More generally, commutative Fourier analysis deals with abelian Liegroups, most of which are direct products of the form R" x Tm x Z' x Fwhere F is a finite abelian group.

We leave aside this part of the theory because the orbit method hasnothing to add to it. But, certainly, we shall use the basic notions andresults of commutative Fourier analysis in our exposition.

Our main object will be the representation theory of non-abelian con-nected compact Lie groups. Such a group K can have a non-trivial centerC. Since the coadjoint representation is trivial on the center, the pictureof coadjoint orbits does not change if we replace _K by the adjoint groupK1 = K/C, or by its simply connected covering K1, or by some intermedi-ate group between K1 and K1. We shall see soon that there are only finitelymany such groups because the center of K1 is finite. In many situations,including the classification problem, it is convenient to consider compactgroups up to local isomorphism.

Therefore, we assume from now on that K is a connected and sim-ply connected compact Lie group with finite center. Then the Lie algebrat = Lie(K) is semisimple and we can use the classification theorems fromAppendix 111.2.3 to get the full list of groups in question.

136 5. Compact Lie Groups

It is known that any simply connected compact Lie group is isomorphicto a product of groups with simple Lie algebras. The latter form two families:

1. The classical compact groups, which are special unitary groupsSU(n, K) over K = R, C, or H, i.e. automorphism groups of real, complex,or quaternionic Hilbert spaces of finite dimension. (In the case K = IR thegroup SO(n, R) has the fundamental group Z2, so we have to consider thesimply connected covering group Spin(n, ]R); see below.)

2. The so-called exceptional groups E6, E7, E8, F4, G2.For classical compact groups almost all of the basic results have been

known since the 1920's and were intensively used in quantum physics, whichappeared just at that time.

The exceptional groups, until recently, were considered a curious but use-less anomaly in the general picture. But the recent development of quantumfield theory showed that these groups may play an essential role in new do-mains of mathematical physics, such as string theory and mirror symmetry.So, the study of these groups and their representations received renewedattention.

At first sight it is hard to expect that one can find something new in thiswell-explored domain. Nevertheless, the orbit method gives a new insightinto known results and even reveals some new facts.

At the same time the idyllic harmony of the correspondence betweenunirreps and orbits starts to break down in this case. The most intriguingnew circumstance can be described as follows. We saw above that thereare essentially two ways to establish the correspondence between orbits andunirreps:

1) via the induction-restriction functors,

2) via character theory.It turns out that for compact groups these two approaches lead to dif-

ferent results!

It is a temptation to consider this discrepancy as a peculiar manifestationof the uncertainty principle: to a given unirrep we can associate an orbitonly with a certain degree of accuracy (of size p = half the sum of positiveroots - see the details below).

1. Structure of semisimple compact Lie groupsThis material can be found in many textbooks (see, e.g., [Bou], [FH], [Hu],[J]). Nevertheless, we prefer to give a short exposition here of the necessaryinformation to make the book reasonably self-contained. We shall use thenotation and results concerning abstract root systems from Appendix 111.3. 1.

§ 1. Structure of semisimple compact Lie groups 137

1.1. Compact and complex semisimple groups.In Appendix 111.3.3 we introduce a root system for a pair (g, l) where

g is a complex semisimple Lie algebra and ry is its Cartan subalgebra. Nowwe use the theory of abstract root systems to study compact Lie groups.

We keep the following notation:

K - a connected and simply connected compact Lie group of rankn

t = Lie(K) - the Lie algebra of K;T a maximal torus (maximal abelian subgroup) in K;t -- the Lie algebra of T;C the center of K, a finite group of order c;g = e OR C -- the complexification of t;G - a connected and simply connected complex Lie group with

Lie(G) = g;h = t &R C - the complexification of t, a Cartan subalgebra of g;H = exp h -- a Cartan subgroup of G;g = n_ + i + n+ - a canonical decomposition of g;R - the root system for the pair (g, h);R+, TI - the set of positive roots and the set of simple roots, re-

spectively;

(X, Y)K the Killing form on g;

A (for \ E the element of El such that (A, H)K = A(H) for allHEIR;

(A, y) := (A, µ)K -- the dual form on 9* restricted to h'.Recall that the Lie algebra t is the compact real form of g and is spanned

by the elements

X. - X-a X. + X_n(1)2 2i

for all a E R+, and id, for a E R.

Here {X,,, a E R} is a part of the canonical Chevalley basis in g with thefollowing properties:

(i) [H, X(] = a(H)X, for H E h;(2) (ii) (X0, X;OK =

(iii) [Xn, X_a]

(iv) [Xe, X,0J = NQ,,3Xa+a if a +)3 E R

where Na,Q are non-zero integers satisfying N_Q,_o = -NQ,Q = No,,.


Lemma 1. a) The Killing form on g and its restriction to j are non-degenerate.

b) The restriction of the Killing form on t is negative definite.

Proof. Any real representation of a compact group K is equivalent to anorthogonal one. Therefore, operators ad X for X E t are given by real andskew-symmetric matrices. Hence, tr (ad X2) = -tr (ad X (ad X)') < 0.

The equality takes place only when ad X = 0. Since t has zero center,it means that X = 0. So, we have proved b).

The first part of a) follows from b). The second part of a) follows fromthe relation (g,,, gp) = 0 unless a +# = 0. 0

Remark 1. On a simple Lie algebra the Ad-invariant form is uniqueup to a numerical factor, hence any such form is proportional to the Killingform.

In particular, for any linear representation it of g there is a constantc2(7r) such that

(3) tr(ir(X)x(Y)) = c2(lr) . (X, Y)K

The constant c2(7r) is called the second index of the representation 7r (see[MPRJ for the tables of the second index).

The etymology of this term comes from the following consideration. Let{Xi} be any basis in g, and let {X'} be the dual basis with respect to theKilling form. The element

(4) 0 = > XiX' E U(g)

does not depend on the choice of basis. It is called the quadratic Casimirelement.

One can easily compute that for any unirrep 7r

J (O) dim 7r = 1(5) c2(ir) = dim g

dim9AX"(e)

where I, is the infinitesimal character of ir. According to formula (14) below,Ina (0) = I ' + p12 - IpI2.

There is one special representation 7rp, p = 2 ER+ a, for which thestructure of weights is well known (cf. the character formula (12) below).Namely, let r be the number of positive roots. Then dp := dim 7rp = 2' andthere are 2' weights of the form 1F_«Er. ±a where all possible choices ofsigns occur.


We mention two remarkable corollaries of the formula for c2(7r):

kT+2 - kTc2(7r(k-1)p) = 24

and IPI2 -dim g

24

Q

Now let K be a simply connected compact Lie group, and let G bethe corresponding simply connected complex Lie group. From the generaltheory of Lie groups (see, e.g., (He!]) we know that K is a maximal compactsubgroup in G and the space M = G/K is contractible.

In the space t`, dual to the Lie algebra of a maximal torus T C K,we have two lattices: i Q C i P C t". We want to describe here therepresentation-theoretic meaning of these lattices. For this we recall thewell-known bijection between equivalence classes of finite-dimensional holo-morphic representations of G and equivalence classes of unitary representa-tions of K.

Starting from any unitary representation Or, V) of K we can definesuccessively a representation (7r., V) of the real Lie algebra t = Lie(K),then the complex-linear representation (7re, V) of g = tc, and finally theholomorphic representation (7rC, V) of G so that the following diagram iscommutative (cf. Theorem 2 from Appendix 111.1.3):

End(V)

exp expI expt

KG,Aut(V).nc

Conversely, any holomorphic finite-dimensional representation of G, beingrestricted to K, is equivalent to a unitary representation of K.

Recall also that a weight \ of a linear representation (7r, V) of G is alinear functional on the Cartan subalgebra h such that for some non-zerovector v E V we have

(6) 7r,(X)v = \(X) v for any X E 1).

This functional takes pure imaginary values on t, hence can be consideredas an clement of it*.

We temporarily denote by P the set of all weights of all holomorphicfinite-dimensional representations of G. It is a subgroup of it', because ifA E Wt(7rl) and µ E Wt(ir2), then A + µ E Wt(7rl ® 7r2) and -A E Wt(7ri ).We shall see in a moment that P actually coincides with the weight latticeP C it' defined in terms of an abstract root system.


For A E h' let the symbol eA denote the function on H given by

(7) ea(exp X) = e"(X) for X E h.

It is not evident (and is not true in general!) that this function is welldefined. Indeed, an element h E H can have different "logarithms" X (i.e.elements X E such that h = exp X).

However, from (6) we conclude that for any A E Wt(ir, V) and theappropriate v E V we have

(8) ir(expX)v = e'* (X) v = e"l v = ea(expX) v.

Hence, for A E P' the function ea is well defined.To go further we recall the notion of a dual lattice. Let V be a finite-

dimensional real or complex vector space, and let V' be its dual space. Toany lattice L in V' the dual lattice L* in V is defined by

VEL' f(v)EZ for all f EL.

It is clear that this notion is self-dual: if we consider V as the dual space toV', then L will be the dual lattice to L'.

Proposition 1. Let P' C Q* C it be the dual lattices to Q C P C it.Then

a) the following relations hold:

(9) 27ri (P')' = h fl exp-1(e), 2lri Q* = h fl exp-1(C);

b) the two notions of a weight coincide, i.e. P = P.

Corollary. The order c of the center C of K (which is also the center ofG) is equal to #(P/Q) = det A where A is the Cartan matrix of the rootsystem corresponding to K.

Using the result of Exercise 9 from Appendix 11I.3.1, we get the followingvalues for the order of the center of all simply connected compact Lie groupswith simple Lie algebras:

c(SU(n, C)) = n + 1, c(Spin(2n + 1, R)) = 2, c(SU(n, H)) = 2,c(Spin(2n, IR)) = 4, 9 - n.

The same method gives c(F4) = c(G2) = 1.


Proof of Proposition 1. We start with the first of the relations in (9).For any X E h fl exp 1(e) we have

1 = ea(exp X) = eA(X) or A(X)E 27riZ.

Therefore h fl exp-1(e) C 27ri (P')`.To prove the inverse inclusion, we use the fact that the group G has a

faithful finite-dimensional representation. It follows that the collection ofall functions e", A E P', separates points of H. Hence, if X E (P')`, thenea(exp21riX) = e2mia(X) = 1 for all A E P' and consequently exp(27riX) = e.

The second relation in (9) is proved by the same argument, but insteadof all representations we consider only the adjoint one. The functions ea, A EQ, do not separate points h1 and h2 from H if Ad h1 = Ad h2, i.e. h1h21

C-

C. Therefore, exp X E C is equivalent to A(X) E 21riZ for all A E Q, i.e.X E 27riQ*.

Let us now prove b). We use

Lemma 2. For the simple 3-dimensional Lie algebra sl(2, C) we have P _P.

Proof of the lemma. It is well known (see Appendix 111.3.2) that irre-ducible representations of sl(2, C) can be enumerated by non-negative inte-gers so that 7r is the n-th symmetric power of the standard (tautological)representation In.

Choose the standard basis E, F, H in sl(2, C) and put h = C H, ga =C C. E, g_a = C C. F. Then in an appropriate basis

71 0 0 ... 00 n-2 0 ... 0

7r,,(H)= 0 0 n-4 ... 0

0 0 0 ... -n

Therefore the evaluation at H identifies the lattice P' with Z. Since theadjoint representation is equivalent to 7r2, the root lattice Q is identifiedwith 2Z.

The weight lattice P for the root system Ai is generated by the funda-mental weight w = 2a. Since w(H) = la(H) = 1, we see that P = P'. 0

We return to the general case. Let A E P'. Consider the correspond-ing representation (7r, V) of G and restrict in. to the subalgebra g(a) C gspanned by Xta and H. Then .\ IB(a) will be a weight of a representation


of g(a). But the expression 2Q' does not depend on the choice of anAd-invariant form on g(a). Hence, it is an integer and P C P.

To prove that P C P it is enough to show that all fundamental weightsbelong to P. For this the explicit construction is usually used, which showsthat all fundamental weights are highest weights of some irreducible rep-resentations of G. This is rather easy for classical groups (see, e.g., [FH],[Hu], [Zh1J and the examples below) and is more involved for exceptionalgroups. Another way is to use E. Cartan's theorem (Theorem 3 below). 0

For future use we define the canonical coordinates (t1, ..., t") on Hby

(10) tk(exp X) = e("'k'X).

These coordinates take non-zero complex values. Thus, they identify theCartan subgroup H C G with the subset (C')' C C". Under this identifi-cation the maximal torus T C K goes to the subset T' C C", which consistsof points whose coordinates have absolute value 1.

At the same time the lattice P is identified with Z" so that the functionea has the form(11) e'(h) = ti'(h)...to (h) for A = (k1,..., k").

Example 1. Let K = SU(N, C), N = n + 1. Then t = su(N, C) andg = t is the simple complex Lie algebra of type A. The maximal torus Tis the diagonal subgroup of K and a general element of T with the canonicalcoordinates (t1, ... , t") has the form

ti 0 0 ... 0

0t2ti1

0 ... 0

0 0 t3t21 ... 0t=

t"t"11 00 to

I1

The central elements have coordinates tk = ek, where e E C satisfies en+1 =1.

The first fundamental representation 7r1 of K is just the defining repre-sentation: 7r1 (g) = g. All other fundamental representations are the exteriorpowers of it: Irk =A k

7r1. Indeed, the highest weight vector in Ak C" isel A .. A ek and the corresponding function eA is given by eA(t) = tk. Thus,A = wk, the k-th fundamental weight. 0

In conclusion we collect the main facts here about the representationtheory for a compact simply connected Lie group K. In the formulae belowp denotes the half sum of positive roots.


Theorem 1. a) Any unirrep 7r of K is finite dimensional and can be unique-ly extended to a holomorphic irreducible representation of the simply con-nected complex Lie group G with Lie (G) = Lie (K)c.

b) A unirrep it is characterized up to equivalence by its highest weight A,which can be any dominant weight, i.e. a linear combination of fundamentalweights with integral non-negative coefficients.

c) The character of the unirrep ira with the highest weight A is given bythe Weyl formula

(12) Xa(t)EWEW(-1)1('w)ew-(P)(t) l/

fort E T.

d) The dimension of zra is given by

(13) d,\ = Tj (A+P,cr)alERR+

(P, a)

e) The infinitesimal character of ira takes the value

(14) IA(02) = (A + 2p, \) = IA + PI2 - IPI2

on the quadratic Casimir element A2.f) the multiplicity of the weight p in the unirrep 7rA is

(15) M'\ (A) = > (-1)1(w)P(w ' (A + P) - (µ + P))wEW

where P is the Kostant partition function on the root lattice definedas the number of ways to write A as a sum of positive roots. It has thegenerating function

(16) > P(.\)ea = JJ (1 - ea)-i.

AEQ aER+

D

To these results we add the Weyl integration formula. For a centralfunction f, i.e. constant on the conjugacy classes in K, it allows us to reducethe integral over the whole group K with the normalized Haar measure dkto the integral over the maximal torus T with the normalized Haar measuredt:

(17) 1 f(g)dg= I''I ff(t)IA(t)12dtK

where(18)

0(t) _ (-1)1(w)e1 *P(t) = e-P [[ (ea(t) - 1) = rj 2sinh a(log t)1

wEW aER+ aER+


1.2. Classical and exceptional groups.In this section we collect some information about exceptional simple

complex Lie algebras. It turns out that any such algebra can be constructedas a (Z/nZ)-graded Lie algebra with some classical Lie algebra in the role ofgo. This allows us to construct matrix realizations of exceptional Lie groups.

Let R be a root system of a simple Lie algebra g of rank n, let II ={al, ..., an} be a set of simple roots, and let rP E R be the maximal root.Let A = II Ail II be the Cartan matrix. Define the extended Dynkin graphI' as follows.

The vertices of I' form the set II = II U {ao} where ao = -Vi. Thevertices ai and aj, as usual, are joined by nij = Ai,j Aj,i edges. Thenumbers ak, 0 < k < n, are defined by

n

(19) ao = 1 and > akak = 0.k=0

Proposition 2. Let rk denote the graph obtained from r by deleting thek-th vertex together with outgoing edges. Then the Lie algebra g can be real-ized as a (7G/akZ)-graded Lie algebra in which go is a semisimple subalgebradefined by the graph I'k.

Proof. Let go be the subalgebra in g generated by {X±a, a E Fk}. Then,according to Theorem 1 c), go is a semisimple Lie subalgebra of g. Relation(19) shows that the root lattice of go has index ak in the root lattice of g.So, g as a go-module is (Z/akZ)-graded. More precisely, the degree of Xa isequal to 2 Wk' ° mod ak.

ak,ak

Note also that the highest weight of the go-module of degree 1 is equalto -ak. 0

Example 2. Consider the exceptional complex Lie algebra G2. Theextended Dynkin graph and numbers {ak} are shown in Figure 1. (Thearrow, as usual, indicates the short root.)

1 2 3o --oE# = o

Figure 1. Extended Dynkin graph I' for G2.

It follows that G2 can be defineda) as a (Z/2Z)-graded Lie algebra g with

go =sl(2,C)®sl(2,C), 91=V10 V3

§1. Structure of sernisimple compact Lie groups 145

where VI is a standard 2-dimensional representation of the first factor andV3 is a 4-dimensional representation of the second factor (the symmetriccube of the standard representation);

b) as a (Z/3Z)-graded Lie algebra g with

9o = sl(3, C), 9i=V, 9-1=V.

where V is a standard 3-dimensional go-module which we realize as the spaceof column vectors and V` is the dual go-module of row vectors.

The corresponding splittings of the root system are shown in Figure 2.

1Bo

Figure 2. Splittings of the root system for G2.

We consider in detail the second approach. The commutators of homo-geneous elements X E gi and Y E g3 are defined as follows.

Let X,YEgo, v,v'EV, f, f'EV*. Then

[X, Y] = XY - YX; [X, V) = Xv; [f, X] = fX;

[v, f] = of - 3fv 1, [v, v'] = f (v, v'), [f, f'] = v(f, f')

Here f (v, v') E V" and v(f, f') E V are defined by the relations

(f (v. v'), v ") = a det l v v' v"I and (f ", v (f, f ')) = b detff'f

where a and b are appropriate constants. The Jacobi identity implies that3ab = 4, so we come to the matrix realization of G2 by 7 x 7 complex


matrices of the form

(20)

E6 :

where A is the linear map from V to so(3, C) given by

v2 -v1 0 vlA(v) = 1 -v3 0 v1 for v = v2

v13- 0 v3 -v2 v3

The compact real form of G2 comes out if we assume X to be a skew-Hermitian matrix and v = -P. To finish with the description of G2, wemention that the 7-dimensional representation (20) identifies the compactreal form of G2 with the Lie algebra of derivations of the 8-dimensionaloctonion algebra 0, restricted to the subspace of pure imaginary elements.The point is that for any matrix X of the form (20) the linear transformationgiven by exp X preserves some trilinear form {a, b, c} in RT. Since it alsopreserves a bilinear form (a, b) = ab, we can define a non-associative bilinearoperation in Ills:

(a, a) * (J3, b) = (7, c) where 'y = a/3 - (a, b) and (c, c) = {a, b, d}.

This is exactly the octonion algebra. O

For the interested reader we show the extended Dynkin graphs and num-bers a k for other exceptional simple Lie algebras in Figure 3.

F4:

X v A(f)X = f 0 -v

A(v) -f -X

1 2 3 4 20--0--0h0--o1 0

1

201 2 1 2 10--0-- o --o--0

3

2 0

1 2 3 1 3 2 1E7: o --o --o-- o --o--o--o

4

Es

3 0

1 2 3 4 5 1 4 20--0--0--0--o-- 0 --0--06

Figure 3. Extended Dynkin graphs.


The reader easily recognizes the rule which governs the distribution ofnumbers ak: there is a maximal value aj = max x ak and all other numbersform an arithmetic progression on any segment of the Dynkin graph startingfrom {j}.

Note also that the sum of all numbers ak is usually denoted by h and iscalled the Coxeter number of the given root system.

Exercise 1.* Describe the graded realizations of all remaining excep-tional Lie algebras.

Answer: For F4:1) 9o = B4, 91 = V,4;

2) go=Cs xA1, 91 =Vwa(9 V.,;3) go4) 9o=AsxAl, 91=Vw,(9Vw,92=Vw20 V2w,93=V,y3(9 Vw.

For Ee:1) go =AsxA1, 91=Vwa®Vw,;2) 9o= A2 x A2 x A2, 91=Vw1®Vw,®V1,11 1

9-1=Vw2(9 Vw2(9 Vw2.

For E7:1) go = A7, 91 = V114;

2) go=D6 xAl,91 =V,6®Vw,;3) go=AsxA2,91=Vw,V,,,2,9-1=Vw2(9 V4114;4) go=AsxAsxA1,91 =Vw,0V.,®Vw,92=Vw2®V.,2®Vo,

93 = V-3 (9 Vwa ® VI"

For E8:1) go As xD5,91=Vw10Vws,92=Vw,®Vws,93=Vw,(9 Vws;2) 9o=A4xA4,01 = Vw1®Vw2, 92=Vw10 VW3,93=Vw4 ®Vw21

94Vw4(9 Vw3;3) go=A5xA2xA1,9, Vw,®Vw2®Vw,92=Vwz®Vw,®Vw,

93 = Vwa ® VO ® Vw, 94 = Vwa ® Vw2 ® Vw, 95 = Vws ® Vw, ® Vw;

4) so =AT xAl,91=Vw,Vu,92=VW4®Vo,93=Vwz®Vw;5) 9o = D8, 91 = Vws;6) 9o = As, 91 = V,,,3, 9-1 = Vw6.

4

2. Coadjoint orbits for compact Lie groups

2.1. Geometry of coadjoint orbits.For a compact Lie group K, in view of the existence of an Ad-invariant

non-degenerate bilinear form on t = Lie (K), the coadjoint representation isequivalent to the adjoint one.


A well-known theorem about compact transformation groups impliesthat for a given compact Lie group K there are only finitely many types of(co)adjoint orbits as homogeneous K-manifolds. We give the description ofthese manifolds here.

It is convenient to identify the functional F E t' with the element XF E tvia the formula (FX, Y) = (X, Y)K for all Y E t. Then the real vector spaceit' spanned by roots will be identified with a subspace of it.

Proposition 3. Let K be any connected compact Lie group. Thena) For any point X E t the stabilizer of X is conjugate to a subgroup S

that contains the maximal torus T C K.b) There are finitely many subgroups intermediate between T and K and

each of them is a stabilizer of some point X E t.c) For all (co)adjoint orbits of maximal dimension the stabilizer of a

point is conjugate to the maximal torus T.

Proof. We deduce all statements from

Lemma 3. Let C+ be the positive Weyl chamber in it*. Any K-orbit int' ^-- t intersects the set iC+ C t in exactly one point.

Proof of the lemma. Let S2 be a K-orbit in t. Pick a regular element H E tand consider the function fH(X) = (X, H)K on Q. Since c is a compactset, this function attains an extremum at some point X0 which must be astationary point for fH. Therefore, for any X E t we have dfH(Xo)(X) =([X, Xo], H)K = 0. But this implies the equality ([Xo, H], X)K = 0 for allX E t. Hence, X0 commutes with H and, consequently, belongs to t. Thus,the intersection i fl t is non-empty.

Besides, it is a W-invariant subset in t. We claim that it is a single W-orbit. Indeed, the subalgebra t C t is the centralizer of H in t. Therefore, ifAd k H E t, then Ad k t = t, hence k E NK(T).

It remains to show that the W-orbit of any point p E it' has a uniquecommon point with C+. Let p E C+ and w E W. Then w(p) is separatedfrom p by l(w) mirrors. Therefore, w(p) can belong to C+ only if it belongsto all these mirrors. But then w(p) = it. 0

We return to the proof of Proposition 3. It is clear that the stabilizerof the point p E t contains T and coincides with T for regular it. Thisproves a). If p is an arbitrary point of t, then the Lie algebra of its stabilizercontains t, hence has the form

e=t+ (R.

X, - X-1(D R

Xa+X-a2 2i

aER'


where R' C R is the intersection of R with the hyperplane n pl. Evidently,there are only finitely many such subsets in R and we denote them byR("'), 1 < m < M.

Let t(m) be the subalgebra spanned by t and the vectors X° 2-° andX<,+X_Q

2i , a E R(m). It is a direct sum of a real semisimple Lie algebrat(m) = [fl"`), !("')] and an abelian subalgebra.

R(') can be considered as a root system related to the complex semisiin-pie Lie algebra g m) = ti") 0C.

Conversely, any subalgebra t(m), 1 < m < M, is the centralizer of anelement p(m) E t. and the corresponding subgroup K(m) is the stabilizer ofµ(m) in K. Hence, the K-orbit of ip(m) E t has the form K/K(m).

The space F = KIT is called the full flag manifold for K. Theremaining homogeneous spaces .") = K/K(m) are called degenerate flagmanifolds: they can be obtained from F by a projection whose fibers areisomorphic to a product of smaller flag manifolds.

Example 3. Let K = U(n), and let T = T(n) be the subgroup ofdiagonal matrices. The Lie algebra t consists of all n x n skew-Hermitianmatrices X. We can define an invariant positive definite bilinear form on gby (X, Y) := -tr(XY).

Part a) of Proposition 3 in this case reduces to a well-known fact thatevery skew-Hermitian matrix can be reduced to the diagonal form via con-jugation by a unitary matrix.

Part b) claims that this diagonal form is unique if we assume that the(pure imaginary) eigenvalues (iAl, ..., iAn) satisfy the condition Al ? A2 >

> An.

Gathering equal eigenvalues of X in blocks of sizes nl , ... , nr, we see thatStab(X) is conjugate to a block-diagonal subgroup Un...... nr ^- Unt X .. X U.The corresponding orbit has dimension d = 2 Ei<j nine, which varies from0 to n(n - 1). The number of different types of orbits in this case is equalto the number p(n) of all partitions of n.

The flag manifold F 1....,n, can be realized as the manifold of all fll-trations, which are also called flags, of type (nl,... , nr):

{0} = Vp C Vl C C Vr = Cn with dim V/Vi_i = ni

(see Figure 4). The most degenerate flag manifolds are Grassmannians

Gn,k = U(n)/Uk,n-k

Geometrically they are realized as manifolds of k-dimensional subspaces inCn. 0


Figure 4. A flag of type (1,2) in R3.

The flag manifolds have a rich geometric structure, which we describehere.

First, being homogeneous spaces for a compact Lie group K, they admita K-invariant Riemannian metric.

Second, being coadjoint orbits, they have a canonical K-invariant sym-plectic structure.

Third, they can be endowed with a K-invariant complex structure, i.e.admit local complex coordinates in which the action of K is holomorphic.

Finally, all three structures can be united into one by saying that flagmanifolds are homogeneous Kihler K-manifolds.

We recall (see Appendix 11.3.1) that a complex manifold M is calledKahler if in every tangent space a Hermitian form h(x, y) is givensuch that

(i) the real part g = Re h defines a Riemannian metric on M;(ii) the imaginary part o = Im h defines a symplectic structure on M.Let z1, ... , z", zk = xk + iyk, be a local coordinate system on M. Then

the local expressions for h, g, and a have the form:

h = ha,Q dz® ® & 0, with hfl,a = ha,R = so,p + iaa,0, so,0, aa,0 E IR,

g = sa, $(dxadxO + dy ady 0) + 2aa, 0 dxady 0,

1a =

2a.,# (dx'a, p(dxa A dxa + dy a A dy Q) - 8,,, t3 dxa A dy

Lemma 4. Let M be a complex manifold with local coordinates zl, ... , z",and let h = ha, pdz adz a be the local expression for a Kahler form on M.

a) In a neighborhood of any given point there exists a real-valued functionK such that the coefficients of h have the form

(21) ha,p = (OQB#K.


b) The function K is defined modulo the summand of the form Re fwhere f is a holomorphic function. It is called the Kahler potential of h.

Sketch of the proof. a) It is convenient to rewrite the symplectic forma = Im h as

zho, pdz Q A dz 0. The condition do = 0 gives

Ohjz

-1Y

o, 0dz-'Adz° Adz/3 + a9z7adzQ AdzY3 Adze = 0.

Since the summands belong to different types ((2,1) and (1, 2), respectively),the equality can be true only if both summands vanish. It follows that

8h7 6 0h. , 9hc .y Oho 0(22)

,, _OzQ

,

Oz7aand

_, ,

oz0 8zy

It is known' that the first condition in (22) is sufficient for the existence offunctions 0,6 such that locally hQ, p = 8¢p/Bz0. Analogously, the secondcondition in (22) ensures the existence of functions rPQ such that locallyh0,a = ftc,/ a.

Finally, the equation'goo O ?P"

azQ 8zR

guarantees that there is a function K such that locally

OK 8K00 =1OZO

So, we have found a function K such that the equality (21) holds. Now ob-serve that the complex conjugate function K also satisfies the same equalitybecause the matrix IIh0,RII is Hermitian. Then Re K is real and again satis-fies (21).

b) This follows from the fact that all real solutions to the system ofequations

a2_ f =0, a,QE {1,2,...,n},8zQ8z0

have the form f = Re g where g is a holomorphic function. 0

'Actually, we use the Dolbeault-Grothendieck Lemma here (the complex analogue of thePoincare Lemma): d'w = 0 locally implies w = d'8. Here d' is the holomorphic part of thedifferential d. Namely, d'(fdz'' A A dz'r A dV' A Adt2) = Of., " dz' A dz" A A dz'o Aez;dill n...Adata.


Example 4. Let K = U(n), and let M = K/Un_k,k = G,,k(C), theGrassmann manifold. For k = 1 we have Gn,1 ^_- pn-1 and this example isconsidered in Appendix II.3.1. The formula obtained there for the Kahlerpotential can be generalized to all Grassmann manifolds.

First, we want to define a convenient parametrization of Gn,k. Considerthe set Zn,k of all rectangular complex matrices Z of rank k and size n x k.It is a left (GL(n, C))-set and a right (GL(k, C))-set with respect to theordinary matrix multiplication.

To any Z E Zn,k we associate the k-dimensional subspace [Z] in Cnspanned by the columns of Z. It is clear that the map Zn,k -' Gn,kZ --. [Z] is surjective and that [Z1] = [Z2] if Z1 and Z2 belong to the sameGL(k, C)-orbit. Thus, the functions on Gn,k are just the GL(k, C)-invariantfunctions on Zn,k

Now, we define the local coordinate systems on Gn,k as follows. Choose asubset I= {i1, i2, ... , ik} of cardinality k from the set of indices {1, 2, ... , n},and let J = { j1 i i2i ... , in-k} be the complementary subset. Denote by ZIthe k x k matrix with elements a,ni = zim,1 and by ZJ the (n - k) x k matrixwith elements b,nt = zjm,(. To each I we define the chart UI on Gn,k thatcontains [Z] for all Z with det ZI 34 0. The (n - k) x k coefficients of thematrix CJ := ZJ(ZI)-1 are by definition the local coordinates on UI.

Exercise 2. Check that the transition functions are rational functionswith non-vanishing denominators.

Hint. Show that for any point [Z] E UI one can choose the representa-tive Z such that ZI = 1, ZJ = CJ. If [Z] also belongs to another chart UI',compute the matrix CJ' in terms of CJ. 4

Define the function KI on UI by

det(Z*Z)KI = -logdet (Z; ZI )

The differences

K I =1odet ZIP + to det ZI,

I g det ZI g det ZI

are real parts of analytic functions. Therefore, the Kiihler form 88KI doesnot depend on the index I and is well defined on G,,,k.

Note that this form and the corresponding metric and symplectic struc-ture are invariant under the action of U(n, C), but not under the groupGL(n, C). 0


There is a simple explanation for why all (co)adjoint orbits, which areflag manifolds, possess a complex structure. Recall that the group K is amaximal compact subgroup in the simply connected complex Lie group G.

Consider the canonical decomposition

g=n-®h®n+ withb =tc-.

Then b± = h ® n± are maximal solvable subalgebras in g. They arecalled opposite Borel subalgebras. Let B± C G be the correspondingBorel subgroups. Often we shall omit the lower index + in the notationn+, b+, B+.

Recall that the subgroup H = Tc is a Cartan subgroup of G, i.e. amaximal abelian subgroup which consists of Ad-semisimple elements.

Consider now the complex homogeneous manifold Y = G/B.

Lemma 5. The compact group K acts transitively on Y and the stabilizerof the initial point coincides with B n K = T.

Proof. Let {Hk, 1 < k < 1; Xa, a E R} be a Chevalley basis in g wherethe Xa satisfy commutation relations (2) and the Hk form an orthonormalbasis in it. The Lie algebra t is spanned by elements

Xa - X_a, iXa + iX_a for all a E R and iHk, 1 < k < 1.

From this we see that

(23) t+b=g and tnb=t.

The first equality in (23) implies that the K-orbit of the initial pointB/B E G/B is open. Since it is compact, it is also closed, hence coincideswith Y = G/B.

The second equality in (23) shows that locally K n B coincides with T.Since T is clearly a maximal compact subgroup in B, this is true globally.0

Thus, we can identify Jcwith Y and obtain a K-invariant complex struc-ture on the flag manifold F.

For degenerate flag manifolds.Ft'i = K/Ki`i the situation is analogous;the role of Y(t) is played by G/P(t) where P(t) is a so-called parabolicsubgroup of G: a minimal complex subgroup in G that contains B and K(t).

Remark 2. A K-invariant complex structure on a flag manifold is notunique. In the case of a full flag manifold F = KIT all possible complexstructures can be described as follows. Let W = NK(T)/T be the Weyl


group corresponding to the pair (K, T). This group acts by automorphismsof F considered as a homogeneous K-space.2

Note that the Weyl group W also coincides with NN(H)/H. Hence, itacts by automorphisms of the homogeneous space G/H 56 G/B. But theaction of w E W on G/B is not holomorphic!

It turns out that W acts in a simply transitive way on the set c(.F) ofall different K-invariant complex structures on Y. So, c(F) is a principalhomogeneous space for W and has cardinality J W I.

Note also that there are exactly I W I different Borel subgroups containingH, hence I W I ways to identify F with Y. CJ

From now on we fix the choice of a complex structure on F or, in otherwords, fix a Borel subgroup B such that B D H 3 T.

Example 5. Let K = SU(2). Then G = SL (2, C), F = S2, Y =P1(C) and the two complex structures on S2 result from the two possibleK-equivariant identifications of S2 with P'(C). 0

Exercise 3.' Describe the six different invariant complex structures onthe flag manifold .F = SU(3)/T.

Hint. The points of the space Y = SL(3, C)/B are geometrically re-alized by flags Vl C V2 where Vi is an i-dimensional subspace in C3. Wecan consider V1 as a point in the projective plane p2(C) and V2, or, more

V-L C (C3)' as a point in a dual projective planeprecisely, its annihilator 2

P2(C)*. We get a K-equivariant embedding of Y into the product of twodual complex projective planes.

In terms of dual homogeneous coordinates (y' : y2 : y3) and (yl : y2 : y3)on these planes the manifold Y is given by the equation

ylyl + 112 y2 + y3y3 = 0.

Now, a K-equivariant map F -* Y is completely determined by the imageof the initial point xo = T/T E KIT =.F. This image must be a fixed pointof the T-action on Y. It remains to show that there are exactly six fixedpoints for T in Y:

1 0(0

1 0

0)' (0 0 1)'

(00 0)'

01

)0 0)' (00 0

(0 0 01)'(0

where the upper row gives the homogeneous coordinates of a point in P2(C)while the second row gives the homogeneous coordinates of a point in p2(C)*.

42C1. Example 13 from Appendix 111.4.2.


2.2. Topology of coadjoint orbits.We are mainly interested in the orbits of maximal dimension, which are

called generic or regular. We already saw that every (co)adjoint orbitintersects the subspace t V. Moreover, for a regular orbit fl this inter-section consists of JW I different points that form a principal W-orbit. Allthese points are regular elements of t, i.e. their centralizer coincides withT. Therefore, all regular orbits are diffeomorphic to the full flag manifold.F = KIT.

We discuss the topology of regular orbits here.

Theorem 2 (Bruhat Lemma). Let G be a complex semisimple Lie group,H a Cartan subgroup, and W = NG(H)/H the corresponding Weyl group.Choose a Borel subgroup B 3 H and for any w E W a representative iv ENG(H) of the class w E NG(H)/H. Then G is the disjoint union of doubleB-cosets:

(24) G = U BivB, w E w.wEW

Exercise 4.' Prove the Bruhat Lemma for the classical simple complexgroups.

Hint. Use the realization of flags as filtrations and associate an elementof the Weyl group to a given pair of filtrations.

In the case of G = SL(n, C) it can be done as follows. Consider a pairof filtrations

f:{0}=V0CV1C...CVn=C', f':{0}=VpCViC...CVn=Cn

with dim V = dim V' = i. We observe that a filtration on a vector spaceV induces a filtration on any subspace, on any quotient space, and on anysubquotient space (i.e. a quotient of one subspace by another). In particular,the first filtration f induces a filtration on WW = Vi'/Vi' 1. This inducedfiltration has the form

o=woW C 14110>C...CW*)=w(2)

with

W`'1 = ((V/nVi) +V' 1) /Vi'-1.

Since W(2) is a 1-dimensional space, exactly one of the quotientsis non-zero. Let it be for j = s(i). We get a map i F- s(i), which in fact is


a permutation of the set {1, 2, ..., n}. But the set S of permutations isexactly the Weyl group for SL(n, C).

Let N = exp n. Then N is a normal subgroup in B equal to the com-mutator [B, B]. It is usually called the unipotent radical of B. In fact,B is a semidirect product H x N.

Let N,,, = N fl iNw-1 where uw is a representative in NK(T) of theclass w E W = NK(T)/T. Define the length of w E W to be l(w) =dim c N - dim E Nw. According to Proposition 6 in Appendix 111.3. 1, thisdefinition is equivalent to each of the following:

a) l(w) is the length of the minimal representation of w as a product ofcanonical generators s1 i ... , s1 of W.

b) l(w) = #(w(R+) fl R_), i.e. the number of positive roots a such thatw(a) is negative.

Corollary. The full flag manifold. = G/B is a union of even-dimensionalcells:

(25) F = U fw , dim 21(w).wEW

Proof. The decomposition (25) shows that there are JWI B-orbits in Flabelled by elements of W. Namely, put F,,,, = Bib mod B. A genericelement x E Fw can be written as x = nhui mod B, h E H, n E N. Sincew E NN(H), we can rewrite it as x = nib mod B. Moreover, the element n inthis expression is defined modulo Nw = iNiu-1. Therefore, Fw = N/NWC!(w) ' R2!(w)

In particular, since 1(w) = 1 only for the generators, we get the followinglist of expressions for the second Betti number b2 (.F) := dimH2(.F, R):

(26) b2(F) = number of simple roots = rk G = dims h = dim T.

This important fact can also be proved as follows.It is well known that for a compact Lie group K with 7r1(K) = 0 we

also have 72(K) = 0. (It follows, for instance, from the absence of anAd-invariant skew-symmetric bilinear form on t.) The exact sequence (seeformula (54) in Appendix 111.4.2)

...-.lrk(G)-7rk(F)-'7rk-1(T)-'lrk-1(G)_'...

shows that the flag manifold F is simply connected and(27) H2(F,7G) ,.. ir2(F) 7r1(T) ., 7+..., ZdimT


Much more precise results can be obtained using the structure of theWeyl group (see [B] or Appendix 111.3.1). For instance, we get the simpleformula for the Poincare polynomial of Yin terms of exponents:

n 2(m,+l)(28) tk dim Hk(F, ][l;) =

I_ t1 - t2

k i=1

Now let ci C t` be a regular coadjoint orbit. The canonical symplecticform a on 11 defines a cohomology class [a] E H2(.F, ]R).

Recall that an orbit 11 C t` is called integral (see Chapter 1, Section2.4) if [a] E H2(F, Z) C H2(F, R). The number of "integrality conditions"for an orbit of a given type is equal to the second Betti number of this orbit.But we have seen that this number is equal to the dimension of the manifold0reg(K) =' Treg/W of regular orbits.

So, for compact groups the integral regular orbits form a discrete set. Amore precise result is given below in Section 3.1.

We conclude this section with a resume of the Cartan theory of highestweights.

Theorem 3 (E. Cartan). Let G be a simply connected complex semisimplegroup.

a) Any finite-dimensional irreducible holomorphic representation (7r, V)of G has a highest weight A (i.e. A is bigger than all other weights in thesense of the partial order introduced in Appendix 111.3.1).

b) Two representations with the same highest weight are equivalent.c) The set of possible highest weights coincides with the set P+ of dom-

inant weights.

Corollary. For a compact simply connected group K the set k of allunirreps is labelled by the set P+ of dominant weights.

Proof of the theorem. a) We shall use the Bruhat Lemma from theprevious section. It implies that the subset G' C G defined as

G'= B_ B=is open and dense in G.

Let (7r, V) be an irreducible finite-dimensional representation of thegroup G. We choose a maximal weight A from among all the weights of thisrepresentation. Here "maximal" means that there is no weight of (ir, V)which is bigger than A. Let v E V be a corresponding weight vector.

Also choose a minimal weight p from among the weights of the dual rep-resentation (lr`, V*) and denote by v' E V' a corresponding weight vector.


Note that we do not suppose that the weights A and p are uniquelydefined.

Consider the function f,,,,, on G (a matrix element of r) given by

.fv,v(g) = (v5, r(g)v)

Lemma 6. The vector v is invariant under 7r(N+), while the vector v' isinvariant under 7r(N_).

Proof. We show that 7r.(n+)v = 0 and r.(n_)v` = 0. Indeed, for a E R+the operators 7r(XQ) increase the weight, while operators 7r(X_Q) decreasethe weight.

Using Lemma 6 we can explicitly compute the function f,,,,, on G' intwo different ways:

(r`(n-)-lv, r(h)r(n)v) = e'(h)(v`, v),(29) fv v(n_hn) _ (7r`(h)-'7r,(n_)-lv*, r(n)v) = e-µ(h)(v', v).

Lemma 7. The quantity (v', v) is non-zero.

Proof. Indeed, otherwise we have fv v. IG,= 0 and, since G' is dense in G,f,,,,, = 0 everywhere. But (r, V) is irreducible, hence the vectors of theform r(g)v, g E G, span the whole space V.

Using the property (g) = we conclude that f,,,,, = 0for all v E V. This contradicts v' 34 0.

From Lemma 7 and (29) we conclude that A = -µ. But we have chosenA as any maximal weight of (r, V) and it as any minimal weight of (r`, V').Therefore, both weights are uniquely defined.

Moreover, since v and v` are chosen arbitrarily in the correspondingeigenspaces, Lemma 7 and (29) imply that both A and A have multiplicity1

Now, we prove that A is not only maximal, but the highest weight ofOr, V). Assume the converse. Then there exist weights A' which do notsatisfy the inequality A > Y. Choose a maximal weight Al among them.It is clear that Al is a maximal element of Wt(ra). (Otherwise we haveA2 > Al for some A2 < A, which is impossible.) But there is only onemaximal weight, hence Al = A, which is also impossible.

Another proof of the statement can be obtained using the universalenveloping algebra. Namely, the space V, is an irreducible U(9)-module.


From the Poincare-Birkhoff-Witt theorem we conclude that the map U(n_)®U(4)®U(n+) U(g), given by multiplication, is an isomorphism of vectorspaces. Since the highest weight vector vN is annihilated by elements of7r.(n+) (cf. proof of Lemma 6) and an eigenvector for 7r.(4), we see thatVA = U(n_)va. It follows immediately that Wt(7ra) C A - Q+.

b) We make use of the following well-known fact.

Proposition 4. An irreducible representation of G is completely deter-mined by any of its matrix elements.

Proof. The same arguments show that the subspace M(7r) spanned by leftand right shifts of contains all other matrix elements of it. The groupG x G acts on G by left and right shifts and this action gives rise to a linearrepresentation II of G x G in M(7r). It is easy to check that II = 7r x it. So,M(7r) under the action of G x 1 C G x G splits into irreducible subspaceswhere G acts by it.

To prove c), we consider the representation (7r., V) of the Lie algebra gand restrict it to a subalgebra 9(a), a E R+. Then it splits into irreduciblecomponents with respect to g(a). Since the multiplicity of A is 1, exactlyone component contains the vector v.

Certainly, v remains a highest weight vector for the restriction. But forthe Lie algebra g(a) ^_- sl(2, C), statement c) is true (see Appendix 111.3.2).Therefore we get 2,' Z+. (Here we use the fact that the quantity inquestion does not depend on the choice of an Ad-invariant bilinear form ong(a).) Since this is true for all positive roots a, the weight A is dominant.

Finally, let A be a dominant weight. We can define the function f.\ onG' by

fA(n_hn) = e\(h).

Considering the restriction of f on the subgroups Gi = expg(a;), one canprove that this function has a continuous extension on all Bruhat cells ofcodimension 1. But an analytic function cannot have singularities on acomplex submanifold of codimension > 1. So, f is regular everywhere. Onecan check that the subspace spanned by left shifts of this function is finitedimensional (in the appropriate coordinates it consists of polynomials ofbounded degrees). Moreover, the corresponding representation is irreducibleand has highest weight A.

The Cartan theorem not only classifies the finite-dimensional irreducibleholomorphic representations of G and the unirreps of K but also providesan explicit construction of them.


Example 6. Let G = SL(n + 1, C), let K = SU(n + 1), and let Hbe the diagonal Cartan subgroup. A general element of H in the canonicalcoordinates introduced in Example 4 has the form

h(t) =

t1 0 0 ... 0 0

0 t2ti 1 0 ... 0 00 0

t3t2... 0 0

... ... ... ...

0 0 0 ... 0

0 0 0 ... 0 tn

I

For A = ( k 1 . . . . . E P = Zn we have ea(h(t)) = ti' tkn Let 'k(9)denote the k-th principal minor of the matrix g E G. Then Ok(h(t)) = tk.It follows that

f a (9) = ' ( g ) .) ... pnn (9),

which is a holomorphic function on G exactly when all ki are non-negative,i.e. when A is dominant.

Consider the space Va spanned by left shifts of fa. It is easy to under-stand that this space consists of polynomials on G that are homogeneousof degree k1 in elements of the first column, homogeneous of degree k2 inminors of the first two columns, etc.

The group G acts on Va by left shifts and the corresponding represen-tation Ira is irreducible because it contains the unique highest weight vectorfa of weight A.

Note that Ira is a subrepresentation of the left regular representation ofG.

The particular case n = 2 is already non-trivial and still very trans-parent. We recommend that beginners (or non-experts) look first at thiscase.

Here we study the next case, n = 3. Let g E SL(3, Q. Introduce thenotation:

xi(9) = gi1, x`(g) = f 3kgj19k2, 1 < i < 3.

These six polynomials on G are not independent and satisfy a unique qua-dratic relation

(30)

The space Vk,1 consists of bihomogeneous polynomials of degree k in xiand degree 1 in x3. If xi and xi are independent, the dimension of Vkj will


be equal to the product (k22) (122). Taking into account (30), we get thegenuine dimension

dim Pk,j _(k + 1)(1 + 2)(k + 1 + 2)

Exercise 5. Show that this result is in agreement with the general Weylformula (13).

Hint. Draw a picture of the root system for SL(3, C), find fundamentalweights wl, w2, and show that positive roots are

a = 2w1 - W2i /3 = -wl + 2w2,

So, if A = kw1 + 1w2, we have

y=p=wl+ W2.

(A+ P, a)k 1

(A+p, 3)1 1

(A+p, y) = k+1+2+ ,

(p, a) =(p, 0)

= ..+(p, y)

2

46

3. Orbits and representations

3.1. Overlook.We keep the notation of the previous sections. In particular, K denotes a

compact simply connected Lie group with a maximal torus T = T" endowedwith canonical coordinates (t1, ..., t").

We have seen that the unirreps (aa, VA) of K are labelled by dominantweights A E P+ C it. So, the set K of unirreps has the form Z', n = rk K.

On the other hand, the set of coadjoint orbits is parametrized by points ofthe positive Weyl chamber C.+ - R. We shall see in a moment that integralorbits form the discrete set O;t(K), which can be identified with the setP+ of dominant weights. The integral orbits of maximal dimension formthe subset OOn9(K) labelled by the set P++ = P+ + p of regular dominantweights.

We want to relate the set K of unirreps to some set of coadjoint orbits.To illustrate the problem, we start with the simplest example K =

SU(2, C). We give a detailed description of all general notions for thisparticular case.

The general element of K has the form k = I u v I , Ju12 + IV12 = 1.\\\ v u/


The Lie algebra t has the canonical basis (X, Y, Z) such that

Al,y, := xX + yY + zZ = 2 (i

xizy )\ y

The dual space t' consists of matrices Fa,b,c (a -i ib -aic 2b) sothat

(Fa,b,e, Ax,y,z) = tr (Fa,b,, A.,y,.) = ax + by + C.Z.

The maximal torus has elements of the form (tt O

where t is the

canonical complex coordinate with I t I = 1.

The lattice exp-1(e) fl t is generated by the element(2lri 0 )

0 27ri

47rZ; the lattice exp-1(C) fl t is generated by 27rZ.The elements of g and g' have the same form Ay,y Z and Fa,b,c as above

but with complex coordinates (x, y, z) and (a, b, c), respectively. In partic-ular, the only simple positive root a, the only fundamental weight w, andthe element p are

a=Fo.o,4, w=P= 1a=F00,5i.

The coadjoint orbits have the form

Qr = { Fa,b,c I a2 + b2 + c2 = r2, r > 0 }

The regular orbits are those for which r > 0.The coadjoint action of t on t' is given by

K.(X)=b8c -cab, K.(Y)=COa-aO, K.(Z)=aab-bOa.

The canonical symplectic form on a coadjoint orbit is given by

adbAdc+bdcAda+cdaAdbo=a2+b2+c2

The symplectic volume of an orbit is vol(S2r) = fn, a. Using the "geo-graphical coordinates" -7r < 0 < 7r, -7r/2 < 9 < 7r/2, we get

a = r cos 9 cos q5, b = r cos 9 sin ¢,

c=rsin9, a=rcos9d4Ad9.


Therefore, vol(St,.) = 4irr. We see that this volume is an integer exactlywhen r E -L Z, i.e. when

Str n t = {±F0,o,ir } c 1 7L w = 1 P.2irz 27ri

Thus, the set Oint of all integral coadjoint orbits for K is naturallylabelled by P+ Z+, while the set O n9 of integral regular orbits is labelledby the set P++ = p + P+ :-- 1 + Z+, which is just the shift of P+.

From Appendix 111.3.2 we know that the unirrep irn of K has n+1 simpleweights nw, (n - 2)w, -nw and that 7r ® 7rm = ®mio(m,n)

Comparing these results with Rules 2, 3, and 5 of the User's Guide (orwith formula (4) of Chapter 3), we see that the unirrep 7rn must be associatedwith the orbit passing through the element 2 . To simplify the formulationswe suggest the following notation:

(N) Sty, C t* denotes the orbit passing through the point zai

Remark 3. This notation needs some comments. According to thePontrjagin duality theorem, there exists a canonical isomorphism betweenany abelian locally compact group A and its second dual A.

But often (e.g., for all finite groups and for additive groups of locallycompact fields) already the first dual A is isomorphic to A. However, in thiscase there is no canonical isomorphism.

In particular, there is no canonical way to label the characters of R byelements of R. Practically, two special ways are used:

1) x,\(x) = e' ' and 2) X ,\(x) = e2ariax

Both ways have their advantages and disadvantages. In this book we usethe second way. It is worthwhile to discuss what happens if we choose thefirst one.

The 1-dimensional representations UF,H, which are used in Rule 2 toconstruct the unirreps, would acquire the simpler form

UFH(exP X) = e'(F'.X).

The notation (N) above would also change to a more natural one:(N') Q,% C r' denotes the orbit passing through the point -iA.

But, on the other hand, the symplectic structure on coadjoint orbitswould be related not to the form BF but to the form The desireto keep the initial definition of the symplectic form a, which is now widelyknown as the canonical or Kirillov-Kostant form, forces us to introducethe rather clumsy definition (N). O


In the case of K = SU(2, C) the notation (N) leads to the correspon-dence 7rn «---i SZ,,,,,,, which is naturally generalized to the correspondenceIra #--+ Q,\ for general compact semisimple groups.

On the other hand, let us try to use Rule 6 to compute the generalizedcharacter of 7rn at the point k = exp Ao,o,z = exp zZ.

The key ingredient is the integral

2a a/2 2 rzX) r rl i2n(F i, rz s ne o = - dO J e a cos OdO = -sin -.I

n 47r o _a/2 z 2

Therefore, taking into account the equality

p(expzZ) =

sinh(ad (zZ/2))-

sin z/2ad (zZ/2) z/2

we get the final formula

But we know that

e2ri(FzZ)Cr = sin rz/2p(zZ) rm sin z/2

X,,,. (exP zZ) _sin (n + 1)z/2

sin z/2

This suggests we assign the unirrep ir,a to the orbit This clearly gen-eralizes to the correspondence Ira F-- Q,\+p for general compact semisimplegroups.

Another argument in favor of this correspondence comes from the consid-eration of infinitesimal characters. We have seen in Section 1.1 that theinfinitesimal character of 7rn is given by I,,(O) = n 82 = n+si 2 - 8 Moregenerally (see Theorem le)), for any compact semisimple group K we haveI.\ (A) = Ia + pi2 - IpI2, an expression which explicitly contains A + p.

For other elements of Z(g) their infinitesimal characters are also betterexpressed in terms of the orbit Sta+P, rather than fa (see Section 3.6).

3.2. Weights of a unirrep.Let (Ira, VA) be the unirrep of K with the highest weight \ E P+. The

restriction ResT Ira is no longer irreducible. Since T is abelian, it splits into1-dimensional unirreps. If we introduce in T the canonical coordinates (10)and identify P with Z' using the basis of fundamental weights, then the 1-dimensional unirreps in question are exactly the functions eA, u E P, definedby (11). For it =(mi,...,mn)EZnwe have

eµ(t) = tm' ... tm' (tM for short).


Denote by Va the isotypic component of type p in VA:

V'\ := {v E Va I ra(t)v = tµv}.

The weights of a given representation (ir, V) form a multiset (i.e. a setwith the additional structure: to every element of the set we associate apositive integer, the multiplicity of this element). We denote this multisetby Wt(7r).

One of the oldest (and most needed for applications) problems in therepresentation theory of compact Lie groups is to compute the multiplicityma(p) of the weight p E Wt(lra), i.e. the dimension of V1.

In terms of the restriction functor we can write:

ResTaa = ma(p)e,.µE P

There are several different formulae for this quantity, but no one of themis efficient enough. The orbit method, unfortunately, is no exception. Theformula suggested by this method is elegant and transparent but (at leastin its present form) not practical.

Example 7. To show the flavor of this problem, we consider the caseG = SU(3). Here the weight lattice is the usual triangular lattice in 1R2generated by two fundamental weights wl and w2. The multiplicity functionma(p), for two specific cases A = 4w1 + 2w2 and A = 3w1 + 3w2, looks like:

1 1 1 1 1

1 2 2 2 2 1

1 2 3 3 3 2 1

1 2 3 3 2 1

1 2 3 2 1

1 2 2 1

1 1 1

1 1 1 1

1 2 2 2 1

1 2 3 3 2 1

1 2 3 4 3 2 1

1 2 3 3 2 1

1 2 2 2 1

1 1 1 1

One can easily guess from these pictures the general principle which isvalid for any unirrep of SU(3).

Namely, let Q c P be the root sublattice. For G = SU(3) it has index 3in P. The support of ma(p), i.e. the set of it E P for which ma(p) # 0, is aconvex hexagon in A+Q, whose vertices form a W-orbit of the highest weightA. The values of ma(p) along the perimeter of the hexagon are equal to 1.On the next layer they are equal to 2 and continue to grow linearly until thehexagonal layer degenerates to an isosceles triangle. Then the growth stops(on the value min(k, l) + 1 for A = kw, + lw2).


We recommend that the reader compare this simple description with thecomputations based on the Weyl or the Kostant formula. 0

For general compact groups the situation is described by the followingtheorem.

Theorem 4. The support of ma has the form

(31) supp ma = n w(A - Q+)WE W

Sketch of the proof. We already mentioned that all weights of 7r,\ arecontained in the set A - Q+. Since supp ma is invariant under the action ofthe Weyl group, we conclude that supp ma c I IWEW w(A - Q+).

The inverse inclusion can be proved by considering the restriction of 7r,\to different 3-dimensional subalgebras g(a).

Remark 4. The convexity of supp ma follows from the remarkablerecent result by Okounkov [Ok] who showed that the function log ma isconcave. G

The ideology of the orbit method relates the restriction functor ResTwith the natural projection p : t` V. The reader can see the relationcomparing Theorem 4 with the following well-known fact.

Theorem 5. The image of fl under the projection p : g' - t` is theconvex hull of the intersection S2a fl V, which consists of points 2nc w(\), w EW.

Sketch of the proof. First, we mention that the statement of the theoremfollows from a more general result about Hamiltonian actions of a torus on asymplectic manifold (see [A] and [GS2]). Here we briefly describe the otherapproach.

We identify t` with t as in Section 2.1 and write the general point X E tin the form

(32) X = X0 + (caXa - Z. X_a),aER+

for X0 E t and c0 C.

Let us study the singular locus of the projection map p : Il -' t : XXo. It is the image by p of those points X E 0 for which the tangent mapp. : TX) -- t is not surjective. Geometrically, it is the set of points X0where the preimage p 1(X0) may differ from the preimage of nearby points.In particular, the boundary of p(Q) is contained in the singular locus.

§3. Orbits and representations

Figure 5

167

The tangent space TX Q consists of elements [Y, X], Y E t. If the imagep. (TX 11) is not the whole space t, then (p. (Tx 9), Z) K = 0 for some non-zero Z E t. This means that for all Y E f we have ([Y, X], Z)K = 0, whichimplies (Y, [X, Z])K = 0, hence [X, Z] = 0. Since

[Z, X] = a(Z)(cQXQ + cQ X_0),aE R+

the coefficients c±o must vanish for all a E R+ with a(Z) 0.

In Section 2.1 we introduced the family of root systems R('"), 1 < m <_M, which are all possible intersections of R with hyperplanes in it. We keepthe notation of that section here.

We say that a singular point p(X) is of type m if X E t(-). Note thatany such point is conjugate by an element of K(m) to a point in t n Sl, i.e.to one of the points of the set S = {2-w(A) I w E W}. So, the set of

1ri

points of type m is a union of projections of finitely many K(')-orbits int. By induction, we can assume that these projections are convex hulls ofW(')-orbits in S where W(1) is the Weyl group of glm).

We see that the set of all singular points forms finitely many convexpolytopes of dimension < n with vertices in S. In particular, the boundaryof p(1,) consists of polytopes of codimension 1 that correspond to rootsystems of rank n - 1 in R.

From this one can deduce the statement of the theorem. D

Example 8. For a group of type B2 C2 the projection of a typicalorbit and the set of singular points is shown in Figure 5. 0


Comparing Theorems 4 and 5, we get

Theorem 6. Let K be a simply connected compact Lie group, and let Tbe its maximal torus. Then Rule 3 of the User's Guide holds for the pairT C K if we associate the orbit Sty to a unirrep Ira and consider in p(Sta)only points which are congruent to A mod Q.

Recall that we use the notation (N) introduced on page 163.

3.3. Functors Ind and Res.Now we consider the decomposition of a tensor product of two unirreps.

This is also an old problem that is important in many applications. Recallthat Rule 5 relates the tensor product of unirreps 7rn, and irn2 with thearithmetic sum III + St2 The most famous and visual case K = SU(2) wasdiscussed above. Here we describe its physical and geometric interpretations.

Example 9. The so-called addition rule for moments or spins inquantum mechanics in mathematical language means that

m+n7rm ®7rn = ® 7rk, k = (m + n) mod 2.

Jm-ni

This is in perfect agreement with the following geometric fact in 1R3: thearithmetic sum of a sphere of radius r1 with a sphere of radius r2 is theunion of spheres of radius r where Irl - r21 < r < rl + r2 0

As was shown recently in [KI] (see also [KT] and [AW]), Rule 5 withthe same amendment holds for all compact Lie groups of type A. It can beformulated as follows.

Proposition 5. Let Sll, St2, St3 be any three integral orbits in su(n), and letirl, 1T2, 1T3 be the corresponding unirreps of SU(n). Then (ll + 112 contains113 iff 7r1 ®7r2 contains 7r3.

All this is strong evidence in favor of the first correspondence: IraSta. However there are some facts that have a better explanation in termsof the second correspondence.

Let us return to the multiplicity problem. It is known that m,\(.) isa piecewise polynomial function, as in Example 7. This follows, e.g., fromthe Kostant formula (15). Moreover, the domains where the multiplicity isgiven by a fixed polynomial are known. They are bounded by hyperplanes

(33) Pw,rn = w(A) + t fl t(-), w E W, rk R(m) = n - 1.

On the other hand, by Theorem 5 the image of Sta under the canonicalw(A), w E W}.projection p : 9' --+ t* is the convex hull of the set 1 1Tri


Moreover, in the proof of this theorem we saw that the set of singular pointsof Sta is the union of polytopes which are exactly the intersections of p(Sta )with the hyperplanes (33).

In other words, the multiplicity formula changes its form exactly wherethe structure of the preimage p-1(X0) changes.

This suggests that the weight multiplicity ma(y) is related to the geom-etry of the preimage p1(iµ) n QA.

This is indeed so, but only if we replace f2a by S2a+p!

Namely, let Xa+p be the quotient of p 1(iµ) f1 1l,+p by the action ofStabK(µ). This is the so-called reduced symplectic manifold (see [AG]and/or Appendix 11.3.4).

It turns out that the most naive conjecture

(34) M,\ (14) = vol Xa+p

is "asymptotically true" (see [Hec]).There is a vast literature surrounding the more sophisticated forms of

this conjecture involving the Todd genus and the Riemann-Roch number ofthe manifold Xa+p. We refer to [GS2], [GLS], and to papers quoted there.

My personal impression is that the right formula for multiplicities isstill to be found. I want to make the following observation here. Let rAdenote the density of the projection p.(vol) of the canonical measure on theorbit S2p+a. Formula (12) for the character together with the Weyl integralformula (17) imply that this density has the form

(35) r,\ (x) = Erna(µ).rp(x -

It

We illustrate it on the simplestExample 10. Let K = SU(2). Then the lattice P can be identified

with2Z

C R. It turns out that r, is simply the characteristic function ofthe segment [-A, A]. (This property of the area of a 2-dimensional spherewas known already to Archimedes.)

In particular, rp = Xj-z2

Equality (35) above takes the form:

n

2 2 2'J xlk--2 rk-22k=0X!-

(See Figure 6.) 0We recommend that the reader consider in detail the next case K =

SU(3). Here the projection of 12p is a regular hexagon in t and the graph of


Figure 6. The projection of the canonical measure on S2.

rp is a right pyramid of volume 1 over this hexagon. Equality (35) in thiscase can be easily interpreted as the rule formulated in Example 7.

3.4.' Borel-Weil-Bott theorem.This is a culminating result in the representation theory of compact Lie

groups. It gives a uniform geometric construction for all unirreps of allcompact connected Lie groups K. At the same time this result requiresmore knowledge of algebraic geometry and homological algebra than otherparts of the book. I do not suggest that the unprepared reader simply skipthis material, but recommend that it be considered as an excursion in aclose but still foreign country.

In fact, the first part of the theory, the so-called Borel-Weil theorem, isjust the modern interpretation of E. Cartan's theory of the highest weight(see Section 2.2).

This theory admits a beautiful generalization due to R. Bott. We wantto show how these results agree with the ideology of the orbit method.

The Borel-Weil theory deals with homogeneous holomorphic line bundlesL on flag manifolds F. i.e. bundles which admit a K-action by holomorphictransformations of the total space and the base. This action automaticallyextends to an action of the complex group G. This allows us to describeeverything in group-theoretic terms.

Namely, let X be a holomorphic character (i.e. complex analytic 1-dimensional representation) of the Borel subgroup B. Let B act on G byleft shifts and denote by Ex the 1-dimensional complex space where B actsvia the character X. Then the fibered product G X B CX is a complexmanifold that has a natural projection on F = G/B with a fiber C overeach point. Hence, we can identify it with the total space of a line bundleL. over F.

A holomorphic section s : F --+ LX is represented by a holomorphicfunction 0, on G satisfying

4s(bg) = X(b) . Os(9).


We denote by rhol(Lx) the (finite-dimensional) space of all holomorphicsections of L. The group G acts on this space by right3 shifts:

09.8(9) = 08(9,9).

The representation thus obtained is called holomorphically induced from(B. X) and is denoted by Ind holB X (cf. Appendix V and Chapter 4).

It is easy to describe the set of homogeneous holomorphic bundles on For, equivalently, the set of all holomorphic characters of B. Namely, thesecharacters are trivial on [B, B] = N and their restrictions on the Cartansubgroup H are exactly the functions ea, A E P, introduced by formula (9)in Section 1.1.

Thus, to any A E P there corresponds a character of the Borel subgroupB, hence a homogeneous holomorphic line bundle on F which we denote byLa.

Theorem 7 (Borel-Weil). The space rhd(LA) is non-zero exactly when A EP+, the set of dominant weights, and in this case G acts on rhol(LA) via anirreducible representation as with highest weight A.

This theorem strongly suggests that the representation ire, should berelated to the integral coadjoint orbit SIB, passing through -L. The point isthat on any orbit fl there is a canonically defined line bundle La associatedwith the symplectic form a on the orbit. The Chern class of L,\ is exactlythe cohomology class defined by a. A more precise statement relates a withthe curvature of a connection in La.

Recall that in the previous section we defined the G-covariant isomor-phism (p : F = G/B -+ 1la. It turns out that L), the inducedbundle.

Example 11. a) The trivial representation po is realized in the spaceof constant functions on F, which is clearly induced by the map po :F{pt} = flo.

So, xo should correspond to the one-point orbit, the origin in t.b) The representation 7rk,,,, of SL(3, C) was realized in Example 6 in the

space of homogeneous polynomials of degree k in (x', x2, x3). According tothe Borel-Weil theorem, this space can be realized as a space of sections of aline bundle Lku,, over F. But from the construction in Example 1 one can seethat Lk,,,, is induced from a canonical line bundle Lk,,,, over SIk,-' P2(C).So, it is natural to associate this representation with S1k",.

Now, consider the arguments in favor of another correspondence.

3ifowever, note that this is a left action: s -. g s.


One of these arguments is the complement by Bott to the Borel-Weiltheorem. It deals with the space Hk(F, La) of k-dimensional cohomologyof F with coefficients in the sheaf LA related to the line bundle L,\.

For k = 0 this cohomology space reduces to the space rh,,I(LA) of holo-morphic sections which appears in the Borel-Weil theorem. The ideology ofhomological algebra suggests that we also consider the higher-dimensionalcohomology each time when the zero-dimensional cohomology appears. Theresult of this consideration is

Theorem 8 (Bott). The space Hk(.F, LA) is non-zero exactly whenA+ p =w(µ + p) for some a E P+, w E W, and k = l(w), the length of w.

In this case the representation of G in Hk(.P, LA) is equivalent to 7rµ.

Example 12. a) The trivial representation 7ro occurs in the spaceH'(F, L_2p) where r = #R+ is the number of positive roots. Indeed,here r is the complex dimension of F and L_2p is the line bundle of the an-ticanonical class since -2p equals the sum of negative roots. Accordingto Serre duality, the r-dimensional cohomology with coefficients in L_2p isisomorphic to the 0-dimensional cohomology with coefficients in the trivialsheaf Lo.

b) The fundamental representation 7r,,,, of SL(3, C) occurs in the follow-ing cohomology spaces:

Ho(.F, & ), H'(F, L-3,,+2,2), H1(F, Z2,,_2112),

H2(F, L-4",2), H2(.F, L-4",+"2), H3(F, L-2W,- )

The first realization is just the standard one (see Examples 1 and 6).Another realization that has a transparent geometric interpretation isH2(F, L_,). Since 4w2 is a singular weight, the line bundle L-4,12 isinduced from a bundle over P2(C). According to Serre duality, the corre-sponding cohomology coincides with H°(1P2(C), L4,, ). 0

The Bott theorem says that the unirrep Ira is related not to one, but tothe whole family of orbits w E W}. Note that these orbits areconsidered to be complex manifolds endowed with a Kahler structure.

On the other hand, any orbit admits not one but several Kahler struc-tures. For regular orbits they are labelled by the elements of the Weyl group(see Remark 2 above).

We see that the correspondence between orbits and representations forcompact groups is much more sophisticated than for solvable groups.


3.5. The integral formula for characters.Another and more visual argument in favor of the correspondence Ira H

9a+p is the equality

(36) vol(1ta+p) = dim Ira,

which is in perfect agreement with the principle of quantization: the dimen-sion of the quantum phase space is equal to the volume of the classical phasespace in Planck units. In short - one dimension per volume unit.4

Equality (36) follows from the integral formula for the characters (themodified Rule 6) which we discuss here in the context of compact groups.

In this case the unirreps are finite dimensional and the characters areregular, even analytic, functions on the group. On the other hand, thefactor per, entering in the modified Rule 6, is defined only in the opensubset e C g where the exponential map is a bijection.

Theorem 9 (see [K1l]). For X E e we have

(37) tr Tra(exp X) = 1e2,,i(F,x)+a

P(X) f2"P

Equality (36) is just a particular case of this theorem when X = 0.

Proof. All known proofs of (37) consist of two parts: 1) show that bothsides are proportional; 2) check that the proportionality factor equals 1.

The first statement follows e.g. from the differential equations for gen-eralized characters (see the next section). This is the way the integral wasfirst computed by Harish-Chandra (Theorem 2 in [H], vol. II, pp. 243-276).

Another possibility uses the properties of the Fourier transform on asemisimple Lie algebra (see [Ver2]).

The second statement basically is equivalent to the comparison of thetwo volume forms on the group K: the one-form dg corresponds to theRiemannian metric on K induced by the Killing form, the other form vol isnormalized by the condition vol(K) = 1.

It also has different proofs. One, used in [Kill], is based on computingthe asymptotics of the integral

f e-t(x,x)dli(expX)

'Here again the two identifications of R with R are related with two choices of the Planckconstants: the usual h and the normalized h = A. . (See Remark 3.)


when t +oo. Another uses equality (36), which can be proved by methodsof algebraic topology assuming the Borel-Weil theorem. 0

Theorem 9, for the simplest case G = SU(2), was already stated in[K1]. But it was observed in [KV] and [DW] only recently that it impliesthe following remarkable property of the convolution algebra on G.

Let C°°(9)' (resp. C°°(G)') denote the space of distributions with com-pact support on a Lie algebra g (resp. on a Lie group G). Define the trans-form - : C°°(g)' C°°(G)' by

(38) (4'(v), f) = (v, p . (f o exp))

Theorem 10 (see [DW]). For Ad(G)-invariant distributions the convolu-tion operations on G and g (the latter being considered as an abelian Liegroup) are related by the transform above:

(39) 4'(µ) *G 4'(v) = 4'(µ *8 v).

0

So, the transform 4? "straightens" the group convolution, turning it intothe abelian convolution on g. This implies in particular the following re-markable geometric fact.

Corollary. For any two (co)adjoint orbits 01i 02 C g the correspondingconjugacy classes CI = exp 01i C2 = exp 02 possess the property

(40) C1 C2 C exp(O1 + 02).

The analytic explanation of this geometric phenomenon can be givenusing the property of Laplace operators which we discuss below (see thenext section).

3.6. Infinitesimal characters.Let Z(t) be the center of the enveloping algebra U(t). An element

A E Z(t) can be interpreted as a distribution on K supported at {e} or asa differential operator DA on K acting by the formula

(41) DAf=A*f=f*A.

Therefore, for any unirrep 7r of K we have the equality

(42) lr(DAI) = 7r(A * f) = lr(A)7r(f) = I,(A)x(f )


It follows that all matrix elements of it and its character are eigenfunctionsfor DA with the eigenvalue I,r(A).

The integral formula (37) and equation (42) for the characters togetherimply the modified Rule 7. Conversely, the modified Rule 7 can be used toderive the integral formula (37).

We use these facts to get an explicit formula for the isomorphism sym :Y(g) -' Z(g). For illustration we give

Example 13. Let G = SU(2) and g = su(2) with the standard basisX, Y, Z obeying the standard commutation relations

[X, Y] = Z, [Y, Z] = X, [Z, X] = Y.

We denote by small letters x, y, z the same elements X, Y, Z consideredas coordinates on g', and by a, 3, y the dual coordinates on g. Also put

r= x2+y2+z2, p= a2+Q2+y2.

It is clear that in our case Y(g) = C [r2J and Z(g) = C [C] where

C := X 2 + Y2 + Z2 = sym(r2) E Z(g).

Unfortunately, the map sym : C[x, y, z] - U(g) is not easily computableeven when restricted to Y(g) = C [r2]. E.g. one can check that sym (r4) =C2 + 3C, but the direct computation of sym (r6) = C3 + C2 +

3Cis al-

ready rather complicated.5 So we choose the roundabout way based on themodified Rule 7.

It is instructive to compare this problem with the computation of thevarious symbols of differential operators (cf. [Ki2], §18, no. 2).

The function p on g in our case takes the form

(sinh(ad(ctX + QY + yZ)/2)

(43)p(, Q, y) _ det

ad(aX +#Y + yZ)/2 ) )sin(p/2) _ p2p/2 24 + .. .

Recall that we identify polynomials on g with differential operators on g'with constant coefficients, so that p2 goes to the operator = a2 +a" +,9;2

and the function j(p) = ei p 22 corresponds to some differential operator Jof infinite order.

5In the lectures by Roger Godement on Lie groups this computation is accompanied by theremark: "Resultat qui n'incite pas a pousser plus loin lea investigations, encore que lee physiciensprecedent tout lea jours depuis le debut des a des calculs de ce genre."


Since the restriction of the operator A to Y(g) is given by the simpleexpression 0 = r-1 odor, we get

a

(JF)(r) = r-1j (f-) (rF(r)) F(r) (r 24r)) + ...

and, in particular.

(44) Jr2 = r2 - 1 J sink ar = sin(a/2) sink ar4' ar a/2 ar

The modified Rule 7 implies that the map sym o J restricted on Y(g) isan algebra homomorphism, hence for any power series f we have

(45) sym((Jf)(r2)) = f (sym(Jr2)) = f (C - 1)4

From (44) and (45) we obtain

(46)sinh ar a/2 sin (a C

syrn ( ) _ .ar sin(a/2) a i - C4

This gives the following explicit expression for the map sym : Y(g) -. Z(g):

(47)

nsyrn(r2n)=

( 4) (2r2k 1)B2k(4k - 2)(1 - 4C)n-k

k=0

where B2k are the Bernoulli numbers. This explicit formula shows, in par-ticular, that there is no simple expression for the map sym in general. Q

4. Intertwining operatorsAnother question important in applications is the structure of intertwin-ing, or G-covariant, operators between two representation spaces. Manyremarkable differential and integral operators can be interpreted (or evendefined) as intertwining operators. For example, the Laplace operator

is all intertwining operator for the natural action of the Euclidean motiongroup in any function space on IR". Other examples are the Fourier andRadon transforms.

§4. Intertwining operators 177

The big deficiency of the orbit method is that, up to this point, it hashelped very little in the study of intertwining operators. (Although there isa beautiful formula for the so-called intertwining number of two repre-sentations in terms of symplectic geometry - see [GS1], [GS2], [GLS].)

The reason is that the natural correspondence exists between orbits andequivalence classes of unirreps rather than unirreps themselves. The con-struction of an individual representation it from a given equivalence classdefined by an orbit Il needs some arbitrary choice (e.g., the choice of arepresentative F E St and a subalgebra fj subordinate to F).

Recently a new approach to the representation-theoretic study of specialfunctions was suggested in [EK] and further developed in [EFK]. It requiresa detailed description of intertwining operators for certain geometric repre-sentations of compact groups. I consider the application of the orbit methodto these questions to be a very challenging problem.

Chapter 6

Miscellaneous

In this chapter we collect information about how the orbit method works insituations different from solvable and compact Lie groups.

There is no general theory here, but we provide some worked-out exam-ples that show how the orbit method can suggest the right answer and givea visual and adequate description of the situation.

1. Semisimple groupsIn a sense, this class of Lie groups is opposite to the case of nilpotent groups,hence, the application of the orbit method is a complicated problem.'

1.1. Complex semisimple groups.This is the most well-understood class of seinisimple groups. The status

of the representation theory for these groups can be found in surveys [Bar],[BV], and [Du2].

The application of the orbit method for these groups was suggested inmy lectures at Institut Henri Poincare (Paris, 1968). I observed that the so-called principal series of unitary representations is in perfect correspondencewith closed coadjoint orbits of maximal dimension. Indeed, it is known thatan adjoint orbit SZ C g is closed if the stabilizer of a point X E 1 is reductive(then the element X itself is ad-semisimple; see, e.g., [Borl).

In the notes of these lectures written by M. Duflo he included his ownimportant result: the integral formula for the generalized characters (i.e.

(Note, however, that for a quantum deformation of a semisimple group the dual quantumgroup is solvable. So, there is another argument in favor of the orbit method (see [8ol for details).

179

180 6. Miscellaneous

Rule 6 of the User's Guide) is true for all representations of the principalseries (see [Dull).

One can associate the so-called degenerate series with degenerate or-bits (i.e. closed orbits of lower dimensions). Indeed, these representationsare induced from 1-dimensional representations of parabolic subgroups of Gin full accordance with Rule 2.

As for non-closed orbits, the most interesting of them are the so-callednilpotent orbits consisting of ad-nilpotent elements. There is a rich anddifficult theory of representations that could be associated with nilpotentorbits; see [Vo2].

Finally, there are the so-called complementary series of unirreps thatare usually constructed using the analytic continuation of principal or de-generate series. It seems that they can be associated with coadjoint orbitsin the complexification g of g'. We say more about this in Section 6.4.

1.2. Real semisimple groups.This class of groups is the hardest for representation theorists. It is

enough to say that the unitary duals of general real semisimple groups re-main unknown despite more than half a century of effort. (The first result,the Bargmann description of the unitary dual for SL(2, R), appeared in1947.)

The deepest results here are due to Harish-Chandra and Langlands, butthe theory is still far from complete. For surveys see [Vol], [Wa], and [Zh2].

We observe that some basic facts from the representation theory of realsemisimple groups can be easily described (but not proved) in terms ofcoadjoint orbits. Consider, for example, the famous Harish-Chandra theo-rem about discrete series. The theorem claims that a real semisimple groupG possesses a discrete series of representations if it has a compact Cartansubgroup.

This can be explained by the geometry of orbits as follows. A closedorbit of maximal dimension has the form S2 = G/H, where H is a Cartansubgroup in G. The number of integrality conditions for S2 is equal tob2(11) = b1(H). This number coincides with the dimension of the orbitspace (which is rk G = dim H) only when H is compact.

2. Lie groups of general typeIt is known that a general Lie group G can be written in the form of asemidirect product G = S a R where S is a maximal semisimple subgroupand R is a maximal solvable normal subgroup. We describe below twoexamples of this type of group.


2.1. Poincare group.This group is well known because it is the symmetry group of the Min-

kowski space, the main object of special relativity theory.Recall that Minkowski space M = R1,3 is a pseudo-Euclidean space with

the quadratic form

Q(x) = (x0)2 - (x1)2 - (x2)2 - (x3)2 or c2t2 - x2 - y2 - z2.

The Poincare group is the group of rigid motions of M, that is, the semidirectproduct P = 0(1, 3, R) X R1"3. We denote by PO the subgroup SOo(1, 3, R)x R1.3, which consists of transformations preserving the space and timeorientations. This group is connected but not simply connected.

The simply connected covering group PO has a convenient matrix real-ization by complex block matrices of the form

P=PoP1, P0=rt g01 9 pl = 10 1

where

g=Ca

C,

0 + 3E SL(2, C), h = h' _

xl2d

XI ix2x0+ x3) E Mat2(C).

The Lie algebra p consists of complex block matrices

a*)' a E sI(2, C), h = h' E Mat2(C).

The space of 2 x 2 Hermitian matrices is identified with M via the coordinatesx0, xl, x2, x3 above. The action of the group PO on M comes from theordinary matrix multiplication and has the form

P'x=PoP1 x=g-1(x+h)(g')-1

We see that po acts as a translation and p1 as a pseudo-rotation in Minkowskispace because the quadratic form det x = (xo)2-(x')2-(x2)2-(x3)2 = Q(x)

is preserved.

Exercise 1. Show that the unitary matrices from SU(2) C SL(2, C)act as rotations in M and Hermitian matrices act by Lorentz transfor-mations. 4

The space p', dual top, can be identified with the space of block matricesof the form

F = 66

with (F, X) = tr (FX) = 2Retr (ba*) + tr (ch).


We write the coadjoint action separately for rotations and for translations:

K(po)(b, c) = (g-'bg, g*cg), K(pi)(b, c) = ((b - hc)o, c)

where (x)o denotes the traceless part of x.Assume that Q(h) = det h > 0. Then, using an appropriate rotation,

we can reduce the Hermitian matrix c to the scalar form c = m 1 with somem>0.

Thereupon, we can apply a translation so that c remains the same and weremove from b its Hermitian part. As a result, b becomes an anti-Hermitianmatrix with zero trace.

Finally, using a unitary rotation, we preserve c and reduce b to the diag-

onal form b = I 0 0s I. So, the orbits with Q(c) > 0 are parametrized

by two numbers` m > 0 and s. A more detailed consideration shows thatthe orbit in question is subjected to one integrality condition: the numbers must belong to Z. In physical applications the parameters (m, s) areinterpreted as the rest-mass (or energy) and the spin of a particle.

2.2. Odd symplectic groups.The groups Sp(2n + 1, R), the so-called odd symplectic groups,

were introduced independently by many authors (see [Pr]). We defineG = Sp(2n + 1, R) as a stabilizer of the first basic vector in the standardrealization of Sp(2n + 2, R) in R2s+2. This group is a semidirect product ofSp(2n, R) and the generalized Heisenberg group If. The unirreps of thisgroup were recently studied in [BS].

The Lie algebra g consists of matrices of order 2n + 2 that have the blockform

0 at bt c

0 A B b

0 C -At -a0 0 0 0

a,bER', cER, A, B, CEMatn(R), B=Bt, C=Ct.It also has a simple realization in terms of canonical commutation rela-tions. Let p',... , p,,, q,. .. , q,, be the canonical operators satisfying CCR(see Chapter 2):

h[p,pj]_[gk,gjI =0, [Pk,4j]=2- ak?

Consider the space P2 of all polynomials in pi, qj of degree < 2. It is easy tocheck that P2 is a Lie algebra with respect to the commutator [a, b] = ab-ba.I claim that this Lie algebra is isomorphic to g.


Indeed, the correspondence

0 at bt c 0 at bt c 1

0 A B b ih t tl O A B b P

()0 C -At -a 2h(- - q p ) 0 C A t- -a)

q

0 0 0 0 0 0 0 0 h

= i (apt + bqt) + ich + 2ih (gtAp + ptAtq + qtBq + ptCp)

establishes the desired isomorphism of g onto P2.

In particular for n = 1 we get the following representation of sl(2, R) x h:

2

H H2

X '-' ip, Y H iq, Z -* ih, E -+ tq i(Pq + qP) F, LP2h 2h 2h

Note that the central element c of hn (corresponding to the monomial1) also belongs to the center of the whole Lie algebra g.

The dual space g' is realized by matrices of the block form

1* * *fx R P *F_ y Q -Rt

Z yt -xtx, y E Rn, Z E R, P, Q, R E Mat, (R), P = Pt, Q = Qt,

where the asterisks remind us that F is not a matrix but an element of thefactor-space Mater+2(R)/gl.

The most interesting representations of G correspond to orbits 1h thatcontain an element F with all entries zero except z = h 0 0. A directcomputation shows that S2h consists of matrices of the form

F= x -xh-1yt -xh-lxt *

y yh-'yt yh-lxt

h yt -xt *

It is clear that the projection of Sth to hn is a single Hn-orbit correspondingto the irreducible representation 7rh of Hr (see Chapter 2).

On the other hand, the projection of Slh to sp(2n, R)' is also a singleorbit of dimension 2n that consists of matrices of rank 1. Actually, thisprojection depends only on the sign of h, so, we denote it by Stf. The orbitSt+ consists of all matrices F for which FJ2n is a non-negative symmetricmatrix of rank 1 and ft_ = -l+.


Both orbits S1± for n > 1 admit no Sp(2n, R)-invariant polarization,neither real, nor complex. Therefore, Rule 2 of the User's Guide does notwork in this situation.

But according to Rule 3 of the User's Guide, we can try to constructthe representations corresponding to Sgt by extending the representation7rt1 of Hn to Sp(2n + 1, IR) and then restrict it to Sp(2n, ]R). This canbe done by expressing the elements of g in terms of canonical operatorspi, , pn; q1, , qn. Note that this method gives us the same expressionfor both orbits f ±.

The resulting representation 7r of sp(2n, R) is called the metaplecticrepresentation. It gives rise to a double-valued representation of Sp(2n, R)which becomes single-valued on the two-sheeted covering group Mp(2n, R),the so-called metaplectic group.

We consider in more detail the simplest case n = 1, h = 1. For thecanonical basis of sl(2, R) we get the following expressions:

ix2 _ p2 _ 1 d2 i(pq + 9p) d 1(E)2i 2 '

7r (F)2i 2i dx2'

x(H) = 2 = x1 +2.

Consider the element X = E - F that generates a 1-parametric compact sub-group exp tX in G with period 27r. We have 7r(X) = I (x2- ). As we have

-dx

seen in Chapter 2, this operator has a point spectrum consisting of eigenval-ues

2, 2t , 2` ..... In particular, 7r (exp (27rX)) = e2i'"(X = -1. Therefore,

the representation in question is correctly defined only on Mp(2n, R).

3. Beyond Lie groupsIn this section we only briefly describe the application of the orbit methodbeyond Lie groups. We refer to original papers for more details.

3.1. Infinite-dimensional groups.Some infinite-dimensional groups can be viewed as infinite-dimensional

smooth manifolds. So, they have a tangent space and adjoint and coadjointrepresentations. The correspondence between coadjoint orbits and unirrepsis more complicated than in the finite-dimensional case, but it still exists.

Example 1. Consider the infinite unitary group. While for a finite nthere is only one unitary group U(n) := U(n, C), there are several differ-ent candidates for the role of U(oo). Let H be a Hilbert space of Hilbertdimension No with an orthonormal basis {xk}1<k<..

The maximal unitary group U(oo) consists of all unitary operators inH. The minimal one is the group Uo(oo) := Un>1 U(n). It is the subgroupof U(oo) consisting of operators that fix all but finitely many basic vectors.


There are many intermediate subgroups between these two. In particu-lar, we have a chain of groups

Uo(JC) C Uf(oc) c . C U(P)(oc) C . C UU(oc) C U(oo)

where all groups are defined by conditions on the operator a = u - 1.Namely:

for U E Uf(oo) : a has a finite rank;

for u E U(P)(oc), 1 < p < oo : a belongs to the Shatten Ideal LP(H)given by the condition tr (a*a)P/2 < oo;

for u E UU(oo) : a is a compact operator.The corresponding tangent space at the unit can be defined for all these

groups using Stone's theorem about 1-parametric groups of unitary opera-tors. Namely:

uO is the space of skew self-adjoint operators annihilating all but finitelymany basic vectors;

uf(oo) is the space of skew self-adjoint operators of finite rank;

u(P)(oo) for 1 < p < cc is the space of skew self-adjoint operators fromLP(H);

u,(oc) is the space of skew self-adjoint compact operators;u(oc) is the set of skew self-adjoint (not necessarily bounded) operators.Note that the set u(oo) is not a Lie algebra and is not even a vector

space. All other tangent spaces have a natural Lie algebra structure definedby the formula [a, b] = ab - ba.

The dual space to u(P)(oo) for p > 1 is identified with u()(oo), q = Pwhile utl)(oo)* is the space of all bounded skew self-adjoint operators. Theelements of uo can be realized as infinite skew self-adjoint matrices with norestriction on matrix entries.

From the subgroups listed above only one is closed in the norm topology:the group Uc(oo).

Let us think about what kind of prediction can be made about the unir-reps of Uc(oo) using the ideology of the orbit method. According to Rule 1of the User's Guide, the unirreps of G are related to integral coadjoint orbitsin g (possibly with additional data if the orbit is not simply connected).

The classification of integral coadjoint orbits for U,. (oo) can be easilyachieved. The point is that every linear functional on the space of skewself-adjoint compact operators, which is continuous in the norm topology,


has the form

f (a) = tr (ab) where b is a skew self-adjoint operator of trace class.

The orbit 11b passing through the skew self-adjoint operator b E L' (H)is integral if the spectrum of 'b is contained in iZ. For an operator of traceclass this is possible only if b has finite rank. Then it can be reduced bythe coadjoint action of UJoo) to the diagonal form with non-zero entriesipj, ivk, satisfying

(1) vn <... <v1 <0<Al <... <ILm,

So, the orbit method predicts that unirreps of UU(oo) are labelled by se-quences of the form (1). And this is indeed so!

To show this, we recall the definition of the Schur functor S(a) inthe category of vector spaces (see, e.g., (FH], §6.1). Let A be an N-tuple(Al > A2 > ... > AN > 0) E ZN where N can be arbitrary. Denoteby Sk(V) the k-th symmetric power of V with an additional agreement:S°(V)=C, Sk(V)={0} fork<0.

Now we define the functor S(a) by the formula

(2) S(")(V) = det IISA'+.i-i(V)IIl<i,j<N.

Instead of the formal definition, we explain how to understand the right-hand side of (2) with examples:

,S1'1(V) = det CSI(V)

Sl(V)= V ®V - S2(V) = A2(V);

S2"1"1(V) = detS2(V)S°(V)

S3(V) S4(V)(V) S2(V)Sl

S-1(V) /S°(V) SI(V)

= S2(V) ® V ® V + S4(V) - S2(V) ®S2(V) - S3(V) ® V

= S2(V) ® A2(V) - (S3(V) (g V - S4 (V)).

In these expressions the difference Wl - W2 is understood as a GL(V)-module W3 such that W1 W2®W3. It is a good (but not trivial) exercise tocheck that the GL(V)-module S(') (V) is simple and that the correspondingirreducible representation of GL(V) is precisely the representation Ira withthe highest weight A.

The unirrep 7r,,;,, of UU(oo) corresponding to the label (1) acts naturallyin the space SM(H) 0 S"(H) where H is the dual space to H.


Notice that for the finite-dimensional group U(n) this representation isgenerally reducible, while for UU(oo) it is always irreducible. To understandthis phenomenon, look at 1x1;1. The representation space is V ® V. In thefinite-dimensional case this is identified with End(V) and splits into a scalarpart and a traceless part. In the infinite-dimensional case it is identifiedwith the space of Hilbert-Schmidt operators in H and does not contain ascalar part. p

Example 2. Kac-Moody algebras and Kac-Peterson groups.A very interesting example of infinite-dimensional algebras was discov-

ered by V. Kac and R. Moody. They form a natural generalization ofsemisimple Lie algebras related to extended Dynkin diagrams. The corre-sponding class of infinite-dimensional Lie groups was introduced by V. Kacand D. H. Peterson. We briefly describe this example referring to [Kal],[KP], and [PS] for the details.

Let L(K) be the loop group of all smooth maps from the circle S'to a compact simply connected Lie group K with pointwise multiplication.It is a connected and simply connected topological group, since i1(K) =1r2(K) = {0}. Moreover, it is an infinite-dimensional Lie group with Liealgebra L(4) = C°°(S', t).

Let {X1, ..., X,,} be an orthonormal basis in t with respect to thenegative Killing form, with the structure constants ck... Then the functions

21rt"fm,k (t) = e .Xk, 1 <k<n,mEZ ,

form a complete orthonormal system with respect to the Ad-invariant bilin-ear form in L(f) given by

1

(f (. ), g(- )) _ - f (f (t), 9(t))Kdt.

This Lie algebra has a non-trivial central extension given by the cocycle

C(fm,k, fm'.k') = m5m,-m'bk.k' Or c(f, J 1 dt.

The description of coadjoint orbits in this case is equivalent to the Flo-quet theory for an ordinary differential equation with periodic coefficients.The representations of Kac-Moody algebras and Kac-Peterson groups havebeen intensively studied, especially by mathematical physicists. We refer to[EFK], [FF2], [GO], and the vast literature on the web.

It turns out that there is a good correspondence between orbits andunirreps. The most impressive result was obtained in [Fr] where the integral


formula for the characters was obtained, which is the direct analogue of Rule7 of the User's Guide.

Example 3. Virasoro-Bott group Vir.This is an interesting example that is related to many physical applica-

tions.

Let G = Diff+(SI) be the group of orientation preserving diffeomor-phisms of the circle. The simply connected covering G of this group canbe realized by orientation preserving diffeomorphisms of the real line, whichcommute with integral translations. Such a diffeomorphism has the form

x -. O(x), J'(x) > 0, c(x + n) = 4(x) + n, n E Z,

where the function 0 is determined modulo the integer summand. The Liealgebra of this infinite-dimensional Lie group is just the space Vect°O(SI) ofsmooth vector fields on the circle, which can be identified with C°O (SI) byv(x) d H v(x).

The group Vir is a non-trivial central extension of the group G. On thelevel of Lie algebras this extension is given by a cocycle discovered around1970 independently by I.M. Gelfand - D.B. Fuchs and M. Virasoro:2

(v, w) = v'(x)dw'(x).fc

On the group level the cocycle was first computed by R. Bott and is

Ic(O, +G) = J log(0 o 0)'d log(V')0

There is a vast literature devoted to the coadjoint orbits of this group(see, e.g., [Ki12], [Wi2]) and its representations, starting with the famousarticle [BPZ], where the representations of the Lie algebra vir were relatedto conformal field theory and to some integrable problems in statistical me-chanics. The mathematical approach can be found in [FF2], [GO], [KY],[Ner].

Unfortunately, this subject is too big to be described here.

3.2. p-adic and adelic groups.Another possibility to extend the orbit method is connected with the

existence of fields different from lit and G. The experience of commutativeharmonic analysis shows that the most promising is the case of locally com-pact non-discrete topological fields. Such fields are completely classified.

21n pure algebraic form this cocycle was discovered earlier by R. Block.


Those that have characteristic zero (i.e. contain the simple subfield Q) arejust algebraic extensions of the p-adic fields discussed in Appendix I.1.2.They are called p-adic fields and play an important role in modern numbertheory.

Still more interesting is the ring of adeles A(K) invented by A. Weil forthe study of number fields K (finite extensions of Q). It appears naturallyif you want to describe the Pontrjagin dual to the additive group of K withdiscrete topology. For K = Q the ring of adeles A(Q), or simply A, can bedefined as a subset of the direct product R x jjp prime Qp that consists ofsequences {ap}, p prime or oo, where ap E Qp, a,,. E IR, and ap E Zp for allbut a finite number of primes.

The field Q is "diagonally" embedded in A in the form of "constant"sequences {r, r, r, . . . } where the first r is considered as a real rationalnumber, the second as a 2-adic number, etc.

Proposition 1. The exact sequence 0 -p Q -- A -+ A/Q -p 0 isPontrjagin self-dual:

Q ^-' A/Q, A ^-' A, A/Q Q.0

For any algebraic group G defined over a number field K one can definethe groups G(A(K)) and G(K) so that the latter group is a discrete sub-group in the former one. Moreover, it is known that on the homogeneousspace X(G, K) = G(A(K))/G(K) there is a canonically defined invariantmeasure z. So, we get a unitary representation it of G(A(K)) in the spaceL2(X(G, K), T). The decomposition of it into unirreps is a very importantand difficult problem in modern number theory.

The orbit method suggests that the spectrum of it is related to the"rational" orbits in g*(A(K)), i.e. those that contain a point from 9* (K).I recommend that the interested reader consider in detail the simplest caseG = H, the Heisenberg group. I refer to [Mo] and [Ho] for some results inthis direction.

A much more difficult and intriguing case of semisimple algebraic groupsover A is related to the so-called Langlands program. But here we enter therealm of arithmetic geometry and I must stop and refer to experts in thisvery deep theory.

3.3. Finite groups.Some finite groups can be viewed as algebraic groups over a finite field

Fq with q elements. Therefore, one can try to apply the orbit method tothese groups and relate the unirreps of G to the coadjoint orbits in g'.


One of the first candidates for this approach is the group Nn of uppertriangular matrices with units on the main diagonal. We consider Nn as analgebraic subgroup in GL(n) given by the equations

(3) 9i,-1 for i = j,

I 0 fori > j.

For any field K we denote by Nn(K) the group of K-points of Nn, i.e.the group of upper triangular matrices with elements from K satisfying (3).The Lie algebra nn(K) is defined as a tangent space to the algebraic manifoldNn(K) and consists of strictly upper triangular matrices with elements fromK. We can also define the dual space n;, and the coadjoint action of Nn(K)on it.

Unfortunately, the classification problem for Nn(K)-orbits in the spacen` is still open despite many attempts to solve it.

It was understood long ago that this problem practically does not dependon the field K. For n < 6, where the full classification is known, the set Onof coadjoint orbits of Nn in nn is a union of simple "cells" Sti of differentdimensions di, which are quasiaffine manifolds of type Kn\Km ^-' Am(K) xGL(1, K)n-m. It seems that a similar, or slightly more complicated, pictureexists in higher dimensions too.

In particular, it is natural to conjecture that for a finite field Fq of qelements the number of orbits is a polynomial in q.3

Moreover, if this conjecture is true and if we denote by On (Fq) the setof all 2m-dimensional orbits4 in nn(Fq), then the sum

Pn(q,t)= tm.#On(Fq)m>0

must be a polynomial in q and t. This polynomial encodes all essentialinformation about the set of orbits.

For small n we have:

Po=1,P1=1,P2 = q,

P3 = (q - 1)t + q2,

P4 = q(q - 1)t2 + (q - 1)q(q + 1)t + q3,

P5 = (q - 1)2t4 + (q - 1)q(2q - 1)t3

+ (q2 - 1)q(2q - 1)t2 + (q - 1)g2(2q + 1)t + q4.

3Recently this problem was solved by I. M. Isaacs (see [I)).4Recall that all coadjoint orbits are symplectic manifolds, hence have an even dimension.


It was shown in [Ki10] that the beginning terms of the sequence {Pf,}written above (and even more detailed information about the orbits) can beobtained by a variant of the so-called Euler-Bernoulli triangle (cf. [Ar1]).But in the seventh row of the triangle a discrepancy arises. So, the generalformula for P is still unknown.

Note that polynomials Y, (q) := P,,(q, q) have a still nicer behavior.To describe it we give here the definition of four remarkable sequences ofpolynomials that were introduced in [KM]:

{An}, {B.}, {Cn}, {Dn}, n > 0, in Z[q].

1. A (q) is by definition the number of solutions to the equation X2 = 0in the space of n x n upper-triangular matrices with elements from Fq. Moreprecisely, let A'(q) be the number of solutions that have rank r. Thesequantities satisfy the simple recurrence relations

Arn+1(q) = q''+1 , An+1(g) + (qn-r _ qr) ' An(q) , An+1(q) = 1, n > 0.

We see, in particular, that An (q) are polynomials in q. Hence, An(q) _Er>o A' (q) are polynomials as well.

2. Consider the triangle

b3o

b04

b02

t-b13

blo

4-b21

4-

boo

4-b11

4----

b22

bo14-b124-

b2o

b31

b03

---+ bao

formed by polynomials bk,1(q), k > 0, 1 > 0, which are labelled in a "shuttleway" and satisfy the recurrence relations

bk,, =q-1bk-1, 1+1 + (q1+1 - 91)b1, k-1 for k > 0;

boj = q'b1-1,o for 1 > 0; bo.o = 1.

Now put B,,(q) := bi_1,o(q), n > 0, Bo(q) = 1. This is our second se-quence.

3. Define the generalized Catalan numbers ck,1 for k > 1, Ilk < k, kmod 2, by

Ck,k-2a = (kS 1) - (k -

11).

It is convenient to set ck,1 = 0 for III > k. The triangle formed by thesenumbers satisfies the same recurrence relation as the Pascal triangle: ck.1 =


ck_l,l + ck,f_1, but with a different initial condition: c1,_1 = -1, cl,1 = 1instead of ca,a = 1.

The numbers c := c2n+1,1, n > 0, form the sequence of ordinary Cata-lan numbers:' 1, 1, 2, 5, 14, 42, 132,....

It is pertinent to remark that for a positive l the generalized Cata-lan number ck,l equals the dimension of the irreducible representation ofthe symmetric group Sk_1 corresponding to the partition (2

kzl ,11-1). In

particular, the ordinary Catalan number cl corresponds to the rectangulardiagram (2').

The third sequence is defined by the formula

.2+.-.2C. (q) _ Cn+l,s . q a 12

s

where the sum is taken over all integers s E [-n - 1, n + 11 that satisfy

s - n + 1 (mod 2), s - (-1)' (mod 3).

As was observed in [KM], the first 26 polynomials Cn(q) coincide with An(q)and Bn(q). Later it was shown by Doron Zeilberger, using the techniquefrom [PWZJ, that An = Cn for all n > 0.

4. The fourth sequence is defined by

D. (q) := (Nn(F9)(-1) = E d(A).AENn(F9)

Here d(A) is the dimension of a unirrep of class A and (o (s) = FIA05 d(A) -'is the so-called group zeta-function of the group G (see, e.g., [Ki9]). In otherwords, Dn(q) is the sum of the dimensions of all irreducible representationsof the finite group Nn(]Fq).

Now we can formulate the remarkable property of the sequence {Yn}.

Proposition 2 (see [KMJ). For 0 < n < 11 we have

A. (q) = B. (q) ° C.(q) = D. (q) -° Y.(q)

'The Catalan numbers are usually defined by the recurrence cn+1 = Ek=p ck cn_k and theinitial conditions co = cl = 1.

§3. Beyond Lie groups

Here are the first dozen of these polynomials:

A11=1,

Al = 1,A2 = q,

A3=2q2-q,A4 = 2q4 - q2,

A5 = 5q6 - 4q5,

As = 5q9 - 5q7 + q5,A7 = 14g12 - 14q" + q7,

A8 = 14g16 - 20g14 +7g12

A9 = 42g2o - 48g19 + 8g15 - q12,

A10 = 42q25 - 75q23 + 35q21 - q15,

All = 132g3o - 165q29 + 44q25 - 10g22

193

To finish the section, we list sortie problems here that look very intriguingand are still unsolved.

1. Give the description of coadjoint orbits of the group N over anarbitrary field.

2. Describe the asymptotics of the number of coadjoint orbits for thegroup N,,(Fq) when n goes to infinity. In particular, compute

log log # onTOlim supn-oo log n

3. Prove (or disprove) that the five sequences An(q), B.(q), CC(q),Dn (q), Yn (q) coincide for all n > 0.

4. The most intriguing problem is to interpret the formula for # On(F9)given in [Kil0):

#On (Fq) = N1 lF) 8((F', [X, Y]))#(q FEn'(Fq)

X,YEn(Fq)

as the partition function of some quantum-mechanical system. Note that inthe natural generalization of this formula to algebraic extensions of 1Fq thedegree of the extension plays the role of the standard parameter f3 (inversetemperature) in statistical mechanics.


3.4. Supergroups.Up to now we have not mentioned one remarkable extension of the no-

tions of smooth manifold and Lie group. This extension is a part of thegeneral idea of supersymmetry advanced by physicists to explain the anal-ogy between bosonic and fermionic particles.

In mathematics supersymmetry requires equal rights for plus and minus,for even and odd, and for symmetric and antisymmetric. In short, the ideol-ogy of supersymmetry means the following. To any "ordinary" (or "even")notion, definition, theorem, etc. there corresponds an "odd" analogue whichtogether with the initial notion form a "superobject".

The formalism of supersymmetry requires a new sort of numbers thatdiffer from the ordinary ones by the property of a product. It is not commu-tative but anticommutative: xy = -yx; in particular, x2 = 0. This new sortof numbers must be used as widely as the ordinary numbers. For example,they can play the role of local coordinates on a manifold. Thus, the notionof a supermanifold arises.

We refer to [Le], [Mn1], [Mn2], [QFS], [Will for more details.Here we only want to say that for the so-called supergroups, which are

group-like objects in the category of supermanifolds, all ingredients of theorbit method make sense. Moreover, for real nilpotent supergroups thereis a perfect correspondence between unirreps and even coadjoint orbits (see[Ka2]).

There are many papers in the physical literature where unirreps of "clas-sical" supergroups SL(mln, R) and OSp(kl2n, 1l) are considered. It wouldbe interesting to relate these unirreps to coadjoint orbits.

4. Why the orbit method worksSome people consider the orbit method a miracle that cannot be explainedby natural arguments. Nevertheless, there exist two independent "explana-tions" of this phenomenon. We discuss these arguments below.

Note, however, that both these explanations fail in the case of finite orp-adic groups, so at least a part of the miracle still remains.

4.1. Mathematical argument.The idea behind this argument goes back to the following simple but

important observation. Let G be a matrix Lie group, which can be a rathercomplicated curved submanifold G C The logarithm map(4) GE) g' logg :_ (_l)k-1(g _ 1)k

k>1 k

§4. Why the orbit method works 195

transforms G into a linear subspace g in Matn(]R) (see Appendix III.1.1).The map log unlike its inverse, the exponential map exp, is well defined

only in some neighborhood of the unit element. But for the moment weleave aside this relatively small inconvenience.

The more important fact is that the group law is not linearized by thelogarithm map. In exponential coordinates it is given by the Campbell-Hausdorff formula:6

(5) log(exp X exp Y)

=X+Y+2[X,Y]+ 12([X,[X,Y]]+[Y,[Y,X]])+....

Of course, it could not be otherwise bearing in mind that the group lawis non-commutative. But Dedekind had already pointed out the remedymore than a century ago. He observed that for non-commutative groupsthe elements 9192 and 9291 can be different but always belong to the sameconjugacy class. We can write it as a principle:

On the level of conjugacy classes, the group law is always commutative.This is a crucial fact in the representation theory of finite groups and is

also a foundation of the Cartan-Gelfand-Godement-Harish-Chandra theoryof spherical functions.

A new aspect of this phenomenon was observed rather recently in [DW](see Chapter 5, Section 3.5). It turns out that using the logarithm map onecan intertwine two kinds of convolution operations: the group convolutionof class functions on G and the Lie algebra convolution of Ad(G)-invariantfunctions on g. In fact, the precise formulations are given in [DW] only forcompact and nilpotent Lie groups (the latter can be apparently extended tothe case of exponential Lie groups).

In both cases the statements are essentially equivalent to the integralformula for characters (Rule 6 of the User's Guide), although quite anothertechnique is used for the proof in the nilpotent case.

The appearance of coadjoint orbits is now very natural. First, we replacethe convolution algebra of class functions on C by the convolution algebraof Ad(G)-invariant functions on g and then note that the latter algebra andthe ordinary algebra of K(G)-invariant functions on g' form the so-calleddual hypergroups, related by the Fourier transform.

I mention also the other, infinitesimal, way to express the same result.The generalized Fourier transform (see Chapter 4) simplifies not only the

'In fact, only the existence of such a formula was proven by E. Campbell and F. Hausdorffand the beautiful (but not very practical) explicit expression for the coefficients was found laterby E. B. Dynkin.


convolution but also the infinitesimal characters, i.e. the action of Laplace-Casimir operators. The result can be roughly formulated as follows.

Proposition 3. All Laplace-Casimir operators become differential operatorswith constant coefficients (in the canonical coordinates) after conjugating bythe operator of multiplication by p(X) and restricting to class functions.

For semisimple Lie groups a theorem of this sort was first obtained inthe thesis of F. A. Berezin in the 1960's.7

For the general case the theorem was suggested in my course at InstitutHenri Poincare in 1968 and was later proved by Duflo [Du3J and by Ginzburg[GiJ.

A much more general statement was recently proved by Kontsevich[Kon] in the framework of deformation quantization. I would like to quotea sentence from his paper:

Now we can say finally that the orbit method has solid background.

4.2. Physical argument.This argument is now widely known under the rather lucky name of

geometric quantization. The idea is to use a correspondence betweenclassical and quantum physical systems.

As we well understand now, there is no canonical and universal corre-spondence: the quantum world is different from the classical one.

Nevertheless, for many particular systems, the so-called quantizationrules were formulated. They allow us to construct a quantum system fromthe classical one. Moreover, the symmetry possessed by a classical systemis often inherited by the quantum counterpart.

There are many ways to translate the physical term "quantization" intomathematical language (e.g., there exist algebraic, asymptotic, deforma-tional, path-integral, and other quantizations). All these theories are basedon the premise that classical and quantum mechanics are just different re-alizations of the same abstract scheme.

The goal of geometric quantization is to construct quantum objects fromthe geometry of classical ones. Historically, there were two origins of thisapproach:

1) the "quantization rules" of old quantum mechanics, which becomemore and more elaborate (but still remain adjusted to rather special Hamil-tonians defined on special phase spaces);

7There were some minor gaps in the Berezin proof noticed by Harish-Chandra during hisvisit to Moscow in 1966. The appropriate correction appeared the next year.

§4. Why the orbit method works 197

2) the functor of unitary induction and its generalizations in the repre-sentation theory, which allowed us to construct explicitly the unitary duals(or at least a large part of it) for many Lie groups.

It was Bertram Kostant who observed in [Kol] that one can merge thesetheories into a new one. At the same time Jean-Marie Souriau suggested thesame idea in [S]. Since then geometric quantization became very popular,especially among physicists.

We have, however, to remark that practically no general results in thenon-homogeneous situation were obtained.8 The quantization rules men-tioned above are usually not well defined and sometimes are even contradic-tory. In the homogeneous situation they are practically equivalent to one oranother variant of the induction procedure (see Appendix V.2).

We refer to [GS1], [Ki7], [Kol], and [S] for the accurate definition andbasic properties of geometric quantization.

The most interesting and important applications of geometric quanti-zation are related to infinite-dimensional systems. Many of them are onlyproved "on the physical level". See [AG], [AS], [GS1], [Vol], and [Wi2]for further details.

Let us now consider physical systems with a given symmetry group G.We call such a system elementary if it cannot be decomposed into smallerparts without breaking the symmetry.

On the classical level the phase space of a physical system with givensymmetry group G is a symplectic G-manifold M. For an elementary systemthis manifold M must be homogeneous.

On the quantum level the phase space of a physical system with givensymmetry group G is a projectivization of a Hilbert space 1[ with a unitaryrepresentation of G in R. For an elementary system this representationmust be irreducible.

Thus, the quantization principle suggests a correspondence between ho-mogeneous symplectic G-manifolds on the one hand and unirreps of G onthe other.

Actually, the situation is slightly more delicate. It is known that theenergy function for classical systems is defined up to an additive constant,while for a quantum system the energy is uniquely defined and is usuallynon-negative.

This shows that the right classical counterpart to quantum systems withthe symmetry group G are Poisson G-manifolds rather than symplectic ones.But we have seen in Chapter 1 that homogeneous Poisson G-manifolds are

8Except some negative ones. For example, in [GS1J it was shown that the general quantiza-tion formula does not work even in a low-dimensional non-homogeneous situation.


essentially coadjoint orbits. So we come to the desired correspondence be-tween orbits and representations.

5. Byproducts and relations to other domainsThe orbit method stimulated the study of coadjoint orbits and turned out tobe related to several other domains that were rapidly developing in the lastfew decades. We briefly describe here the three most important directions.

5.1. Moment map.The first general definition of the moment map was given by Souriau

[S], although its particular cases (e.g., related to the Galileo and Poincaregroups) were known to physicists long ago.

In particular, the famous E. Noether theorem describing the connectionbetween symmetries and invariants in Hamiltonian formalism is simply themoment map for a 1-dimensional Lie group of symmetries.

Most of the new applications of the moment map are related to the no-tion of symplectic reduction (see Appendix 11.3.4). This procedure, whichalso goes back to classical Hamiltonian mechanics, is naturally formulatedin terms of the moment map.

Let G be a connected Lie group, (M, a) a symplectic G-Poisson mani-fold, and p : M g' the associated moment map (see Chapter 1). For anycoadjoint orbit f C g' the set Mn = µ-1(f) is G-invariant.

Suppose that this set is a smooth manifold and that G acts on it so thatall orbits have the same dimension.9 Then the set (Mn)G of G-orbits in Mnis also a smooth manifold and possesses a canonical symplectic structure.

Indeed, one can easily check that the restriction of a to MF = µ-1(F)is degenerate and ker aIiy, at a point m coincides with the tangent spaceto the Stab (F)-orbit of m. Therefore, a induces a non-degenerate form on(AIF)Stab(F) (Mf2)G

Exercise 2. Let M = Pr'(C) with the symplectic structure induced bythe Fubini-Study form. The torus T"+1 that acts on M by the formula

(to, t1, , t,) . (zo : zl : ... : zn) = (tozo : tjzi : ... : tnzn).

Describe the symplectic reduction of M with respect to the whole torus andwith respect to its subgroup Tk+1, 0 < k < n. 4

91n practical situations these conditions are often violated on a submanifold of lower dimen-sion. Then one has to delete this submanifold or consider so-called orbifolds and more generalmanifolds with singularities.

§5. Byproducts and relations to other domains 199

This procedure allows us to reduce the study of a mechanical system withthe symmetry group G to the study of another system with fewer degreesof freedom (and fewer degrees of symmetry).

Sometimes, it is worthwhile to reverse this procedure and consider acomplicated low-dimensional system as a result of the reduction of a simplehigher-dimensional system.

We refer to [AG[ for a survey of the symplectic geometry and its ap-plications and to [GS21 for the description of geometric properties of themoment map.

Here we give only the simple algebraic interpretation of the symplecticreduction. In the algebraic approach, the submanifold Mo is replaced by thealgebra C°O(Mn) of smooth functions on this manifold. It is the quotientalgebra of COO(M) by the ideal I formed by functions which vanish on Mo.

Unfortunately, I is not a Poisson algebra: it is not closed with respectto the bracket operation.

Consider the subalgebra A C C°°(M) _ { f E COO(M) I { f, I} C 1}.Then I is a Poisson ideal in A, hence A/I inherits a Poisson structure. Itturns out that A/I ^ C°O((Mc)G).

Note that the same manifold (M0)G can be obtained as a submanifoldof the quotient manifold MG. In an algebraic approach this means that wetake a subalgebra CO°(M)G C COO (M) and factorize it over the appropriateideal J.

5.2. Integrable systems.This huge domain was intensively developed during the last 30 years.

Before that isolated examples were known and no general theory existed.The new era began with the seminal discovery that the so-called Korteweg-de Vries (KdV for short) equation

Pt = PPx + Pxxx

is a completely integrable system which possesses an infinite series of conser-vation laws. Since then a lot of important examples of classical and quantumintegrable systems were found and several schemes were proposed to explaintheir appearance (see, e.g., [DKN]).

The orbit method is a natural source of homogeneous symplectic man-ifolds (coadjoint orbits) that can be considered as phase spaces of classicalmechanical systems. Note that most of them are not isomorphic to cotan-gent bundles and therefore do not correspond to a traditional mechanicalsystem. On the other hand, this new kind of phase space includes the fol-lowing remarkable example.


Example 4. Consider a 2-dimensional sphere S2 with a symplecticform such that the total volume is an integer n > 1. As a coadjoint orbit forsu(2) it admits a quantization via the corresponding unirrep of dimensionn. The physical interpretation of this quantum system is a particle withspin s =21, which has no other degree of freedom. So we get a classicalcounterpart of the notion of spin that for a long time was not believed toexist. 0

To construct an integrable system, we need either a big family of Poissoncommuting functions (in the classical picture) or a big family of commutingoperators (in the quantum picture). The so-called Adler-Kostant scheme(see, e.g., [Ko2l and [RS]) provides such a family.

The most simple version of this scheme is based on the decomposition ofa Lie algebra g into a direct sum of subspaces 9± that are in fact subalgebrasin g. In this case we can define on g the new commutator

(6) [X, Y] [X+, Y+) - [X-, Y

where Xf denotes the projection of X E g to 9±.The commutator (6) defines a new Lie algebra structure on g and a new

Poisson structure on g'.The remarkable fact is that central functions with respect to the first

bracket are still commuting in the sense of the second bracket.Moreover, the Hamiltonian systems corresponding to H E P(g')G admit

an explicit description in terms of the factorization problem:

(7) 9 = 9+ 9-, 9 E G, gf E G±-

Theorem 1 (Adler-Kostant-Semenov-Tyan-Shanski). a) All functions inP(9* )G form a Poisson commuting family with respect to the new structure.

b) For a function H E P(9*)G define the curves g±(t) in G by theequation

(8) exp(tdH(F)) = g+(t)-lg_(t).

Then the trajectory of a point F E g' under the Hamiltonian flow corre-sponding to the function H is given by

(9) F(t) = K(g+(t))F = K(g_(t))F.

0The application of this scheme to different Lie algebras g and differ-

ent points F E 9' gives a uniform construction for most known integrable

§6. Some open problems and subjects for meditation 201

systems, including infinite-dimensional systems (such as KdV and its super-analogues, which are related to the Virasoro algebra and its superextensions;see [OPRJ, (KO]).

The only deficiency of this approach is that it appeared post factum,when almost all interesting examples were discovered by other methods.

6. Some open problems and subjects for meditation

6.1. Functional dimension.It is well known that:1) All separable, infinite-dimensional Hilbert spaces are isomorphic.

2) The spaces C°O(M) are isomorphic Frechet spaces for all smoothcompact manifolds M of positive dimension.

3) All infinite countable sets are equivalent.But there is no natural isomorphism between1) L2(IR, dx) and L2 (R2, dxdy).

2) C°G(S') and

3) Z and Z2.

C°°(S2).

The non-formal problem is to define the functional dimension f-dimof an infinite-dimensional space so that, for example, we have

(10) f-dim L2(lR", d"x) = f-dim C' (R') = n.In order to do this we have to restrict the set of morphisms between ourspaces and allow only natural morphisms. There are several possible waysto impose such a restriction.

One way is to define some "basic" morphisms and consider as naturalonly those morphisms that are compositions of basic ones.

For the spaces of smooth functions on manifolds the set of basic mor-phisms should include

a) multiplication by non-vanishing functions,

b) diffeomorphisms of the underlying manifolds,c) some integral transformations such as Fourier or Radon transforms.Another way is to introduce an additional structure on our linear spaces

and consider only morphisms that preserve this structure. For example, itis not difficult to show that a compact smooth manifold M is completelydetermined by the associative algebra A = C°°(M) (see Chapter 1) or by theLie algebra L = Vect(M) of smooth vector fields on M (see below). Indeed,the points of M correspond to maximal ideals in A or to Lie subalgebras ofminimal codimension in L.


Here we discuss one more non-formal problem related to the notion offunctional dimension:

Show that if dim Ml > dim M2, then the Lie algebra Vect(MI) is, first,"bigger" and, second, "more non-commutative" than Vect(M2).

The answer to one of the possible rigorous versions of the first questionwas obtained in [KK] and [KKM]. We describe it below.

Let 1;, rl be a pair of vector fields on M. Consider the Lie subalgebraL(t;, 77) C Vect(M) generated by these fields. It is a bigraded Lie algebra ofthe form

(11) rl) = FL(x, y)/I r!)

where FL(x, y) is a naturally bigraded free Lie algebra with two generatorsx, y and I(t:, 77) C FL(x, y) is the kernel of the map qS : FL(x, y) --r L(t;, 77)defined by O(x) = , 0(y) = r/.

It turns out that for generic . , 77 the ideal I (t , 17) depends only on dim M.So, for each n E N we get a distinguished bigraded ideal In C FL(x, y) andwe define the bigraded Lie algebra L(n) := FL(x, y)/I,,.

The growth of the numbers ak,1(n) := dim Lk,l (n) can be considered ascharacteristic of the size of Vect(M) for n-dimensional M.

Theorem 2 (Conjectured in [KK], proved later by Molev).

(12) ak,1(1)=pk(k+l-1)+pl(k+l-1)-p(k+l-1)where p(n) is the standard partition function and pk(n) is the number ofpartitions of n into < k parts (or into parts of size < k).

An interesting corollary is that the sequence a,,,(n) = Ek+l=m ak,l(n)for n = 1 has intermediate, or subexponential, growth. Namely,

1 V-2-J 3m(13) am.(1) " 4f which implies log log a,,, (1)

2I2 log M.

A more general result has been obtained by A. I. Molev:

(14) log logam(n)n

n+llogm.

Hence, all Lie algebras of vector fields on smooth manifolds have in-termediate growth. In particular, contrary to a widespread delusion, theynever contain a free Lie subalgebra that has exponential growth:

1 1:dim FLk,I(x, y) =

m

Et(d)2m/d __ 2m

k+l=m dim

Many interesting problems arise in connection with the second question,i.e. with the algebraic structure of the algebras L(n) and ideals 1(n). Wemention only the following.


Conjecture. The ideal 1(1) is spanned by expressions

(15) E sgns ad(xe(l))ad(x3i2i)ad(x3i3i)ad(xsl4i)xO, xi E FL(x, y).SES4

There is some evidence in favor of this conjecture. In particular, it seemsthat L(1) admits a faithful representation as the Lie algebra of vector fieldson an infinite-dimensional manifold, tangent to a 1-dimensional foliation.

In conclusion we repeat the main question:Give a rigorous definition of the functional dimension of a unirrep so

that Rule 9 of the User's Guide holds.

6.2. Infinitesimal characters.The general proof of the modified Rule 7 was obtained independently in

[Du3] and [Gi]. The proof is rather involved and analytic in nature. But thestatement itself is purely algebraic and certainly can be proved algebraically.

My own attempt to do it was broken by the discovery (cf. Appendix111.2.3) that the manifold A,, of structure constants of n-dimensional Liegroups is highly reducible. So, it is doubtful that one can prove the statementjust using the defining equations of A.

Another approach was suggested in [KV] but the problem is still open.Quite recently I learned that M. Kontsevich has found a new proof based

on the computation of a functional integral suggested by a variant of quan-tum field theory.

This is probably the right solution to the problem (cf. the quotationin the end of section 4.1). In very general terms, the advantage of thisnew approach is that one considers general Poisson manifolds (and not onlythose which are related to Lie algebras). So, the situation is similar to thefamous Atiyah-Singer Index Theorem for differential operators, which wasproved only after passing to the more general context of pseudo-differentialoperators.

6.3. Multiplicities and geometry.Let G be a compact simply connected Lie group, let T be its maximal

connected abelian subgroup, and let g D t be the corresponding Lie algebrasthat are identified with their duals via an Ad(G)-invariant scalar product.Let P C it' be the weight lattice, Q C P a root sublattice, p E P the sumof the fundamental weights, and S2,\ C g" the coadjoint orbit of the pointiA E t' C g'. We denote by p the natural projection of g" on V.

It is known that for any A E P+ the set Ca = p(SZa+p) is the convex hullof I W I different points {iw(A + p), w E W }.


Let us call an elementary cell in t' the set Co = p(Qp), as well as allits translations by elements of P. One can check that Ca is the union ofelementary cells centered at the points of (A + Q) flp(Sl,\). For more preciseresults see Chapter 5.

A difficult and non-formal question is:

How is the multiplicity m.\(M) of a weight 1L in the unirrep 7a related tothe geometry of the sets p-1(Co + p) and p -'(A)?

6.4. Complementary series.The orbit method apparently leaves no place for the complementary

series of representations of semisimple groups. Indeed, according to theideology of the orbit method, the partition of g' into coadjoint orbits cor-responds to the decomposition of the regular representation into irreduciblecomponents. But unirreps of the complementary series by definition do notcontribute to this decomposition.

One possible solution to this paradox is based on a remark made in one ofthe early papers by Gelfand-Naimark. They observed that for non-compactsemisimple groups there is a big difference between L1 (G, dg) and L2(G, dg)due to the exponential growth of the density of the Haar measure.

The effect of this difference can be explained as follows. One of theingredients of the orbit method is the generalized Fourier transform (seeChapter 4) from the space of functions on G to the space of functions on g',which is the composition of two maps:

1. the map from functions on G to functions on g :

exp X)f - where O(X) = d(dX

f (exp X);

gS

2. the usual Fourier transform that sends functions on g to functions on

The image of L2(G, dg) under the generalized Fourier transform consistsof square integrable functions (at least if we consider functions with thesupport in the domain E where the exponential map is one-to-one). But theimage of L1 (G, dg) consists of much nicer functions that admit an analyticcontinuation from g' to some strip in gZ.

So, one can try to associate complementary series of unirreps with thoseG-orbits that lie inside this strip and are invariant under complex conjuga-tion. One can check that in the simplest case G = SL(2, R) this approachleads to the correct integral formula for the generalized character of a rep-resentation of the complementary series.

I believe that the problem deserves further investigation.


6.5. Finite groups.The orbit method can be developed in a very interesting direction.

Namely, one can try to apply it to some finite groups of the kind G(Fq)where G is an algebraic group defined over Z and ]Fq is a finite field of qelements. For instance, we can consider the sequence G of classical groups:GL(n), SO(n), Sp(n).

We are mainly interested in the asymptotic properties of harmonic anal-ysis on Gn(Fq) when q is fixed and n goes to infinity. In particular I wantto advertise here some principal questions:

1. What is the asymptotics of the number of coadjoint orbits for Gn(]Fq)?

2. Can one describe the "generic" or "typical" coadjoint orbit?3. More generally, which characteristics of orbits and representations

can one deal with for the groups of "very large matrices", say of order 1010(or even 20) over a finite field?

(Note that the simplest numerical questions about these groups are outof the range of modern computers.)

Of course, these questions make sense not only for triangular groups andtheir analogs (unipotent radicals of classical groups). For instance, one cantry to find the answer for GL(n, ]Fq), using the results from [Ze].

6.6. Infinite-dimensional groups.This is perhaps the most promising generalization of the orbit method.

In the situation when there is no general theory and many very importantand deep examples, the empirical value of the orbit method is hard to over-estimate.

The most intriguing and important question is related to the Virasoro-Bott group. It can be formulated as follows:

How one can explain the rather complicated structure of discrete series ofunirreps of Vir in terms of the rather simple structure of the set of coadjointorbits?

Note that the very interesting paper [AS] gives an answer to this ques-tion, although it is written on the physical level of accuracy and needs atranslation into mathematical language.

Following the example of D. Knuth, I conclude the section by the fol-lowing general

Exercise 3.' Formulate and solve other problems concerning the appli-cation of the orbit method to infinite-dimensional groups. 4

Appendix I

Abstract Nonsense

In this appendix we collect some definitions and results from topology, cate-gory theory, and homological algebra. They are formulated in a very abstractform and at first sight seem to claim nothing about everything. That is whyhomological algebra has the nickname "abstract nonsense".

I hope, however, that the examples in the text will convince you thatthis material is really useful in many applications and, in particular, inrepresentation theory.

Note that in the examples we use some notions and results describedelsewhere. So, the reader must consult the index and look for the informationin the main text or in the other appendices.

1. Topology

1.1. Topological spaces.The notion of a topological space is useful, but not strictly necessary

for the understanding of the main part of the book. We briefly discuss thisnotion here to make the reader more comfortable.

Roughly speaking, a topological space is a set X where the notion ofa neighborhood of a point is defined. In the ordinary Euclidean space 1R"the role of neighborhoods are played by the open balls.

It is well known that all basic concepts of analysis can be formally definedusing only the notion of a neighborhood:

interior point: x E X is an interior point of S C X if it belongs to Stogether with some neighborhood;

207

208 Appendix I. Abstract Nonsense

exterior point: x E X is an exterior point of S C X if it is an interiorpoint of the complement -S = X \S;

boundary point: x E X is a boundary point of S C X if it is neitherinterior nor exterior for S;

open set: a subset S C X is open if all of its points are interior;closed set: a subset S C X is closed if it contains all its boundary

points;

connected space: a topological space X is connected if it cannot berepresented as a union of two disjoint non-empty open subsets;

limit of the sequence {xn} C X: a point a E X, such that everyneighborhood of it contains all members of the sequence except, maybe,finitely many of them;

continuous map: a map f : X ---* Y is continuous if for any opensubset S C Y its preimage f -1 (S) is open in X;

sequentially continuous map: a map f : X -* Y is sequentiallycontinuous if it commutes with limits:

f (lim xn) = lim f (xn) for any convergent sequence {xn} in X.n-oc n_00

The primary object of abstract topology is the notion of an open set.From this we can derive the definition of a neighborhood of a E X as anyopen subset 0 C X that contains a.

So, to define a topology on a set X we have only to specify the collectionO of open subsets in X. It turns out that 0 can be chosen arbitrarily subjectto three very simple axioms:

1. The union of any family of open sets is open.2. The intersection of a finite family of open sets is open.3. The empty set 0 and the whole space X are open.It is a miracle how from this simple set of axioms one can derive deep

and important consequences, such as the existence and/or uniqueness of so-lutions to very complicated systems of algebraic and differential equations.)

1.2. Metric spaces and metrizable topological spaces.Of course, the most interesting topological spaces are those which nat-

urally appear in different domains of mathematics and its applications. Formany of them the topology can be defined in a special way. Namely, assumethat the space X in question admits a natural notion of the distance d(x, y)between the points x and y. It is supposed that the distance satisfies thefollowing three conditions:

As a byproduct we also get a lot of papers of minor interest, but this is not a miracle.

P. Topology 209

1. Positivity: d(x, y) > 0 for x 0 y and d(x, y) = 0 for x = y.2. Symmetry: d(x, y) = d(y, x) for all x, y E X.

3. Triangle Axiom: d(x, y) < d(x z) + d(z, y) for all x. y, z E X.The space X with a distance d : X x X R+ satisfying the above

conditions is called a metric space. It is usually denoted by (X. d).In a metric space (X, d) we can define for any c > 0 the E-neighborhood

of a point a as the set

U,(a) = {x E X I d(x, a) < f}.

The set U,(a) is also called an open ball with center a and radius E.So. every metric space (X, d) can be endowed with a topology. Namely,

the subset A C X is called open if it is a union of an arbitrary family ofopen balls. Topological spaces of this kind are called metrizable. Mosttopological spaces used in our book are metrizable, although there are a fewexceptions.

Exercise 1. a) Prove that for any topological space a continuous mapis sequentially continuous.

b) Show that for metrizable spaces the converse is also true: any sequen-tially continuous map is continuous. 46

Among all metric spaces we can distinguish a very important class ofcomplete metric spaces defined as follows. Call a sequence {x } in a metricspace (X, d) a fundamental sequence (or Cauchy sequence) if

(1) lim d(.r,,,, a:,,) 0.m.ra-x

More minutely: for any f > 0 there exists N = N(e) E N such that

d(x,,,, f for all m > N, n > N.

A metric space (X. d) is called complete if every fundamental sequenceconverges, i.e. has a limit in X.

Two useful properties of complete metric spaces are:

Theorem on nested balls. Assume that a sequence of closed balls

B,,={xEX I d(x,a,,)<r,t}

has the properties: a) B for all n, b) r -+ 0 when n oo.Then the intersection n,, B contains exactly one point. 0

210 Appendix L Abstract Nonsense

Theorem on contracting maps. Assume that for some A E (0, 1) a mapf : X -+ X has the property

(2) d(f (x), f (y)) 5 A d(x, y).

Then f has a unique fixed point xo E X (i.e. such that f (xo) = xo).

Not all metric spaces are complete, but every metric space (X, d) hasa so-called completion (X, d). By definition, (X, d) is a complete metricspace such that k contains a dense subset X0 which is isometric to X.

Informally speaking, X_ is obtained from X by adding some "ideal"points, which are limits in X of those Cauchy sequences that have no limitsin X.

Example 1. The real line R with the standard distance d(x, y) = Ix-ylis the completion of the set Q of rational numbers.

Here the ideal points are irrational numbers. The infinite decimal rep-resentation of x:

j=nx = xn...x1x0,x-1 ...x-k.. _ Exj - 10j

-00

provides a Cauchy sequence of rational numbers converging to x.

One can define other distances on Q which are translation-invariant andin a way compatible with multiplication. Namely, for any prime number pthe so-called p-adic norm is defined as lirOOp = p -k if the rational numberr has the form r = pk n where m and n are coprime with p. We define thep-adic distance by

(3) dp(rl, r2) = Pin - r2lIp

It turns out that the completion of the metric space (Q, dp) is a field Qpwhere all arithmetic operations are extended by continuity from Q. Theelements of Qp are called p-adic numbers; they can be written as a semi-infinite string of digits {0, 1, ... , p - 1}:

+00(4) a=...an...ala0,a-I...a-k= E aj - pi.

j=-k

Note that the series is convergent in (Qp, dp).

It is worthwhile to mention that the closure Zp of the subset Z C Qcoincides with the closure of the subset N C Z and is compact and homeo-morphic to the Cantor set. The elements of Zp are called p-adic integers.In the notation of (4) they are characterized by the property k = 0. 0

§2. Language of categories 211

Exercise 2. Show that for any a E Zp the limit lim ap exists. It isn-.oo

called the p-adic signum of a and is denoted by signp(a). Prove the followingproperties of this function:

a) signp(ab) = signp(a)signp(b);

b) the number s = signp(a) satisfies the equation sp = s, hence is either0 or a (p - 1)-st root of unity;

c) the value of signp(a) depends only on the last digit ao in the notationof (4). 4

Example 2. The Hilbert space L2(It, dx) is the completion of each ofthe following spaces:

a) the space .A(R) of smooth functions f with compact support;

b) the space Co(R) of continuous functions f with compact support;

c) the space of step-functions f with compact support;

d) the space of functions f (x) = P(x)ex2where P is a polynomial in

X.

All are endowed with the distance

d(fi, f2) =JR If I (T) -

f2(x)I2dx.

Here the ideal points are Lebesgue measurable functions on R with an in-tegrable square that do not belong to the subspaces listed above. In mostapplications of the Hilbert space L2(R, dx) these elements play only a sub-sidiary role to make the whole space complete. Q

2. Language of categories

2.1. Introduction to categories.Since the beginning of the 20-th century all mathematical theories have

been founded on the set-theoretic base. This means that every object ofstudy is defined as a set X with some additional structures. The notion of astructure itself is also defined in the set-theoretic manner as a point in someauxiliary set constructed from X.

Example S. A map 0: A -+ B can be defined in set-theoretic termsas a subset r C A x B, the graph of.0, such that the projection ro - Ais a bijection;

a group is a set G with a distinguished point in G(c"C) which specifiesthe multiplication law;

a topological space is a set X with a special point in22x describing the

collection of open subsets in X;


a vector space over a field K is a set V with a point in V(vuK) x v, which

determines the addition law and the multiplication of vectors by numbers;etc.

(In this example we used the following, now practically standard, nota-tion: BA denotes the set of all maps from A to B and 2X denotes the set ofall subsets of X that is in a natural bijection with the set of all maps fromX to a two-point set.) Q

There exists, however, a quite different approach to the foundation ofmathematics: one of a more sociological2 nature. The point is to considermathematical objects as members of a certain commonwealth and charac-terize them by their relations to other objects of the same nature. The basicnotion of this new approach is the notion of a category.

To define a category C we havea) to specify a family (not necessarily a set!) ObC of objects of C;b) for any pair of objects X, Y E ObC to define a set of morphisms

from X to Y, denoted Morc(X, Y);c) for each triple X, Y, Z E ObC to define a map

Morc(X, Y) x Morc(Y, Z) -+ Morc(X, Z) : (f, g)'--+ g o f,

which is called the composition of morphisms.It is assumed that1. The composition law is associative: (f o g) o h = f o (g o h) whenever

defined.

2. For each object X there exists a unique unit morphism lx EMorc(X, X), which plays the role of left and right unit:

f o lX = ly o f= f for any f E Morc(X, Y).

For many important categories their objects are sets (usually with someadditional structures), morphisms are maps (preserving the additional struc-tures), composition is the usual composition of maps, and 1X is the identitymap Id. In other words, they are subcategories of the category Sets.

In particular, this is true for the following categories:Sets: objects are sets, morphisms are maps, so that Morset,(A, B) _

BA;

lkctK: objects are vector spaces over a given field K, morphisms areK-linear operators, so that Morlbd,(V, W) = HomK(V, W);

21 borrowed this epithet from Yu. I. Manin.


Gr: objects are groups, morphisms are group homomorphisms;Man: objects are smooth manifolds, morphisms are smooth maps;GG: objects are Lie groups, morphisms are smooth homomorphisms.But there are categories of a quite different kind and we consider some

useful examples below.

Note that the family of all categories3 also forms a sort of categoryCat: objects are categories and morphisms are the so-called functors

which we now describe.

To define a functor F from a category Cl to another category C2 wehave to specify:

a) an object F(X) E ObC2 for any X E ObCI andb) a morphism F(O) E Morc2(F(X). F(Y)) for any ¢ E Morc,(X, Y) so

that

F(lx) = lF(X), F(r¢ o th) = F(¢) o F(+/,) whenever it makes sense.

For any category C one can define the dual category C°. It has thesame class of objects as C. A morphism from A to B in C° is by definitiona morphism from B to A in C. The composition fog in C° is defined as thecomposition g o f in C.

Example 4. There exists a natural duality functor * : 1kctK(1kctK)° which sends a space V to the dual space V" := HomK(V, K)and a linear operator A : V W to the adjoint operator A* : W* - V.In Appendix VI this functor is extended to the category of Banach spacesand (in a different way) to the category of Hilbert spaces.

Often a category is represented graphically: objects are denoted bypoints or small circles and morphisms by arrows. The transfer to the dualcategory means just "reversing the arrows".

A diagram constructed from objects and morphisms of some category iscalled commutative if the following condition is satisfied:

The composition of arrows along any path joining two objects X and Ydoes not depend on the choice of the path.

Example 5. The commutative diagram

V,

QI

A

B

Vl

tQ

V2

3More precisely, here we have to restrict ourselves by the so-called small categories, whichwe do not define here since it is not essential for understanding the material.


in 1kctK means the relation QA = BQ between the linear operators A :V1 -+ V1, B : V2 -- + V2, and Q : Vi - V2. 0

2.2. The use of categories.The categorical approach allows us to unify many definitions and con-

structions that are used independently in different domains of mathematics.A simple example: the equivalence of sets, homeomorphism of topological

spaces, isomorphism of groups or algebras, and diffeomorphism of smoothmanifolds are all particular cases of the general notion of an isomorphism ofobjects in a category, which is defined as follows.

Two objects X, Y E ObC are called isomorphic if there exist mor-phisms f E Morc(X, Y), g E Morc(Y, X) such that fog = ly, g o f = 1X.

Another useful categorical notion is the notion of a universal object.An object X E ObC is called a universal (or initial) object if for anyY E ObC the set Morc(X, Y) contains exactly one element. (Graphically:there is exactly one arrow from the point X to any point Y.)

The dual notion of a couniversal (or final) object is obtained by re-versing the arrows: Y is final if there is exactly one morphism from anyobject X to Y.

Note that any two universal objects X, X' in a given category C arecanonically isomorphic. Indeed, by definition there exist unique morphismsa : X' X and 0 : X X'. The composition a o,3 is a morphismfrom X to itself which is forced to be lX since X is universal. The sameargument implies 3 o a = IX'.

As an application we define here the direct sum and direct product oftwo objects X1 and X2 in a category C. To do this, we construct an auxiliarycategory C(Xi, X2). The objects of C(X1, X2) are triples (al, a2, Y) whereY can be any object of C, al E Morc(Xi, Y), and a2 E Morc(X2, Y).

Graphically, an object of C(X1, X2) looks as in the following diagram:a, a2Xl I

A morphism from the object (al, a2, Y) to the object (b1, b2, Z) is de-fined as a morphism 0 from Y to Z such that the following diagram iscommutative:

X1 Y 1-a2 X21X,1 ml

11x2

X1 --+ Z` X2b, b2

If the category C(X1i X2) has a universal object (i1, i2, X) (recall that allsuch objects are isomorphic), then X is called the direct sum of X1 and X2


in C and morphisms il, i2 are called canonical embeddings of summandsinto the sum.

Exercise 3. Show that direct sums exist for all couples of objects inthe following categories and describe them.

a) Sets, b) 1kctK, c) Gr, d) Man.Answer: a) and d) the disjoint union, b) the direct sum of vector spaces,

c) the free product of groups. 4The definition of the direct product in C is obtained from the above

definition by reversing the arrows. In other words, the direct productis the direct sum in the dual category Co. It looks as in the diagramXl

,

Pi X ?2 i X2. So, the direct product X always comes togetherwith canonical projections of a product to the factors.

Exercise 4. Describe the operation of direct product for the categorieslisted above.

Answer: a) and d) the direct (Cartesian) product, b) the direct sum ofvector spaces, c) the direct product of groups. 4

Exercise 5. Extend the definitions of direct sum and direct product toan arbitrary family {XQ}QEA of objects.

Hint. Use an auxiliary category e whose objects have the form (X, { f , }),where X E ObC and fQ E Morc(XQ, X) for the direct sum, whereasfQ E Morc(X, XQ) for the direct product. 46

Let C be any category and X E Ob C. The correspondence A HMorc(X, A) can be extended to a functor FX : C -' Sets. Namely, for0 E Morc(A, B) we define Fx(0) : Morc(X, A) Morc(X, B) by FX(a) =¢ o a. Often the object X can be uniquely reconstructed from this functorFX.4

When a functor F : C Sets has the form F = FX, it is called a rep-resentable functor and the object X is called the representing object.Even when this is not the case, it is convenient to consider non-representablefunctors from C to Sets as "generalized objects" of C.

For a more detailed introduction to the categorical ideology we refer to[Gr] and [M].

2.3.` Application: Homotopy groups.Topology plays an essential role in the representation theory of Lie

groups. In particular, low-dimensional homotopy groups are used. Weassume that the reader is acquainted with the notion of the fundamental

4The analog of this statement in ordinary life is the principle: "tell me who your friends areand I will tell you who you are".

216 Appendix L Abstract Nonsense

group al. The quickest way to introduce the higher homotopy groups xk isvia category theory.

Consider the category 7{T whose objects are pointed topological spaces(i.e. topological spaces with a marked point) and morphisms are homotopyclasses of continuous maps which send a marked point to a marked point.

Recall that two continuous maps fo, fl : (X, x) - (Y, y) belong to thesame homotopy class [1] if there exists a continuous map F : X x [0,1] -Y such that FIxx{o} = .fo, FI xx{1} = fl and Fixx(o,li = y. Two topologicalspaces have the same homotopy type if they are equivalent objects of 7{T.

Sometimes it is convenient to consider these homotopy types as trueobjects of 7{T.

Let (Se, s) be the standard unit sphere in Rn+1 given by the equation(x°)2 + + (xn)2 = I with the marked point x° = 1, xk = 0 for k 34 0.

The collection of sets

(5) 7rn(X, x) := MorxT((S", s), (X, x)), n > 0,

has a very rich algebraic structure. It is the main object of homotopytheory.

In particular, for n > 1 every set a group law,which comes from the natural morphism S" - S" + S" (direct sum in thecategory 71T). Therefore, 7rn(X) is called the n-th homotopy group of X.

It is known that the group is abelian for n > 2. Moreover, if X itself isa topological group, then all n > 0, have a group structure, whichis abelian for n > 1.

A very useful tool is the so-called exact sequence of a fiber bundle.

Proposition 1. For any fiber bundle F E -L+ B there is an exactsequence of homotopy groups:

(6) ... .1 xk(F) 7rk(E) p xk(B) b' 7rk-1(F) - ..

3. Cohomology

3.1. Generalities.The general idea of homology and cohomology is very simple. A sequence

of abelian groups and homomorphisms

(7) - G+1 -- C. 4 CA.-,

§3. Cohomology 217

is called a chain complex if

(8) 8ko8k+1=0 forallk.

Denote the kernel of 8k by Zk and the image of 8k+1 by Bk.

The elements of Ck are called k-chains, the elements of Zk are calledk-cycles, and the elements of Bk are called k-boundaries.

The relation (8) implies that every boundary is a cycle. The quotientgroup HA; = Zk/Bk is called the k-th homology group of the complex (7).The image of a cycle z E Zk in Hk is usually denoted by [z] and is calledthe homology class of z.

Sometimes it is convenient to consider the direct sum C = ®k Ck andreplace the family {8k} by the single boundary operator 8 : C Csuch that 8Ick = 8k. Then (8) takes the simple form 82 = 0.

Reversing the arrows, we get the definition of cochains, cocycles,coboundaries, and cohomology groups. Traditionally, the following ter-minology is used:

Ck - the group of k-cochainsdk : Ck Ck+1 - coboundary operatorBk = im(dk_1) - the group of k-coboundaries

Zk = ker(dk) - the group of k-cocycles

Hk = Zk/Bk - the k-th cohomology group.

This general scheme has many concrete realizations. We describe belowonly three of them that are especially used in representation theory.

3.2. Group cohomology.Let G be a group and M a G-module, i.e. an abelian group on which G

acts by automorphisms.

Define a homogeneous k-cochain as a G-equivariant map

Z:

G acts on Gk+1 by the left shift: g (go, ... , gk) = (ggo, ... , 99k)In other words, a homogeneous k-cochain Z is an M-valued function on

Gk+1 with the property

(9) .9 k ) -

T h eThe set Ck(G, M) of all G-equivariant maps is a group under the usualaddition of functions.


Define the differential, or coboundary operator

d : Ck(G, M) , Ck+l(G, M)

by the formula

(10)

_ k+1

i=O

where the hat sign - means that the corresponding argument is omitted.It is clear that d2 = d o d = 0. Indeed, the term

6(90, .., A,, ..., 9k+1)

enters in dc twice with opposite signs.Technically, it is more convenient to deal with cochains written in non-

homogeneous form. Note that the condition (9) implies that the elementc E Ck(G, M) is completely determined by the values c(g1, ... , gk)Z(e,91,9192, Namely, we have

6(9o,91, ,9k119k)-

We call the quantity c the non-homogeneous form of c. It is justan arbitrary M-valued function on Gk. The coboundary operator in thenon-homogeneous form looks like

(11)

do (.41, , 9k+1) = 91 6(92, , 9k+1)k

+1: (-1)' 6(91, ... , 99i+1, ... , gk+1) + (-1)k+16(91, ... , 9k).i=1

This formula looks less natural than (10) but in return non-homogeneouscochains have one argument less. Later on we shall always use the non-homogeneous form for cochains.

Let us study the cases of small k in more detail.The case k = 0. Here C°(G, M) = M and the differential is (dm)(g) _

g m - m. So, H0 (G, M) = Zo (G, M) = MG, the set of G-invariant elementsof M.

The case k = 1. Here dc(g1, g2) = 91 c(g2) - c(g1g2) + c(g1). So, a

1-cocycle is a map c : G M satisfying the cocycle equation:

(12) 6(9192) = 6(91) + 91.6(92)

§3. Cohomology 219

We shall see several appearances of this equation in the main part of thebook.

The cocycle c is trivial (i.e. [c] = 0) if it has the form

c(g) = g m - m for some m E M.

The trivial cocycles form a coboundary group isomorphic to M/MG.In the special case when M is a trivial G-module, the coboundary

group reduces to {0} and H1(G, M) = Z1(G, M) coincides with the groupHom(G, M) of all homomorphisms of G to M.

The case k = 2. Here we have

dc(g1, 92, 93) = 91 . c(92, 93) - c(9192, 93) + c(91, 9293) - c(92, 93)

The trivial cocycles have the form

c(91, 92) = g1 b(92) - b(9192) + b(g1) for some b : G -+ M.

We shall see the interesting example of a 2-cocycle in Appendix V when wediscuss the Mackey Inducibility Criterion.

Remark 1. There is a metamathematical statement:All cocycles are essentially trivial.

This means the following: if we suitably extend the collection of cochains,then any cocycle becomes a coboundary.

Another formulation of this principle is due to well-known physicistL. D. Faddeev who used to say:

Cohomology are just the functions but with singularities.

V

Many important theorems (including those that were discovered beforethe cohomology era) can be reformulated as a claim that certain cohomologygroups are trivial or become trivial after an appropriate extension of thecochain group.

3.3. Lie algebra cohomology.This and the next sections use a basic knowledge of smooth manifolds

and Lie groups. The reader who has doubts about possessing this knowledgemust consult Appendices II and III.

Let G be a connected and simply connected Lie group, and let H be aclosed connected subgroup. It is well known that all geometric and topo-logical questions about the homogeneous manifold M = G/H can be trans-lated into pure algebraic questions about the Lie algebras g = Lie (G) and


h = Lie (H). In particular, we can ask about the cohomology H(M). Inthe case when M is compact and we consider the cohomology with real co-efficients5 H(M, IR) the answer can be formulated in terms of the so-calledrelative cohomology H(g, h, IR), which we now define.

By definition, the group of relative k-cochains is (Ak(g/4)*)H, i.e. con-sists of H-invariant k-linear antisymmetric maps c : 9/4 x ... x g/ry R.

The coboundary operator acts as

dc(Xo,... , Xk) = F,(-1):+jc(lxi, X,J, Xo, ... . XJ .... Xk).i<j

Proposition 2. Let G be a connected compact Lie group, and let H beits closed connected subgroup. Let g = Lie (G) and h = Lie (H). ThenHk(G/H, R) ^' Hk(g, l , R). o

This statement can be naturally generalized to the case when R is re-placed by a finite-dimensional vector space V with a linear H-action. Thenwe can define a G-vector bundle E = G X H V and also a relative cohomologygroup Hk(g. , V). It turns out that this group coincides with the k-th Cechcohomology group of G/H with coefficients in the local system of sectionsof E.

3.4. Cohomology of smooth manifolds.The general definition of cohomology groups in topology can be formu-

lated as follows. It is a functor from the category Top of topological spacesto the category S91Z°, dual to the category Sgi of supercommutative Z-graded rings.

For smooth compact manifolds the most simple (and most popular) co-homology groups are those with real coefficients. This is a functor fromthe category Man of smooth manifolds to the category SgA°, dual to thecategory SGA of supercommutative graded algebras over R.

Put simply, to any smooth compact manifold M we associate a gradedalgebra

dim M

H'(M,IR)= ® Hk(M,IR)k=0

and for any smooth map f : M N we associate the homomorphism

f': H'(N,R)-'H'(M,IR)

preserving the grading.

SSee the next section for the definition of real cohomology of a manifold.

§3. Cohomology 221

There are several equivalent ways to define this cohomology functor. Weconsider two of them.

The de Rham cohomology HDR(M) of a smooth manifold M is de-fined as follows. The cochain complex is the algebra fl(M) of smooth realdifferential forms on M endowed with the usual grading. The role of thecoboundary operator is played by the exterior differential d : flk(M)ck+1(M).

The cocycles are the closed forms and the coboundaries are the exactforms.

Example 6. Let M = S' be the unit circle with the parameter 9 E 1!emod 27rZ.

The 0-cycles are smooth functions f (9) satisfying f'(9) = 0, i.e. constantfunctions. Since there are no 0-coboundaries, we see that HDR(S') = R.

The 1-cycles are all smooth 1-forms w = b(9)d9 and 1-coboundaries areexact forms w = df = f'(O)dO. To decide which cocycles are coboundarieswe have to know when a 21r-periodic function 0(9) is a derivative f'(9) ofsome 27r-periodic function f. It is well known that the antiderivative of 0 isthe function f (9) = f 0eo O(r)dr. It is 27r-periodic if fow b(9)d9 = 0. So, thecoboundary space B'(S') has codimension 1 in the cocycle space Z' (Sl )and HDR(S1) =111;.

Exercise 6. Compute the Betti numbers bk(M) = dim HDR(M)a) for the n-dimensional sphere S";b) for the n-dimensional torus Ti'.Answer: a) bk(S") = bk,o + ak,n;

b) bk(T") = (k)It follows that S" can be diffeomorphic to T' only for n = 1. 46

The definition of the tech cohomology R) is more involved, buthas an important advantage. It makes sense for any topological space andis manifestly invariant under all homeomorphisms.

We start with a covering of our topological space X by a family U ={UQ}aEA of open subsets. Denote by the intersection nk0 U.

The k-cochain c associates the real number cQO...Qk to any (k + 1)-stringof indices {aO... ak} with non-empty intersection UaO... so that

Cog(O) ...Qe(k) = Sign (s)Ca0...Qk for any permutation s E Sk+1.

Denote by Ck(X,U, IR) the vector space of all k-cochains and by R)the direct sum of all C'(X,U,R).


The coboundary operator is defined by the rule

(dC)ao...ak+l

k+I

- (-I)tCa0...Q,...ak+1i=O

where, as usual, the hat means that the corresponding argument must beomitted.

The cohomology of this complex is denoted by U, ]R). It maydepend in an essential way on the covering U.

Assume that the covering U' _ {U', Q E B} is finer than U = {Ua, a EA}. It means that every element U E U' is contained in some elementUa(g) E U where /3 a(,3) is a map from B to A. Then there is a naturalmap Ou.u' from U, R) to U', R):

(Ou,u, c)go...,sk _. Co(Bo)...((gk)

This linear map commutes with the differential, hence gives rise to a map ofcohomology which we denote again by OU U, : H (X, U', R) --- H (X, U, R).

To proceed further, we recall here the categorical definition of a director inductive limit. We recommend that the reader compare it with thedefinition of the direct product in Exercise 4.

Let A be a directed set, i.e. a partially ordered set with the property:for any two elements a, fl in A there is an element -y E A which is biggerthan both a and /3.

Suppose that a family {Xa}aEA of objects in C is given and for everyordered pair a < /3 in A a morphism a g E Morc(Xa, Xg) is defined, sothat we have

cpg,.y o Va,,3 = W,,-y for any ordered triple a < 3 < 'y in A.

Consider the auxiliary category e defined from the previous data in thefollowing way.

An object of e is an object X of ObC together with a family of morphismsfa E Morc(X, X) for every a E A such that

fa = fg o cpa R for any pair a < /3.

A morphism from (X, {fa}) to (Y, {ga}) in the category C is a morphismE Morc(X, Y) such that for all a E A we have ga = fa o

Assume now that the category e has a universal object (X, {ma}). Thenthe object X is called the direct or inductive limit of the family {Xa} along

§3. Cohomology 223

the directed set A and the morphisms ma are called canonical maps ofX,,, to X.

The dual construction gives the definition of the inverse or projectivelimit.

The tech cohomology of a topological space X is defined as a directlimit HHech(M, R) of graded vector spaces H'(X, U, R) along the directedset of all coverings.

This definition is convenient in theory but is highly non-constructive.To get a practical prescription for computing the cohomology, the followingresult is used.

Leray's Theorem. Assume that the covering U has the property: all non-empty intersections Uao...a,k are contractible sets.6 Then U, R) _H&ch(X, R).

Example 7. Let us compute the (tech cohomology of the circle X = S1.In this case there is a simple covering of X by three open sets that satisfiesthe condition of the theorem. Namely, choose any three different pointsPo, P1, P2 on M and denote by Uk the arc joining P, with P3 and containingPk (here {i, j, k} is any permutation of {0, 1, 2}).

The space of 0-cochains consists of triples (co, c1i c2) and the space of 1-cochains consists of triples (co,1i c1,2, c2,o). The boundary operator is definedby

(dc)i,j = c, - cj.

It is clear that 0-cocycles are the triples of the form (c, c, c) and 1-cobounda-ries are the triples (p, q, r) with p + q + r = 0. We get HO (S1) = R =

Hiech ('SL)Oech

0Exercise 7. Compute the (tech cohomology of S" and Tn.Hint. For S" use the covering by n + 2 open balls such that any n + 1

have a non-empty intersection but all n + 2 have no common points.For T" use the fact that T" s- S1 x .. x S' (n factors). 4r

Exercise 8.s7 Use the Cech cohomology to prove the following general-ization of the famous Helly theorem on convex sets.

Claim. Assume that a family {Uk}o<k<"+1 of n + 2 open subsets in R" hasthe property: all m-intersections Uk,...km :_ f i `_1 U. are non-empty andcontractible for 1 < m < n + 1. Then all n + 2 sets {Uk}o<k<n+1 have acommon point. 46

'Cf. the definition of Leray atlas in Appendix 11.1.2.7The idea of this exercise is due to D. Kazhdan.


We see in the examples above that de Rham and (tech cohomology ofspheres and tori are the same. It is not a coincidence.

de Rham's Theorem. For any compact smooth manifold there is a canon-ical isomorphism

HDR(M) HOech(M, R).

Sketch of the proof. Let c E HDR, and let w E Ilk(M) represent the classc. Consider a Leray covering U = {Ua}QEA and for any m > 0 construct thefamily of closed (k - m)-forms wao ,,_ am on Ua,,,,, am as follows.

1. Denote by wa the restriction of w to Ua.2. Since w is closed and Ua is contractible, we have wa = dOa for some

(k - 1)-form ea on Ua. Put wa p = Oa - Bp on U0,p.3. The form wa,p is a closed (k - 1)-form because dwa,p = d(90 - 0,3) _

wa - wp = 0. Since Ua,p is contractible, the form wa,p is also exact and wecan write it as dO0,p for some (k - 2)-form ea,p on ,,,,p. Then we definethe (k - 2)-form wa,p,.y on Ua p y as ea p + 90,.y + 0.y a. It is closed becausedwa,p,.y = dO0,p + d00,1 + d8-y,. = wa,p + wp,.y + wry,a = 0, etc.

Eventually we come to the family of closed 0-forms w00....,0k on U0O.... k

But a closed 0-form on a connected domain is just a constant function,i.e. a real number. Moreover, by construction, the numbers wao,,..,ak areantisymmetric with respect to permutation of indices. So, we get a Cechcochain cao,...,ak = wao,...,0k in C k, th(M, U, R).

We omit the verification that this k-cochain c is actually a cocycle andits cohomology class [c] depends only on the class [w]. So, we have defineda map HDR(M, R) -i HHeCh(M, R).

F o r the definition of the inverse map H k th(M, R) C eCh(M, U, R)HDR(M, R) we need a partition of unity {0a} subordinated to the

covering U. Using this partition and starting from a (tech k-cocycle cao,...,akI

we construct for any m > 0 the family of closed m-forms wao,,,,,ak_m onU0O,....0k_r.

As wao,...,ak we take just the family of constants Cao,..-,ak.

Then we define the m-form wao,...,ak_m on U00.....ak_m by the formula

w0O,...,ak - m E Ca,....,akd4ak-m+t A ... A dOak.ak-m+1 ....,ak

Finally, we associate to a cocycle c the k-form w such that

w IUao= E C0O,...,akd4Oa, A ... A dOak.

§3. Cohomology 225

One can check that w is closed and the correspondence [c] )-+ [w] is theinverse to the map [w] H [c] constructed above.

In conclusion we mention the connection between the homology andhomotopy groups.

Proposition 3. If irk-(X) = 0 fork < n, then

7r (X) forn>1 orn=0.Hn(X, ) = { ni(X)/[iri(X), iri(X)] forn = 1.

Appendix II

Smooth Manifolds

1. Around the definitionIn this appendix we collect some information about smooth manifolds in theform used in the book.

There are three different ways of thinking about manifolds: geometric,analytic, and algebraic. It is important to combine all three and to be ableto switch from one to another. As an illustration, I will tell you an old jokeabout two math students who met after one had a geometry class and theother had an algebra class.

First student: I have finally understood why the system of linear equa-tions has a unique solution: it is because two lines intersect in one point!

Second student: And I have finally understood why two lines intersectin one point: it is because a system of linear equations has a unique solution!

The conclusion for a reader: there are three (not just two!) importantfacts:

1. Two straight lines intersect in one point.2. A system of two linear equations with two unknowns has a unique

solution.

3. The two statements above are actually the same.

1.1. Smooth manifolds. Geometric approach.Smooth manifolds appear in geometry as collections of geometric objects:

points, vectors, curves, etc. (or equivalence classes of these objects).

227

228 Appendix H. Smooth Manifolds

Usually, these collections come together with a natural topology: wecan define what a neighborhood of a given geometric object m is in a givencollection M.

The characteristic property of a manifold is the ability to describe theposition of a point m E M at least locally by an n-tuple of real numbers(xt x2 x"). These numbers are called local coordinates and identifya neighborhood of m E M with some open domain in lR".

The first important class of manifolds is formed by smooth submani-folds in Euclidean spaces. A smooth submanifold M C RN can be given asthe set of all solutions to a system of equations

(1) Fk(x)=0, 1<k<s,

where FA, are smooth real functions on IRN

Geometrically, the collection of function {Fk, 1 < k < s} can be viewedas a smooth map F : RN 1R8 and the set (1) is precisely the preimageF-1(0) of a point {0} E R.

In fact, not every system of type (1) defines a manifold. A simple suffi-cient condition is provided by

Proposition 1. Assume that the following condition is satisfied:

(2)The rank of the s x N matrix DF = II

8Fk

IIox.,is equal to a constant r in a neighborhood of M.

Then the set M defined by the system of equations (1) is a smooth manifoldof dimension n = N - r.

A local coordinate system in a neighborhood of a given point is providedby a projection of M on a suitable n-dimensional coordinate subspace in RN.

Scheme of the proof. For a given point x E M let us choose the indicesi1, ... , it so that the submatrix formed by corresponding columns of DF(x)is of rank r. Then the Implicit Function Theorem ensures that in someneighborhood U(x) C M the coordinates xi, , ... , x;r can be expressed assmooth functions of the remaining n coordinates.

Remark 1. Actually, this type of manifold is a general one: accord-ing to the famous Whitney theorem, every abstract' smooth n-dimensionalmanifold M can be realized as a submanifold in lR2n.

ISee the next section for the definition of an abstract manifold.

§1. Around the definition 229

Moreover, if M is a compact n-dimensional manifold, then randomlychosen 2n + 1 smooth functions on M give an embedding of M in R2n+l.

(The precise meaning of this informal statement is that for k > 2 the set ofembeddings is open and dense in the space of all k-smooth maps 0: M --R2n+1 endowed with a suitable topology.)

Example 1. Let MM be the subset in Matn(R) defined by the equation

det A = c.

We show that for all c # 0 the set MM is a smooth submanifold in Matn(R) ^-'R n2. For this, according to Proposition 1, we have to check that the partialderivatives

AteBAS,

det A

form a non-zero matrix A. But from linear algebra we know that A A =det A 1. Therefore, for c # 0 and A E MM we have A = c A-' 56 0.

There is another, more general, way to construct a manifold geometri-cally. Let X be some set of geometric objects and consider a quotient setM = X/ - with respect to some equivalence relation -.

Often this equivalence relation is defined via the action of some group Gon X: two points of X are equivalent if they belong to the same G-orbit. Ascoordinates on the set of equivalence classes we can use G-invariant functionson X.

Proposition 2. Let X be a smooth N-dimensional manifold, and let G bea group of smooth transformations of X. Assume that all G-orbits in X areclosed smooth submanifolds of the same dimension k. Then the set of orbitsM = X/G is a smooth (N - k)-dimensional manifold, called the quotientor factor manifold.

Scheme of the proof. Let x E X. Consider an (N - k)-dimensional sub-manifold S C X passing through x and transversal to the G-orbit S2x. Thenfor a sufficiently small neighborhood U x all G-orbits f passing through Uintersect S n U exactly in one point (the Implicit Function Theorem appliedto the map G x S -p X). So, the set of such orbits is identified with S n Uand provides a local coordinate system on M = X/G. 0

Warning. It seems that this proof does not use the assumption thatthe orbits are closed submanifolds. Actually, this assumption is needed toshow that M/G is Hausdorff (see the next section).

Recall the standard example when the assumption does not hold: X isthe torus T2 = R2/Z2 with the action of the group G = R by the formula

t (x, y) = (x + t, y + at)


where x mod 1, y mod 1 are the usual coordinates on T2. If a is rational,the orbits are circles and X/G is a smooth manifold (actually, also a circle).But when a is an irrational real number, all G-orbits are everywhere densein X and X/G is not Hausdorff. 4

Example 2. The n-dimensional real projective space Pn(R) is the quo-tient of the set X = 1Rn+1\{0} modulo the equivalence relation:

x - y x is proportional to y (i.e. x = c y for some real number c).

Clearly the equivalence classes are just the orbits of the multiplicative groupIIY" acting on X by dilation-reflections. Note also that lPn(R) can be viewedgeometrically as the set of all 1-dimensional subspaces in Rn+1

Denote by [x] the equivalence class of x E X and by (xo, x1, ..., xn)the coordinates of x. Then a point [v] E Pn is specified by the so-calledhomogeneous coordinates (xo : x1 : : xn) where the colons are usedto show that only the ratios of the coordinates matter.

Let Ui be the part of Pn where xi # 0. On Ui we introduce n localcoordinates

xi , 0 < j < n, j # i.

The transition functions between the two coordinate systems on U1 n Ukare rational with non-vanishing denominators:

u(k) = u(ii/u(i) for j , i, k, u(k) =

One can embed P"(R) into IRN, N = n+12n+2 using the map

xi xjx H Ekn=0 (xk)2

0Combining the notions of submanifold and of quotient manifold we can

introduce a manifold structure on many sets of geometric origin. This is thereason why we call this approach the "geometric definition of a manifold".

1.2. Abstract smooth manifolds. Analytic approach.In calculus courses we usually consider smooth functions defined on Eu-

clidean spaces Rn or on open domains D C Rn. The crucial role is playedby coordinate systems that allow us to consider a function f on D justas a function of n real variables. There exist more complicated sets thatalso admit coordinate systems at least locally. They are called abstractmanifolds and we give their definition below.


EDFigure 1

Definition 1. A k-smooth n-dimensional manifold is a topologicalspace M that admits a covering by open sets UQ, a E A, endowed with one-to-one continuous maps 6Q : UQ VQ C R' so that all maps cpQ,a = qare k-smooth (i.e. have continuous partial derivatives of order _< k) when-ever defined.

We call the manifold M smooth (resp. analytic) when all the mapsare infinitely differentiable (resp. analytic).

The following terminology is usually used:

the sets UQ are called charts;the functions xR = x' o 0Q are called local coordinates;2the functions cpQ are called the transition functions;the collection {UQ, qQ}QEA is called the atlas on M.

Remark 2. For technical reasons two additional conditions are usuallyimposed.

1. A topological space is called Hausdorff if any two different pointshave non-intersecting neighborhoods.

2. A manifold is called separable if it can be covered by a countablesystem of charts.

In our book we always assume that both conditions are satisfied unlessthe opposite is explicitly formulated.

An example of a non-Hausdorff (and non-orientable!) 1-dimensional man-ifold can be seen in Figure 1.

Note also that the sets of orbits for a non-compact Lie group are usuallynon-Hausdorff. V

Definition 1 above still needs some comments. Namely, a given topo-logical space M can be endowed with several different atlases satisfying allthe requirements of the definition. Should we consider the correspondingobjects as different or as the same manifold? To clear up the situation, weintroduce

Definition 2. Two atlases {UQ}QEA and {U }fEB are called equivalent ifthe transition functions from any chart of the first atlas to any chart of the

2Here {x'}1<;< are the standard coordinates in R^;

232 Appendix 11. Smooth Manifolds

second one are k-smooth (whenever defined). The structure of a smoothmanifold on M is an equivalence class of atlases.

Remark 3. When dealing with a smooth manifold, we usually followa common practice and each time use only one, the most appropriate, atlaskeeping in mind that we can always replace it by an equivalent one.

In practical computations people usually prefer minimal atlases. Forexample, for the n-dimensional sphere S' and for the n-dimensional tonesT" there exist atlases with only two charts.

Sometimes it is convenient to assume that each chart is contractible(i.e. homeomorphic to an open ball). An interesting geometric characteristicof a manifold is the cardinality of a minimal atlas with contractible charts.(For example, for S" it is equal to 2 while for T" it is n + 1.)

In homology theory the special role is played by the so-called Lerayatlases that have the property: all non-empty finite intersections fl.1 Uk

are contractible. The existence of such atlases is not evident but it is knownthat any compact smooth manifold admits a finite Leray atlas. For example,one can use convex open sets with respect to some Riemannian metric.

The maximal atlases are also useful in some theoretical questions. Thereason is that in each equivalence class there is exactly one maximal atlas.It consists of all charts that have smooth transition functions with all chartsof any atlas from the given class. C7

If M1, M2 are k-smooth manifolds, then for any in < k there is a naturalnotion of an m-smooth map f : M1 1b12. Namely, for any point xo E M1and any charts U 3 x, V 3 f (xo) the local coordinates of f (x) in V mustbe m-smooth functions of local coordinates of x in U.

A smooth map from one manifold to another which has a smooth inversemap is called a diffeomorphism. Two manifolds are called diffeomorphicif there is a diffeomorphism from one to another.

Warning. Different smooth manifold structures on a given topologicalspace M can define diffeomorphic manifolds. For instance, any continu-ous invertible map 0 : IR ----+ IR defines a one-chart atlas, hence a smoothmanifold structure on R.

All smooth manifolds obtained in this way are diffeomorphic to the stan-dard R. On the other hand, the atlases corresponding to functions 0 andare equivalent only if 0 o jr' is smooth. 4

Remark 4. For topological (i.e. 0-smooth) manifolds Al of small di-mensions the following phenomenon is observed:

there is exactly one (up to diffeomorphism) smooth manifold M that ishomeomorphic to M.


This statement fails in higher dimensions. The first counterexample forcompact manifolds is the 7-dimensional sphere S7, which is homeomorphicto 28 different smooth manifolds.

The first non-compact counterexample is still more intriguing. Thereexists a smooth manifold that is homeomorphic but not diffeomorphic tothe standard space R4.

Finally, there exist topological manifolds that are homeomorphic to nosmooth manifold at all. The simplest known compact example has dimension8. O

We conclude this section by introducing the notion of orientation. Twocharts with coordinates {x1} and {y' } are called positively related if thefollowing condition is satisfied:

(3) det II8Oyi

II > 0 whenever defined.

We say that two charts are negatively related if the opposite inequalityholds.

Warning. It can happen that two charts are both positively and nega-tively related (when they are disjoint), or neither positively nor negativelyrelated (when their intersection is not connected; see Example 3 below, thecasen=2).

Definition 3. A manifold M is called orientable if it admits an atlaswith positively related charts. We call such an atlas oriented, as well asa manifold endowed with it. The choice of an oriented atlas is called anorientation of M.

Example 3 (Mobius band). Let us consider n copies3 of the openrectangle 0 < x < 3, 0 < y < 1 and glue them together as follows.

Let xk, Yk denote the coordinates on the k-th rectangle. We identifythe part 2 < Xk < 3 of the k-th rectangle with the part 0 < xk+1 < 1 of the(k + 1)-st rectangle according to the transition functions

xk+1 = xk - 2, Ylk+1 = yk, 1 < k < n - 1,

and the part 2 < xn < 3 of the n-th rectangle with the part 0 < x1 < 1 ofthe first rectangle according to the transition functions

xl =xn-2, Yi = -Yn

3The description below makes sense for n > 2. In the case n = 1 it should be slightlymodified. We leave the details to the reader.


rule:

f(m) 1 if UQ(m) and V7(+m) are positively related,

- { -1 if and Vryi,,,l are negatively related.

We leave it to the reader to check that this function is correctly defined(i.e. does not depend on the choice of the charts covering m) and is locallyconstant. Since M is connected, f must be constant.

It follows that the atlas V is equivalent to one of the atlases U, U. 0

1.3. Complex manifolds.The analytic definition of a smooth real manifold has a natural modifica-

tion. Namely, let us consider the charts with complex coordinates z1, ... , z"and require that the transition functions be 1-smooth in the complexsense (which implies that they are analytic). What we obtain is the defini-tion of a complex manifold.

Example 4. The complex projective space lF'1(C) is defined exactly inthe same way as U"1(R) but with complex homogeneous coordinates (z° :zl : ...: zn). We use the fact that the transition functions are rational,hence analytic in their domains of definition. Q

Of course, any complex n-dimensional manifold can be viewed as a realmanifold of dimension 2n. The converse is far from being true.

First, not all even-dimensional real manifolds admit a complex structure.For example, on a sphere Stn such a structure exists only for n = 1.

On the other hand, there could be many non-equivalent complex mani-folds that are diffeomorphic as real manifolds.

Example 5. Let T2 be the 2-torus. Every complex structure on it comesfrom the realization of T2 as the quotient manifold C/L, where L, =is a lattice (discrete subgroup) in C generated by 1 and r E C\R.

It is known that all complex analytic automorphisms of C are affinetransformations. It follows that C/L,1 and C/L,.2 are equivalent as complexmanifolds if

mn + nrlprl + qri where C m

n)E GL(2, ?L).

\\\ P 9 /

So, the set of all complex manifolds homeomorphic to T2 is itself a complexmanifold, the so-called moduli space M1 = (C\R)/GL(2, Z). Q

Warning. Note that the analogue of the Whitney theorem no longerholds for complex manifolds. For example, no complex compact manifolds ofpositive dimension can be smoothly embedded in CN. In the simplest case

236 Appendix II. Smooth Manifolds

Al = P1 (C) it follows from the well-known fact: any bounded holomorphicfunction on C is constant.

One can try to realize a general complex manifold as a submanifold ofpN (C) for large N. But this endeavor also fails: according to the Kodairatheorem all compact complex submanifolds of IPN(C) are algebraic, i.e.given by a system of polynomial equations. Therefore, even complex tori7' = C"/L, where L is a lattice of rank 2n in C", can be embedded in aprojective space only for very special lattices L. 4

1.4. Algebraic approach.There is a more algebraic way to define a smooth manifold M. Namely,

let COC(M) be the algebra of smooth functions on M. The support of afunction f, denoted by supp f , is by definition the closure of the set {m EM I f (m) # 0}.

Denote by A(M) (the notation CC(M) is also used) the algebra of allsmooth real functions with compact support on M. In particular, if M itselfis compact, then A(M) is the algebra of all smooth functions on M.

A remarkable fact is that a smooth manifold M is completely determinedby the algebraic structure of A(M).

Theorem 1. Any non-zero R-algebra homomorphism X : A(M) - R hasthe form

(4) X=X.":f'-'f(m) forsomemEM.

So, we can reconstruct the set M from the algebra A as a set of allnon-zero algebra homomorphisms of A to R.

It turns out that smooth maps from one smooth manifold to anothercan also be described algebraically. In the case of compact manifolds thesituation is very simple: smooth maps 0: M -p N correspond to non-zeroalgebra homomorphisms 1 : A(N) -p A(M) by the formula (5) below. Toinclude the case of non-compact manifolds we need two additional defini-tions.

1. A smooth map 46: N --+ Iv! is called proper if the preimage of anycompact set K C M is a compact subset in N.

This condition is automatically satisfied if N is compact. Note thatthere are no proper maps from a non-compact manifold to a compact one.

2. An algebra homomorphism 4 : A B is called essential if itsimage 4(A) is not contained in any ideal of B different from B itself.


If A has a unit, then fi is essential iff B also has a unit and fi is a unitalhomomorphism.

In particular, there is no essential homomorphism from an algebra withunit to an algebra without unit.

Theorem 2. Let M and N be any smooth manifolds. Then the relation

(5) 'DW = f 0 0

establishes a bijection between the proper maps 0: N - M and the essen-tial algebra homomorphisms fi : A(M) -' A(N).

In particular, there is a bijection between points of M and non-zerohomomorphisms 0: A(M) - R.

The main step in the proofs of Theorems 1 and 2 is

Lemma 1. Let X : A(M) R be a non-zero homomorphism. Consider theideal Ix C A(M), which is the kernel of X. Then there exists a point m E AIsuch that all functions f E Ix vanish at m.

Proof. Assume that there is no such point. Then for any m E M there isa function fn E Ix such that frn(m) 0 0. But in this case f. is also non-vanishing in some neighborhood Urn of m. Let g E A(M) be an arbitraryfunction. Since the set S = suppg is compact, we can choose finitely manypoints m1i ... , mN so that the corresponding neighborhoods Urm;, 1 < i <N, cover all of S. Then the function F(x) :_ > i f, (x) belongs to I. andis everywhere positive on S. Hence, - E A(M) and g = F f E Ix. acontradiction to x # 0.

In conclusion we give an example of a non-essential homomorphism.Example 6. Let A = A(R), B = A(S' ). Choose on R the real coor-

dinate x and on S1 the complex coordinate z, Izj = 1. For f E A(R) weput

t1'(f)(z)=fIi-l+zf(tan if z=e't,Itl<n.

Then the image 4'(A(R)) consists of functions on S' that vanish in someneighborhood of the point P = -1 and is a non-trivial ideal in A(S'). So,0 is non-essential.

The homomorphism 0 does not correspond to any map ¢, : S' - R.Indeed, the map xp o 4' is identically zero, hence the image O(P) is notdefined. 0


The algebraic definition given above can be formulated in categoricallanguage as follows. Let Aig (K) be the category whose objects are com-mutative associative algebras over a field K and whose morphisms are K-algebra homomorphisms. Then the category Man of smooth manifolds isjust a subcategory of Alg (R)°.

It is tempting to consider all objects of Alg (K)° as generalized mani-folds. In other words, the following principle is proclaimed:

Any commutative associative algebra A

is an algebra of functions on some "manifold" Spec(A).

This principle suggests the geometric interpretations of many other purealgebraic notions, e.g.,

jecti d l LA f ti f t b dl Lvepro -mo u e space o sec ons o a vec or un eover Spec(A);

Lie algebra Vect (Spec(A)) -- Lie algebra of derivations of A; etc.

This point of view has been basic to modern algebraic geometry sincethe seminal works by Grothendieck.

Last time it became popular to also consider non-commutative alge-bras in this context (non-commutative algebraic geometry, or the theory ofnon-commutative manifolds). Two special cases are of greatest importancebecause of the role they play in modern mathematical physics:

a) supermanifolds, where the set of functions forms a Z2-graded super-commutative algebra;

b) quantum groups, which can be defined as group-like objects in thecategory of non-commutative manifolds.

We only briefly mention them in our lectures but the extension of theorbit method to these new domains is a very challenging problem that isonly partly resolved (see, e.g., [QFS], [So], [Ka2]).

2. Geometry of manifoldsWe assume that the reader is acquainted with the elements of differentialgeometry. Here we only recall some notions and facts using all three ap-proaches: geometric, analytic, and algebraic, which were described above.

2.1. Fiber bundles.A manifold X is called a fiber bundle over a base B with a fiber F

if a smooth surjective map p : X ---' B is given such that locally X is adirect product of a part of the base and the fiber.

More precisely, we require that any point b E B has a neighborhood Ubsuch that p-1(Ub) can be identified with Ub x F via a smooth map ab so that


the following diagram (where pl denotes the projection to the first factor)is commutative:

p-1(Ub) Ub x F

Pt tPt

Ub Ub.

From this definition it follows that all sets Fb := p-1(b), b E B, called fibers,are smooth submanifolds diffeomorphic to F.

We use the notation

(X, B, F, p) or F X --!+ B or else X F+ B

for a fiber bundle with the total space X, base B, fiber F, and projection p.Sometimes fiber bundles are called twisted or skew products of B and

F. The reason is that a direct product X = B x F is a particular kind offiber bundle: one can put U = B, a = Id. We observe also that B and Fplay non-symmetric roles in the construction of a skew product.

The collection of all fiber bundles forms a category where the morphismsfrom Xl L Bl to X2 IL B2 are pairs of smooth neaps (f : Xl -i X2, 6:B1 - B2) such that the following diagram is commutative:

X1 1 X2

1P2

B1 B2.

The objects that are equivalent to a direct product are called trivial bun-dles.

A map s : B X is called a section of a fiber bundle X a B ifp o s = Id. In the case of a trivial bundle the sections can be identified withfunctions f : B --- F. So, the sections of a fiber bundle X - B give anatural generalization of the notion of a function from B to F.

Example 7. The subcategory of fiber bundles with given base S' andgiven fiber R1 has only two equivalence classes of objects: the cylinderS1 x RI (a trivial bundle) and the Mobius band M (a non-trivial bundle).

0A fiber bundle (X, B, F, p) is called a vector bundle if F and all

Fb, b E B, are vector spaces (real or complex) and all maps at,, b E B, arelinear on the fibers.

We now consider in more detail the tangent bundle TM, the most impor-tant example, which was the source of the whole theory of vector bundles.


A tangent vector to a manifold M at the point in E M can bedefined in three different ways.4

a) Geometrically, as an equivalence class of smooth parametrized curvesin Al passing through m. Namely, two curves x(t) and y(t) are called equiv-alent if

1. x(0) = y(0) _ in;

2. lx(t) - y(t) I = o(t) in any local chart covering m.b) Analytically, as an expression En 1 a` a

,,in some local coordinate

system centered at in.c) Algebraically, as a linear functional on A(M) satisfying the condition

(6) M(fg) = W)g(m) + f(m)Vg)

The explicit correspondence between these definitions looks as follows.A vector a = {a`} E Rn corresponds:

a) to the equivalence class of curves x(t) for which dx'(0) = a';

b) to the expression a` a 57;

c) to the linear functional f H E 1 ai e (m).We denote by Tm M the set of all tangent vectors to M at in and call it

the tangent space. The union TM := Umem TmM forms a fiber bundleover Al -with the projection p that sends all elements of Tm.M to in.

Indeed, if U C Af is a local chart with coordinates (x', ... , xn), thenany tangent vector a E TxM can be written as a = ak8k where ak = ak(x)are numerical coefficients and 8k := 8/C xk denotes the partial derivative(which is a tangent vector in the sense of the third definition).

So, the set p-1(U) is identified with U x Rn. It is by definition a localchart on TM with coordinates (x', xn: a', ,

an).

The (smooth) sections of TM are called (smooth) vector fields on M.They also admit several interpretations:

a) Geometric: as generators of flows, i.e. one-parametric groups {4t} ofdiffeomorphisms of M defined by the ordinary differential equation (wherethe dot denotes the time derivative)

-i't(m) = Vin)-

Here one has to suppose M is compact or restrict the behavior of the vectorfield at infinity: otherwise the flow {fit} exists only locally.

'We suggest that the reader compare the mathematical definitions below with the physicaldefinition of a tangent vector as a velocity of a point moving along a given surface.


b) Analytic: as differential operators E a' (x) ei, in each local coordinatesystem on M with the natural transformation rule under the coordinatechanging.

Namely, the vector field F_ a' (x) dx, in another chart with local coordi-

nates {yl, 1 < j < n} looks like E, an(y)-, where &4 (y) a a'(x(y)).c) Algebraic: as linear operators v on A(M) satisfying the Leibnitz

rule:

(7) v(f9) = v(f) . g + f - v(9)Such operators are called derivations of the algebra A(M).

We denote the space of all smooth vector fields on M by Vect(M). It isa module over the algebra A(M) in an obvious way.

Exercise 2. Check if this is really obvious for you. Describe explicitlythe module structure in the analytic and algebraic interpretations. 46

The second important example of a vector bundle is the so-called cotan-gent bundle T*M, which we now discuss.

A differential 1-form, or a covector field on Al, is defined in twoways:

a) Analytically, as an expression FL 1 bi(x)dx' in each local coordinatesystem on M with the natural transformation rule under the coordinatechanging. Namely, in terms of other local coordinates {y', 1 < j < n} our1-form looks like >j b,(y)dy2 where ba(y) = >i bi(x(y)) ;.

b) Algebraically, as an A(M)-linear map from Vect(M) to A(M).Indeed, any such map has the form Ei a'(x)az, H Ei a'(x)bi(x), hence

corresponds to the form E , bi(x)dx'.We denote the space of all smooth covector fields on M by f'(M). It

is also a module over A(M) that is naturally dual to Vect(M). In any lo-cal coordinate system (U, {x'}1<j<n) the quantities {8i := a,, 1 1<i<n and{dx'}1<i<n form dual bases in the A(U)-modules Vect(U) and Q '(U), re-spectively.

We can interpret a local expression w = widx' of a covector field as asection of some vector bundle 1R -) T*M - M. The fiber of this bundleover a point m E M is naturally identified with the dual vector space toT,,,M. This is why T*M is called the cotangent bundle to Al.

The space Slk(M) of smooth differential k-forms on Al can be definedas the k-th exterior power of the A(M)-module Q1 (M). In a local coordinatesystem a k-form w looks like

(8) w = E wi...... ik (x)dx" A ... A dx'kii....,ik


where all indices is, 1 < s < k, run from 1 to n and the wedge product A isbilinear, associative, and antisymmetric.

The geometric meaning of differential forms is related to integration overmanifolds and will be discussed later.

We also define the space Vectk(M) of smooth polyvector fields on Mas the k-th exterior power of the A(M)-module Vect(M). Locally, a k-vectorfield c is given by an expression

(9)C = [1 et,,...,tk(x)aj1 A ... A Oik.

Both IZk(M) and Vectk(M) can be interpreted as spaces of smooth sectionsof vector bundles AkT(141) and AkT*(M), respectively.

Remark 5. We shall usually follow Einstein's rule and omit thesummation sign in expressions like (8) or (9) where a letter occurs twice asa lower and an upper index. For example, we write a vector field v as viOiand a differential 1-form w as widxi. The value of w on v will be the functionw(v) = wivi.

This notation not only saves space but sometimes suggests the rightformulation of a theorem or a solution to a problem. Consider, for exam-ple, Newton's Second Law, which relates the force 1, the mass m, and theacceleration a:

(10) f =m.a.

If we take into consideration that geometrically the force is a covectorfield (i.e. a differential 1-form) and the acceleration is a vector field (thetime derivative of the velocity), we see that (10) is a "wrong" equation: theleft-hand side has the lower index while the right-hand side has the upperone.

A way to correct equation (10) is to replace the scalar m by the tensorfields mi3 with two lower indices and rewrite it in the form

(10') fi = mija3.

The most common geometric example of a tensor with two lower indicesis the Riemann metric tensor gig. We conclude that the mass must be relatedto the metric, which is the main principle of General Relativity! C1

We finish this section with a pure algebraic definition of a vector bundle.

SSee the next section for the definition of tensor fields.


Lemma 2. Let M be a compact manifold, and let L vi M be a vectorbundle over M with a finite-dimensional fiber V. Then the space 1'(L, M)of all smooth sections of L is finitely generated as an A(M)-module.

Proof. For any point m E M we can find a neighborhood U such thatL is trivial over U. Then r(L, U) f-- C°°(U, V) is a free module of rankn = dim V over COO(U) with the basis vi, ..., v,,. Let W be a function inA(M) with supp p C U and cp(m) # 0. The sections si = cp v=, 1 < i < n,of L over U can be extended by zero to the whole M and take linearlyindependent values in any point of a smaller neighborhood U' C U whereWOO.

Since M is compact, we can choose finitely many, say N, such neigh-borhoods that cover all of M. So, we construct nN sections whose valuesat any point m E M span the whole fiber V,,,. It follows that they generateI'(L, M) as an A(M)-module. 0

The lemma implies that the A(M)-module I'(L, M) is a submodule of afree module of finite rank. (Geometrically, it means that our vector bundleL can be considered as a subbundle of a trivial bundle L with a finite-dimensional fiber V.)

Actually, it is not only a submodule, but a direct summand of L. Suchmodules are called projective. So, we get the algebraic definition of avector bundle L: the space F(L, M) is just a projective module over A(M).

2.2. Geometric objects on manifolds.We start with a general and somewhat abstract definition. Soon it will

be clear that actually it is quite a working approach. Consider the cate-gory C whose objects are smooth manifolds and whose morphisms are openembeddings.

A natural vector bundle is a functor L from C to the category ofvector bundles such that L(M) is a vector bundle over M.

Elements of natural vector bundles are called geometric objects anda section of a natural vector bundle L(M) is called a field of geometricobjects on M.

Vectors and covectors are examples of geometric objects while polyvectorfields and differential forms are examples of fields of geometric objects.

It follows from the definition that a canonical action of the group Diff(M)of all diffeomorphisms of M is defined on sections of L(M). More simply,there exists a canonical way to change variables in the local expression fora given geometric object.


For the examples above the algebraic approach allows us to define theaction of Diff(M) in a uniform way. Namely, the group Diff(M) acts on thealgebra A(M) by algebra automorphisms:

(11) O*(f) = f o¢ for 0 E Diff(M), f E A(M).

Warning. Actually, (11) is a right action and should be written asf ¢. But we prefer to keep the traditional notation 0*(f ). Note, however,that the confusion between left and right actions may lead to a discrepancyin notation. For example, in some books the definition of Lie brackets oftwo vector fields has the sign opposite to ours. 4

All other transformation laws follow from (11) because our geometricobjects are defined in terms of the algebra A(M). Recall that we definedVect(M) as the set of all derivations of A(M) (with the canonical A(M)-module structure), Q1 (M) as a dual A(M)-module to Vect(M), and f (M)and Vectk(M) as corresponding exterior powers. So, any automorphism ofA(M) automatically defines automorphisms of all these objects.

There are more general geometric objects called tensors. Analytically,we define a general tensor field of type (k, 1) on M as an object T, whichis locally given by coefficients tai:::i'.(x) with the following transformationlaw:

(12)8xP1 axPk syJt syJtJ°Jt q,...91

i,...ik (y) = tP,...Pk(. WO syi, ...ayik axgi

...ax,,,

P1 ... Pt

To remember this rather involved rule, just keep in mind that from thecoefficients of a tensor using the Einstein rule we can construct the expression

a aT = t-i" *' k (x)dxi' ®... ®dxik

®axj-,

®...®axj,'

which has an invariant meaning independent of the local coordinate system.Algebraically, the set T(k, 0)(M) of all tensor fields of type (k, 1) on M is

the tensor product over A(M) of k copies of 01(M) and 1 copies of Vect(M).It can also be defined as the A(M)-module

HomA(M)(Vect(M) x . . . x Vect(M) x S21(M) x . . . x Sl1(M),A(M))

k times I times

of all maps that are multilinear over A(M).Observe that k-vector fields and differential k-forms are particular cases

of tensor fields (antisymmetric tensor fields of types (0, k) and (k, 0), re-spectively).


All tensor fields possess the following properties:

1) They are transformed linearly under the change of variables.2) In their transformation law only the first derivatives of the new vari-

ables occur with respect to the old ones or vice versa.Therefore tensor fields are called linear geometric objects of first

order. Actually, they are almost all objects of this type.More general linear geometric objects of first order are the so-called

tensor densities (of two kinds) of degree A E C. The precise definition is asfollows.

A tensor density of the first kind that has type (k, 1) and degree Ais a geometric object with the same type of coordinates as a tensor field oftype (k, 1). But the transformation law of these coordinates, as comparedto (12), contains the additional factor: the A-th power of the absolute valueof the Jacobian J: = 1 1 .

The definition of a tensor density of the second kind is obtained byadding one more factor in the transformation law: the sign of the Jacobian.

If we consider the category of oriented manifolds with orientation pre-serving diffeomorphisms, then there is no difference between tensor densitiesof first and second kind.

We discuss now in some detail the classification problem for linear geo-metric objects of first order. For technical reasons it is convenient to startwith connected oriented manifolds.

It is known that for any connected manifold M the group G = Diff(M)acts transitively on Al. So, any natural vector bundle L = L(M) is ahomogeneous G-bundle. In Appendix III we show that any homogeneousG-bundle is determined by the action of the subgroup H = Stab(in) C Gon the fiber over a point m E Al. For geometric objects of first orderthis problem is reduced to the description of all finite-dimensional linearrepresentations of the connected Lie group GL+(n. R) = {g E GL(n, R)det g > 0}. The most interesting and important objects are related toirreducible representations. So. for the reader's convenience, we collect thenecessary information here.

Here the following notation and terminology are used:G = GL+(n, R),H - the abelian subgroup of diagonal matrices in G;Or. V) - a finite-dimensional complex irreducible representation of G;weight vector - a vector v E V that is a common eigenvector for all

operators ir(h), h E H;


weight - a vector p = (µi, ... , pn) E C' satisfying the condition µk -pj E Z for all k, j;

dominant weight - a weight satisfying the additional condition

(13) Pk-Ai>0 forallk>j;we say that a weight p. is bigger than v if p. - v is a dominant weight;

eµ(h) or. simply, hu - the homomorphisrn of H to CX given by

n

(14) eµ(h) = 11 hr-'`" (det h)'`" for It = diag(hi,..., hn) E H;k=i

a weight vector v E V is said to be of weight p if ir(h)v = hµv for allh E H.

Proposition 4. Let (7r, V) be a finite-dimensional complex irreducible rep-resentation of the group G = GL+(n, R). Then

a) the space V has a basis consisting of weight vectors; the correspondingweights form the multiset Wt (7r) of weights of ir; the multiplicity of a weightp, is denoted by m,y(p);

b) among the weights of it there is exactly one maximal weight; it hasmultiplicity 1 and is called the highest weight; the corresponding weightvector v E V is characterized by the property of being invariant under alloperators sr(g) where g is a strictly upper triangular matrix:

g12 ... gin

I ... 92n

c) equivalent representations have the same highest weight; the represen-tation (ir', V*), dual to (2r, V), has the highest weight

(15) A" = (-An, ..., -AI);

d) for any dominant weight ,\ = (Ai, ..., An) there is exactly one (up toequivalence) irreducible representation Ira with the highest weight A;

e) any real N-dimensional irreducible representation (ire. Va) of C pro-duces a complex representation (ir, V) with V = Va OR C; the representation(7r, V) is either irreducible with a real highest weight A (then the initial rep-resentation (7ro, VO) is called absolutely irreducible), or a sum of twoirreducible representations with complex-conjugated highest weights A and1 0


It follows from the above proposition that the equivalence classes ofabsolutely irreducible real representations of GL+(n, 1R) are labelled by realhighest weights. It turns out that all geometric objects corresponding tothese representations are tensor densities of certain symmetry type withrespect to permutations of indices.

We list here the highest weights of representations related to the mostimportant geometric objects:

geometric objects highest weights

functions (0. 0. ... , 0, 0)

vector fields (0, 0, ... , 0. -1)k-vector fields (0, 0. ..., 0,-1, -1, ..., -1)

k times

differential k-forms (1, 1, ... , 1, 0, 0, ... , 0)k times

symmetric tensors of type (k, 0) (k, 0, ... , 0)

.A-th power of a volume form (A, A, ... , A)

The general situation, when we consider non-orientable manifolds andgeneral diffeomorphisms, is related to the representations of the full lin-ear group GL(n, IR). Here again the absolutely irreducible representationscorrespond to tensor densities of the first or second kind.

An example of a geometric object with e = 1 is the so-called axial vectorfield (magnetic vector field in physics), which changes the sign when theorientation is changed.

2.3.* Natural operations on geometric objects.A powerful tool used to study geometric objects is the natural op-

erations on these objects, i.e. algebraic and differential operators whichcommute with the action of all diffeomorphisms. Therefore it would be veryuseful to know all the natural operations on geometric objects.

But this problem is rather complicated. An even more specialized prob-lem, to describe all polylinear natural operations on tensor fields, is solvedonly in the simplest cases of linear and bilinear operations.

Let us discuss some properties of natural operations.

Using translation invariance, one can show that in any local coordinatesystem a polylinear natural operation on tensor fields is given by a polydif-ferential operator with constant coefficients.

Moreover, if we consider polylinear natural operations on tensor fieldsof given types -r1 = (k;, 1;), then the following Finiteness Theorem holds.


Theorem 3 (see [Ki8] or [KMS]). The vector space £(r1, r2, ..., TM; r)of all natural polydifferential operators from TT' x x T" to T r is finitedimensional.

Now we list some known polylinear natural operations.

The only example of a linear natural operation of differential order 1 isthe exterior differential d acting on differential forms. There is no invariantlinear operation of differential order > 2. (In particular, this implies thatd2 = 0.) There are many bilinear natural operations of differential order 1.The full classification was obtained in [Grz].

Here we discuss three of the most important examples.1. The Lie derivative. For any vector field v E Vect (M) and any

space of geometric objects T(M) the operator L in T(M) is defined asLt, = d4 (t) It=o. Here the diffeomorphism fi(t), denoted also by exp tv, isdefined as a map x '-. x(t) where x(t) is the solution to the equation

i(t) = v(x(t)), x(0) = X.

(Actually, the solution exists only locally, but it is enough to define thediffeomorphism 1(t) in a neighborhood of a given point x E M for small t.This in turn is enough to define L,,.)

Note the three particular cases of the Lie derivative:a) the ordinary derivative v f of a function f along the vector field v is

justb) the so-called Lie bracket of two vector fields is [v, w] =

-Lwv;c) the Lie derivative L defines a derivation of degree zero of the graded

algebra 1(M) = ®k'_oMQk(M). There are two other derivations: theexterior differential d of degree +1 and the interior multiplication i, by avector field v of degree -1. These three derivations are related by the

Cartan Formula.

(16) L = [d, i o d.

Proof. The right-hand side is exactly a supercommutator (see item 3below for the definition and properties of "superobjects") of two odd deriva-tions d and i,,. Therefore, it is an even derivation. Since both sides arederivations of 1(M), it is enough to check (16) on generators, i.e. on formsof type f E Q0 (M) and df E 1ll(M).


Using the relations i f = 0 and d2 = 0 we obtain

L,,f =vf(do iv +i o d)df.

0

A very useful property of the Lie derivative is the

Generalized Leibnitz rule. If F is any k-linear natural operation ongeometric objects, then

k

(17) Ak) _ F(Al1 .... L,Ai, ..., Ak).i=1

Example 8. Consider the differentiation of an object A along the vectorfield v as a bilinear operation F(v, A) = Then the generalized Leibnitzrule gives:

L,,F(v, A) = F(L,,,v, A) + F(v, L,,,A) or L[w,v] = LwL,. -

So, the Lie derivative defines the action of the Lie algebra Vect(M) on thespace of geometric objects of a given kind. 0

Exercise 3. Using the generalized Leibnitz rule, derive the followingformula for the exterior differential of a k-form w:

(18)

k

dl(t'O,...,vk)=E(-1)'viw(vo,...,2'i,...,vk)i=O

+ E (-1)i+jw([Z'i. vj], v0, ... , vi, ... , vk).O<i<j<k

For future use we write the important particular cases k = 0, 1, 2 sepa-rately:

(19)df f. 77) = 0(i7) - r70() - 9(1., 771),

dw(C 71, C) = O fw(7i C)- C) w([ . 771, C)

where the sign 0 denotes the sum over cyclic permutations of t;, 77, (. 42. The Schouten bracket is a bilinear operation on polyvector fields on

a manifold. It defines on the graded space of polyvector fields the structureof a so-called graded Lie algebra, or Lie superalgebra, which extendsthe structure of a Lie algebra on Vect(M).


Proposition 5. There exists a unique bilinear operation [ , J on the spaceof polyvector fields which has the properties:

a) For any v E Vectk(M), w E we have

[u, v] E Vertk+t-1(M) (Homogeneity)

b) [u, v A w] _ [u, v] A w + (-1)kv A [u, w] (Distributive Law)

c) For k = l = 1 the operation is the ordinary Lie bracket of vector fields.

Moreover, this operation is natural (i.e. commutes with all difeomor-phisms) and satisfies the identities:

[u, v] = -(-1)(k-1)(1-1)[v, u.] (Super antisymmetry)

[u, [v, w]] = [[u, v], w] + (-1)(k-1)(1-1)[v, [u, w]] (Super Jacobi iden-tity).

Proof. The uniqueness follows immediately from properties a), b), and c).Indeed, these properties imply that for decomposable polyvector fields

v and w the operation is defined by the formula

(20) [vi n...AVk, wi A...Aw!]_[-`(-1)i+i+k-1[v:, wj]Avi ...Av,A...AVkAwi

e.j

A...AgA...AW1.

We could use (20) as a definition of our operation, but we need to checkthat it is correct, i.e. does not depend on the way a polyvector field as asum of wedge products of vector fields is written.

For this end one can use an alternative definition of the Schouten bracket.Fix a volume form vol on our manifold and define a map 5 : Vect k(1111)

Vectk-1(M) by the formula

(21) 5(v=i.",k8=1 A ... A 8Zk) = 81v" ...=k813A ... A 8=k

in the special local coordinate system where vol = dx1 A. Adz'. (Such coor-dinate systems are called unimodular.) For k = 1 the map S : Vect(M)COO(M) is the usual divergence of a vector field.

The second definition of the Schouten bracket is

(22) [v, wJ = 5(v A w) - 5(v) A w - (-1)kv A 5(w) for v E Vect'(111).

This definition formally depends on the choice of a volume form vol butactually it does not.

Exercise 4. Show that the right-hand side of (22) does not change ifthe volume form vol is multiplied by a function p.


Hint. Check that the map 6' corresponding to the form vol' = p vol isrelated to the initial map 6 by the equation

k

6'(v) = 6(v) - j:(-l)-vi A ... A i A ... A Vk

i=1

where the hat means that the factor must be omitted. 4It remains to check that (20) and (22) define the same operation and

this operation has the announced properties. We leave this to the reader.0

Remark 6. Observe that the homogeneity property of the Schoutenbracket is related to the grading in the space of polyvector fields in a non-standard way.

To get the standard homogeneity property, we have to assign the gradingk - 1 to k-vector fields.

It is also tempting to extend the bracket operation to a bigger space byincluding S1k(M) as a homogeneous component of the grading -1- k. 4

3. The Nijenhuis bracket is defined for so-called vector differen-tial forms, i.e. the elements of Vect(M) 00.(m) I(M). The raison d'etreof this operation is the algebraic structure of I(M) as an associative super-commutative differential algebra.

Recall that a Z-graded associative algebra A is called supercommuta-tive if for homogeneous elements a, b E A we have ab = (-1)°'0 ba wherea=deg a, 6 =degb.

A derivation of degree k of a commutative superalgebra A is a linearmap D : A -' A which sends A' to Am+k and satisfies the super Leibnitzrule:

D(ab) = D(a)b+(-1)k'aD(b), a = dega.

We now return to differential forms. Put A = S1(M) and observe thatthis is a commutative superalgebra with the derivation d of degree +1.

We shall consider only those derivations D of S1(M) which supercommutewith d:

D(dw) = (-1)deg Dd(Dw).

From this property and the Leibnitz rule we see that D is completelydetermined by its restriction to 11°(M) = C°°(M). The image of this restric-tion is contained in f1deg D(M). Moreover, from the Leibnitz rule applied tothe case of the product of two functions, we get

(23) D f = wi8i f where wi E Qdeg D(M), 1 < i < n.


One can verify that any collection of k-forms {wt}1<i<n gives rise to a deriva-tion of degree k whose restriction to Q°(M) is given by (23). So, these deriva-tions can be identified with geometric objects from Vect(M)®Coo(M)1lk(M) :D «-+ 8®®w' .

The derivations of a graded superalgebra form themselves an algebrawith respect to the supercommutator

(24) [DI, D2] = Dl o D2 - (-1)degD1 degD2D2 o Dl.

The right-hand side of (24) is a derivation of degree deg Dl + deg D2 andthe supercommutator satisfies the super Jacobi identity:

(25) [DI, [D2, D3]] = [[D1, D2], D3] + (-1)degD1degD2[D2, [Di, D3]]

In our case this supercommutator is exactly the Nijenhuis bracket.When deg Dl = 0 or deg D2 = 0 we regain the Lie derivative. The sim-

plest new operation arises when deg Dl = deg D2 = 1. The correspondinggeometric objects can be interpreted as fields of linear operators in the tan-gent spaces TIM. They can also be considered as A(M)-linear operators onVect(M).

The Nijenhuis bracket of two such operators is a vector 2-form that canbe interpreted as a family of bilinear operations ("multiplications") in TIMor as an A(M)-bilinear operator from Vect(M) x Vect(M) to Vect(M).

In this case the operation is symmetric: [A, B] = [B, A]. Therefore itis determined by the corresponding quadratic operation: A i--+ [A, A]. Thelatter admits a simple explicit formula:

(26) [A, rl) = [Ac, An] - A771 - A[Ae, 17] + A2[e, 77].

It shows that the resulting bilinear operator [A, A] is antisymmetric in l;, 77.Exercise 5. Check directly that the right-hand side of (24) is bilinear

in 1; and 77 not only over R but also over A(M).

Hint. Use the formula [ f , rl] = f 71] - i7f l;, which follows from thegeneralized Leibnitz rule. 4

Many important geometric theorems use the just described case of theNijenhuis bracket in their formulations. We mention two examples here.

a) Recall that an almost complex structure on M is defined by atensor field J E T" satisfying

JjJk = -4 (or simply J2 = -1).

It introduces a complex structure on every tangent space TIM.


Clearly, each complex manifold M possesses an almost complex structurebut the converse is true only under an additional integrability condition.

b) The so-called almost product structure on M is given by a tensorfield P E T""l satisfying

P Pk - Pi (or simply P2 = P).

It defines on every tangent space TTM the structure of a direct sum: TTM =P(TXM) ® (1 - P)(TTM). This structure arises each time M is a productof two manifolds: M = Ml x M2. Here again, the converse is true (evenlocally) only under an additional integrability condition.

Proposition 6. The integrability condition for an almost complex structureJ (respectively for an almost product structure P) has the form

(27) [J, J] = 0 (resp. [P, P] = 0).

Proof. The necessity of (24) is easy: if J comes from a complex struc-ture (respectively P comes from a product structure), then in appropriatecoordinates this tensor has constant coefficients. Hence, [J, J] = 0 (resp.[P. P] = 0).

Conversely, let [J, J] = 0 (resp. [P, P] = 0). Consider the complex geo-metric distribution A = ker (J - il) (resp. the real geometric distributionB = ker (P - 1)). From (26) we derive that the space Vect(M, A) of admis-sible vector fields is a Lie subalgebra in Vectc(M) (resp. Vect(M, B) is aLie subalgebra in Vert(M)). Hence, the distribution satisfies the Frobeniuscriterion (see Section 3.4 below). 0

2.4. Integration on manifolds.Here we recall the basic facts about integration of differential forms

and densities over manifolds.

In calculus courses you are taught to integrate functions. Actually, thisis not quite correct. The object you are dealing with is not a functionintegrated over a set, but a differential form of top degree integrated overan oriented manifold or a density of the first kind and degree 1 integratedover an arbitrary manifold.

This is stressed by the notation suggested by Leibnitz 300 years ago (andsurvived to the present time!):

Jbor ff(x)IdxI.


In the first case we have an integral of a 1-form f (x)dx over the orientedsegment [a, b]. In the second case we integrate a density f (x)l dxl over anon-oriented real line.

Both types of integration have their advantages and disadvantages.The first one relates the analytic notion of the exterior differential of

a differential form with the geometric notion of the boundary of a smoothmanifold. The famous Stokes formula,

(28) fm dw=. w'

is a very powerful method both to compute integrals and to study the ge-ometry of smooth manifolds.

In this formula we assume that the orientations of M and 8M are relatedin a definite way. Namely, if a neighborhood of a boundary point of M iscovered by a positive chart with coordinates (x1, ... , x'a), satisfying theinequality x' < 0 for interior points and the equality x' = 0 for boundarypoints, then the coordinates (x', ..., x"-1) form a positive chart for 8M.This definition does not work for n = 1 and needs to be slightly modified.We leave it to the reader.

The second one has a natural generalization where a smooth manifoldM is replaced by an arbitrary set X and the density IdxI is replaced by ameasure p defined on some collection of subsets in X. In most applications Xis assumed to be a locally compact topological space and u a Borel measuredefined on all Borel subsets.6

Remark 7. The expressions fX f (x) and f f (x), often occurring instudent's notes, do not make any sense. Try, for example, to define whatthe integral of the function f - 1 over the edge of the table is. The possibleanswer: "the length of the table" leads immediately to the next question:in which units is the length measured?

The missing part dx just shows the scale, introducing a parametrizationof the geometric object (e.g. the edge of the table) by a segment of the realline.

Now we give the general definition of the integral of the first kind. LetM be a smooth oriented manifold and w a differential form of the top degreem = dim M. We assume that w has compact support (i.e. vanishes outsidesome compact subset K C M). Our goal is to define the integral fm w.

6Recall that Borel subsets form a minimal collection of subsets in X that contains all opensets and is closed under set-theoretic operations: countable union, countable intersection, andtaking the complement.


First, we reduce the general problem to the simple case when our man-ifold is a bounded domain in R'. To this end we choose a covering of ourmanifold M by local charts {Ua}aEA with the following properties:

1. all Ua are pairwise positively related and define the orientation of M;

2. all Ua have compact closures in M;3. any compact subset K C M intersects only finitely many Ua.We do not prove the existence of such a covering here. In all practical

situations it is rather obvious.Further, it is known that in this situation there exists a system of func-

tions {qa}aEA such that

0.EA(M), 0a>0, supp0aCUa,aEA

This system of functions is called a partition of unity subordinated to thegiven covering {Ua}aEA

We define the integral in question as a sum over a E A of the integrals

Ia= f 0a-W.Ga

Note that actually only finitely many summands are different from zero.Using local coordinates (x1.... , in Ua, we write Oa -w as f (x)dx,7

and define Jr., as the integral of a smooth function f (x) over a domainVa C R". The latter integral is defined as the limit of Riemann integralsums. Namely, we split Va into small pieces Vi, choose a point xi E Vi, andintroduce the sum

S(f; {Vi}, {xi}) = > f (xi) vol(V ).i

Since the integrand is continuous and has compact support, the integralsums indeed have a limit when the maximal diameter of the parts { V } tendsto zero.

It remains to check that the sum EaEA Ia does not depend on the choiceof a covering {Ua} and a partition {0a}. Let {U,}QEB be another cover-ing with the same properties, and let {?I'$}REB be the partition of unitysubordinated to it.

Lemma 3. Put I" = w. Then

1: Ia=EI$.aEA IEB

7Here d"`x is a short expression for dxi A ... A dxm

256 Appendix IL Smooth Manifolds

Proof. Consider the third covering {Uy }.yEr, which is finer than either ofthe previous ones. It is convenient to put r = A x B and U.'y' = UaflUU for y =(a,#). Let X, = d 'a for y = (a, 0). Then {X,},Er is the partition ofunity subordinated to {U4'}yEr Put I, = fe,, X.) - w.

We live it to the reader to check the equalities:

Ia = 1 (Q,J)'8EB

Ia = 2(a.8)' I:= F, I'ry=EIa=Eio.aEA ,YEr aE.4 $EB

11

3. Symplectic and Poisson manifolds

3.1. Symplectic manifolds.By definition, a symplectic manifold is a pair (M, a) where M is a

smooth manifold and a E 112(M) is a non-degenerate closed differential 2-form on Al. In a local coordinate system a has the form aij(x)dxi A dx'where a = Ijai,j(x)II is a skew-symmetric non-degenerate matrix. Sincesuch matrices exist only in even-dimensional vector spaces, all symplecticmanifolds are even-dimensional.

Darboux Theorem. Any non-degenerate closed differential 2-form in anappropriate local coordinate system (pl, .... p, q1, ..., q") has the follow-ing canonical form:

(29) a=dpindq'0

This theorem shows that the notion of symplectic manifold is an oddanalogue of the notion of a Riemannian manifold with a flat metric.

The coordinates (pi, qJ) above are called canonical coordinates. Theyare defined up to so-called canonical transformation, i.e. diffeomorphism,preserving the basic form a. Such transformation% are also called symplec-tomorphisms.

We call a vector field v symplectic if it defines an infinitesimal auto-morphism of (Al, a), i.e. if L,,a = 0. In other words, the flow generated onAl by a symplectic vector field consists of canonical transformations of M.

We can use the basic form a to raise and lower the indices of tensorfields, exactly as in the Riemannian case. In particular, we can establish abijection between Vect (M) and fl1 (Al) (also denoted by a):

(30) v = viOi ° 4 0 = 9kd?A where 9 = al,;r= or 9 = -ia.


The skew-gradient of a function f EC°O(M) is a vector field s-grad f= a-ldf, corresponding to the 1-form df (i.e. i,,a = -df). In canonical

coordinates we have

(31)

In particular,

s-grad f = of (9 - as aape Oqi a9= a

s-grad pi =aQi

, s-grad 9' -IT.

Exercise 6. Let S be a smooth oriented 2-dimensional surface with ametric gel and a symplectic form wig. Assume that both structures definethe same volume form on S, i.e. det Ilgii II = det Ilwij II. Show that the skew-gradient s-grad f is obtained from the ordinary gradient grad f by a rotationon the right angle.

Hint. In appropriate local coordinates at a given point we have

1 1gii = (0

1)' wii = 4 0)

46

Usually, skew-gradient vector fields are called Hamiltonian fields andthe function f is called the generating function, or simply the generatorof s-grad f. We denote the collection of all symplectic vector fields on Al byVect(M,a) and the collection of Hamiltonian vector fields by Ham(M. a).

Theorem 4. a) All Hamiltonian vector fields are symplectic.b) If the manifold M is simply connected, the converse is also true.c) The commutator [v, w] of two symplectic vector fields is a Hamiltonian

vector field with the generator a(v, w).

Proof. a) Let us recall the Cartan formula (see formula (16) in Section 2.3):

Applying it to the form or, we get L,,a = d9 where 0 = -a-lv. If v =s-grad f is a Hamiltonian field, we have 0 = -(If' and -dz f = 0. So,v is symplectic.

b) If v is symplectic, we get d6 = 0, i.e. the form 0 is closed. On asimply connected manifold every closed form is exact. Hence, 0 = cf forsome smooth function f and v = s-grad f is a Hamiltonian vector field.

c) Let v, w be symplectic vector fields and f = a(v, w). Then f =(i,,, o iv )o,. Therefore df = (d o i., o iz,)a = (L,,, - i,,, o But ia is closed,


hence df = iLW,,a = -i[ W1a, i.e. [v, w] = s-grad f. (Here weused the generalized Leibnitz rule (17) from Section 2.3.) 0

A submanifold S of a symplectic manifold (M, a) is called isotropic ifor Is= 0. It is clear that for any isotropic submanifold S we have dim S <

dim M. If the equality holds, S is called a Lagrangian submanifold.In representation theory and in geometric quantization an important

role is played by so-called Lagrangian fibrations M ---P-+ B, i.e. fibrationswith Lagrangian fibers.

Let us now turn to the algebraic approach and characterize a manifoldM by the algebra A(M). On any symplectic manifold we have an addi-tional algebraic operation on A(M) (which is also defined on the biggerspace C°O(M)). It is the so-called Poisson bracket defined in threeequivalent ways:

(32) {fi, f2} = (s-gradfi)f2 = -(s-gradf2)fi = 010-gradfi, s-gradf2).

In any canonical local coordinates we have

Ofi IM _ Oh aft(33) {h, f2} = api aqi 8q= api

In particular, the following canonical relations hold:

(1 ifi=j,(34) {pi, pj } _ {q`, qj } = 0, {pi, q3 } = 81 : Sl 0 otherwise.

Theorem 5. a) The Poisson bracket defines the structure of an infinite-dimensional Lie algebra on the space C°O(M).

b) The map f t-+ s-grad f is a Lie algebra homomorphism from COO(M)to the Lie subalgebra Ham(M, a) C Vect(M, a) C Vect(M).

c) The kernel of the above homomorphism coincides with the center ofthe Lie algebra COO(M) and consists of all locally constant functions (whichare constant on every connected component of M).

Proof. a) The antisymmetry of the Poisson bracket is evident. We leave itto the reader to check the Jacobi identity (in canonical coordinates it is asimple calculation).

b) This is part c) of Theorem 4.

c) Let f belong to the center of the Lie algebra C°O(M). Then s-grad fannihilates all smooth functions, hence must be a zero vector field. Thereforedf = 0 and f is locally constant. 0


The algebraic incarnation of the notion of a Lagrangian fibration is amaximal abelian subalgebra A in the Lie algebra C°O(M) of a special kind.Namely, we require that for any m E M the Lie algebra A contains n =

dim M functions which have independent gradients at m.

Indeed, let A = p' (C' (B)). It can be viewed as an algebra of functionson M that are constant along the fibers. Let P(m) C TmM be a tangentspace to a fiber of P passing through the point m E M. If a function f isconstant along the fibers, then its gradient df (m) is a covector vanishing onP(m). Therefore, its skew-gradient has the property:

s-grad f (m) E P(m)1 = P(m).

Hence, for any two such functions we have I fl, f2} = a (s-grad fj, s-grad f2)=0.

Conversely, if the family of functions { fQ }QEA has the property (f",, f}= 0 for all Q, ,O E A, then their skew-gradients at any point m E M spanan isotropic subspace in T,n(M) of dimension at most n =

2dim M. Hence,

there is at most n functionally independent functions with vanishing Poissonbrackets. If fl, ..., fn is such a family, then the system of equations

fl(x) = cl, ..., fn(x) = c

defines a Lagrangian fibration of M.There are three main sources of symplectic manifolds: cotangent bun

dles, complex algebraic manifolds, and coadjoint orbits. The last source isconsidered in detail in the main part of the book. We briefly discuss theother two.

Example 9. Let M be any smooth manifold, and let X = T*M bethe cotangent bundle on M. We can define in a purely geometric way adifferential 1-form 9 on X. Namely, let p : X -> M be the canonicalprojection. For any t; E TTX we denote by p.(C) the corresponding vectorin Tp(x)M. Now recall that x itself is a covector in Tp(Z)M. We define theform 9 by

(35) 9(x)() = (x, P.W).

For any chart U with local coordinate system (q', ... , qn) on M we candefine the coordinate system (ql, ... , qn; p', ... , p,) on T* (U) C T*(M) sothat

(36) pi (x) = (x, 8i) where 8j :=e80.


In these coordinates the form 0 looks like

(37) 9 = pidq`

since 0(x)() = pi, 0(x)(-) = 0.Therefore, the 2-form w = dO is given by the formula (29) and, con-

sequently, is closed and non-degenerate. We see that T*(M) is indeed asymplectic manifold.

The functions on X that have the form f = qop (i.e. functions dependingonly on q-coordinates) form a maximal abelian subalgebra in C°°(X). Thecorresponding fibration is just the canonical fibration of X = T*M over M.

There is a convenient description of Lagrangian submanifolds L C X forwhich the projection p is a bijection of L to M. Namely, such a manifold isexactly the range (i.e. set of values) of a closed 1-form on 11'1 viewed as asection of the cotangent bundle T*M.

Indeed, the intersection L fl T*U in the coordinate form is given by theequations

pi = ai (q), 1 < i < n.

The tangent space TL is spanned by vectors Ej = 0 + ' '. Hence, L isLagrangian if w(1;; ) = 0 for all i j i eE if - = 0 Let 0L be the, , . ., 3

OV.

09,associated 1-form on M given by

OL(v) = (p-'(m), v) for v E Tm(M), or 9L = ai(q)dgi.

It is clear that the condition above is equivalent to dOL = 0.Note in conclusion that all symplectic manifolds of the type X = T*M

are non-compact. Q

Example 10. Complex algebraic geometry provides a lot of examplesof compact symplectic manifolds. We start with a reminder of the elementsof complex geometry.

Let M be a complex manifold, and let U C M be a chart with a localcoordinate system (zl, ..., z"). A Hermitian form on Al is given locallyby the expression

h = hij(z)dzz 0 dzj

where hij are complex-valued functions on U satisfying hj;(z) = hij(z).The real part g = $2h is a real bilinear form on TU. We keep the same

notation for the corresponding quadratic form. Let us introduce the realcoordinates uk = Rzk, VI = c zk and the real-valued functions a;j = aji =lthij, bij = -bji = !ahij. Then we have

g = aij(duiduj + dvidvj) + b;j(duidv).

+ dvjdui).


The imaginary part of h defines a real differential 2-form w on M:

w = aij (dv' A dud) + bi j (du' A duj - dv' A dvj ).

A Hermitian form h is called Kahler if its real part g = Rh is positivedefinite and the imaginary part w = 3h is closed. A complex manifoldendowed with a Kahler form is called a Kahler manifold.

In this case w is always non-degenerate, since in an appropriate localcoordinate system we have hij = bij, hence aij = bij, b,, = 0 and, conse-quently,

g = (du)2 + (dv)2, w = dv` A du'.

So, every Kahler manifold possesses a symplectic structure.Now, the restriction of a Kahler form to a complex submanifold is again

a Kahler form, since both the positivity of g = Ih and closedness of w = 3'hare preserved by a restriction. Therefore, any complex submanifold of aKahler manifold is itself a Kahler manifold, hence possesses a canonicalsymplectic structure.

The closedness of 3`h implies that locally there exists a real-valued func-tion K such that

htJ 8zi 8zjK.

This function K is called a Kahler potential of the form h. It is not uniqueand is defined modulo a summand of the form R f where f is a holomorphicfunction.

Conversely, if K is any real-valued function, we can define a Hermitianform

8a K(z) l dz' ®dzj.h= 070-K:= Gzi5z=_j /

The imaginary part of this form is always closed. If, moreover, the real partis positive definite, we get a Kahler form.8

It is well known that the complex projective space PN(C) has a remark-able Kahler form, the so-called Fubini-Study form h, which is uniquelydefined by two conditions:

1. The form h is SU(N + 1)-invariant.2. For any P'(C) naturally embedded in PN(C) we have f p,lcl w = 1.

We give here the explicit expression of this form. Let (x° : xl :...: xN)be the homogeneous coordinates in PN(C). The real functions

kN xk2Ki = log Elt l

slf the real part is non-degenerate, but not positively defined, we obtain a so-called pseudo-Kahler form.


have the property

K3 - Ki = log -Ilog (\\

/Xi\+log (xi-).'I2X'/ X'

We see that the differences K? - Ki are real parts of analytic functions onUi n Up. Therefore the Hermitian forms hi = 88Ki and h,j = 98KK coincideon the intersections Ui n Up. Hence, they define a single Hermitian 2-formh on the whole space PN(C).

In terms of of ine coordinates z = (z', ... , zN), Zi = , this form looksas follows:

(38) h = 881og (1 + Izl2) =Idzl2 zdz'

1+IzI2 - I 1+Iz12

2

This form is obviously invariant under the group SU(N) C SU(N + 1)acting linearly on affine coordinates and also under the group SN+1 actingby permutations of N + 1 homogeneous coordinates. It follows that it isinvariant under the group SU(N + 1) acting by projective transformationson PN(C).

The real part of this form is an SU(N+1)-invariant metric g = hkjdzkdzjand the imaginary part is an SU(N + 1)-invariant symplectic form

(39) w= 2 hkjdzk n dxj2

where hk,j =(1 +(1I+ Iz12)2xkxj

It remains to multiply this form by an appropriate constant in order tosatisfy the normalization condition 2 above. We shall do it after some dis-cussion.

The restriction of the Fubini-Study form on any smooth algebraic sub-manifold M C PN(C) is again a Kiihler form whose real part gives a metricon M and the imaginary part defines a symplectic structure on M.

Let n be the complex dimension of M. Then the form

wn nwvol = i (n factors)

n!

is a volume form on M and the integral of this form over M is an importantcharacteristic of the algebraic manifold M, which is called the degree anddenoted by d(M). The geometric meaning of the degree can be seen fromthe following proposition.


Proposition 7. The intersection of M with a generic projective subspaceof the complementary dimension N - n consists of d(M) points.

In particular, the degree of a smooth hypersurface given by the equationP(x) = 0 is equal to d where P is a homogeneous polynomial of degree d.

The normalization condition means that the degree of P'(C) C PN(C)is equal to 1. So we can determine the normalization factor by computingthe integral of the form (39) over the submanifold P'(C) C PN(C). Theresult implies that the Fubini-Study form on PN(C) is equal to

(40) hF's = -2 88 log (1 + Iz12).

0Symplectic manifolds form a category Sym where the morphisms are

symplectomorphisms. This category admits a very interesting extensionSym. We observe that the graph r. of a symplectomorphism cp : MlM2 is a Lagrangian submanifold in Ml x M2. Here Mi denotes the symplec-tic manifold (Ml, -al) and the symplectic structure on a product MI x M2is defined as plat +p4a2.

The objects of Sym are again symplectic manifolds, but morphisms fromMl to M2 are all Lagrangian submanifolds in MI* x M2. The composition ofmorphisms f : M2 -. M3 and g : Ml M2 is defined as a submanifold1" fog c Ml x M3 that contains all pairs (ml, m3) for which there exists apoint m2 E M2 such that (ml, m2) E lg, (m2, m3) E r f-9 We refer to [We]for the applications of the category Sym.

3.2. Poisson manifolds.

Definition 4. A smooth manifold M is called a Poisson manifold if it isendowed with a bivector field c = ci78i8j such that the Poisson brackets

(41) {fi,f2} = c'jaifla,f2define a Lie algebra structure on C°°(M).

The Jacobi identity for Poisson brackets imposes on c a non-linear dif-ferential condition [c, c) = 0 where [ , ] is the Schouten bracket on polyvectorfields (see Section 2.3).

In this particular case the Schouten bracket [c, c] is a trivector tijk givenby

(42) t'?k19 I19if2akf3 =0 {fi, {f2, f3}}

where j denotes the summation over the cyclic permutations of fl, f2, f3.

9Actually, this composition law is only partially defined. So, Sym is not a true category.


Proposition 8. Any symplectic manifold (M, a) has a canonical Poissonstructure c such that in some (and hence in every) local coordinate systemthe matrices II c`j II and II ail II are reciprocal: c`1ojk = 6k'-

Proof. Let If,, f2}p be the Poisson brackets defined by the bivector c =(7-1 via (40), and let {fl, f2}S denote the Poisson brackets defined by the

form a via (32).In local canonical coordinates x1, ... , xn, yl, ... , yn the form a looks

like >i dxi A dyi where ai = ai = a/ay,. The bivector c = Q-1 in thesame coordinate system takes the form 1 ai AO'. From this we easilyobtain the equalities

{fl, f2}P = aihaif2 - 19ihaif2 = {h, f2}s

Thus, the bivector c = a-1 defines the same Poisson bracket as the sym-plectic form a.

We return now to general Poisson manifolds. The main structure theo-rem about Poisson manifolds (see, e.g., [Ki4]) is the following

Theorem 6. Any Poisson manifold (M, c) is foliated by the so-called sym-plectic leaves, i.e. can be uniquely represented as the disjoint union ofsubmanifolds {Ma}«EA such that the bivector c has a non-degenerate re-striction cQ on each M,, and (M0, c;') is a symplectic manifold for eachaEA.

In other words, on each Ma there is the Poisson bracket I. , }C.

relatedto a symplectic structure aQ = (c0)-1 such that the value of If,, f2} at thepoint m E Ma is equal to {f1IM., f2IMQ}Q (m).

Example 11. Define a Poisson structure on R2 by

c = f(x, y) alas A alas, f E C-(12).

Then the plane R2 splits into 2-dimensional and 0-dimensional symplecticleaves. The former are connected components of the open set f 54 0, andthe latter are points where f = 0.

3.3. Mathematical model of classical mechanics.Symplectic geometry is the mathematical counterpart of the Hamilton-

ian formalism in classical mechanics. We adduce here the corresponding

§3. Symplectic and Poisson manifolds

physics-mathematics dictionary:

Physical notions

configuration space

position coordinates

phase space

impulse coordinates

state of the system

physical observable

the value of an observablein a given state

kinetic energy K

potential energy V

total energy H = K + V

equations of motion

265

Mathematical interpretations

smooth manifold N

local coordinates q1, ... , q" on N

cotangent bundle M = T ` N

coordinates pl, ... , p" in fibers of T*Ndual to q1, , q"

a point m E M

a smooth function f on M

the value f (m)

a positive quadratic form Kon the fibers of T*M

a smooth real-valued function V on N

the function H = K + p` (V) on M

f= If, H}

The remarkable fact, discovered by Hamilton, is that the final result, theequations of motion, are invariant under a very big group of symplectomor-phisms of M.

Moreover, the whole theory can be formulated in the much more generalsituation where AM is an arbitrary symplectic manifold and H is an arbitrarysmooth function on M.

In particular, among these more general phase spaces one can find theclassical analogue of the spin particle in quantum mechanics. It is the usual2-dimensional sphere of integral area. We refer to [Ki7] for details.

3.4. Symplectic reduction.In classical mechanics there is a useful procedure which allows us to

reduce the number of degrees of freedom in the presence of symmetry.Mathematically speaking, it is a prescription to construct a new sym-

plectic manifold from a given symplectic manifold with a symmetry.The initial data are the following:G - a Lie group with the Lie algebra g;


(M, a) -- a symplectic G-manifold which is also a G-Poisson manifold;p : M g' - the moment map;SlF - the coadjoint orbit in g' passing through the point F E g`;Stab(F) - the stabilizer of the point F in G, and stab(F) is the Lie

algebra of Stab(F);M0 - the reduced manifold p-1(F)/Stab(F) ^_- p-1(SZF)/G.We assume that the following equivalent conditions are satisfied:

a) the group Stab(F) acts freely on p-1(F).b) the group G acts freely on 1A-1(QF)).

Proposition 9. The coset space Mo is a symplectic manifold with respect tothe form ao defined by p*ao = a, where p : p-101F) -p Mo is the canonicalprojection.

Proof. From our assumptions it follows that the coset space Mo is a smoothmanifold. Consider the geometric distribution P = kera on p-1(1lF) CM. The fibers of the projection p are exactly the leaves of the foliation ofp-101F) C M associated with P.

The following fact is important for application to mechanics. Assumethat the function H is G-invariant. Then it defines a function Ho on Mo.

Proposition 10. The flow generated on M by the Hamiltonian vector fields-grad H preserves p-1(QF) and is projected to the flow on Mo generated bythe Hamiltonian vector field s-grad Ho.

This approach is based on the general principle of symplectic reduc-tion. In pure mathematical terms this notion was introduced independentlyby Marsden-Weinstein and by Arnold in 1974, although its origin is in theclassical mechanics of the 18th and 19th centuries.

The idea of symplectic reduction is very simple. A submanifold N of asymplectic manifold (M, a) in general is not symplectic because the 2-forma' = a IN can be degenerate.

But a' is still closed, therefore the kernel of a' is an integrable distribu-tion on N (it clearly satisfies the Frobenius criterion in the second formula-tion). If it has a locally constant rank and if the leaves of the correspondingfoliation are locally closed, then we can view N as a fibration over somemanifold Mo. It is clear that the form a' can be "descended" to Mo, i.e.a I N= p'(aO) where ao is a non-degenerate 2-form on Mo and p : N -+ Mois the projection.

Thus, we get a smaller symplectic manifold (MO, ao), which is calledreduced.


In practical applications (M, a) is usually interpreted as a phase spaceof a Hamiltonian mechanical system with the energy function H E C°°(M).The time development of the system is given by the Hamiltonian flow 4 (t) _exp tv generated by the vector field v = s-grad H.

Suppose that the submanifold N can be given by a system of equations

(43) F2=O, 1 < i < k,

where the Fi are the first integrals of the system, i.e. smooth functions onM that Poisson commute with H.

Then the flow fi(t) preserves the submanifold N and its foliation, hencecan be descended to a flow lo(t) on the reduced manifold (Mo, oo). More-over, the function H is constant along the fibers and so defines a function Hoon Mo. Finally, one can check that the restriction of the initial Hamiltonianflow to the reduced manifold is itself a Hamiltonian flow generated by thevector field vo = s-grad Ho.

The construction described is especially effective when the functions F1in (43) span a Lie subalgebra g in C°°(M), viewed as a Lie algebra withrespect to the Poisson bracket on M. Let us consider the Fi as coordinateson the space g`, dual to g. Then we can interpret the collection (F1)1<i<kas a map µ from M to g'. Following mechanical terminology, it is called -amoment map.

This map has many nice and useful properties, which we discuss later.Here we list only some simple facts which are needed in this section.

First, this map is by definition g-equivariant. (Recall that g acts onM via vector fields vi = s-grad Fi, 1 < i < k, and on g' via the coadjointrepresentation: K.(Fi)Fj = {Fi. F;).)

Second, assume that the Lie algebra action of g on M can be lifted toan action of the connected Lie group G with Lie(G) = g. Then the momentmap y will be G-equivariant.

Third, choose any G-orbit St C g` and let N = µ-1(9). It is a G-invariantsubmanifold of M and one can check that the corresponding reduced sym-plectic manifold (Mo. vo) is just N/G, the set of G-orbits in N.

Appendix III

Lie Groups andHomogeneousManifolds

1. Lie groups and Lie algebras

1.1. Lie groups.A Lie group is a smooth manifold G endowed with a multiplication law

that is a smooth map G x G G satisfying the usual group axioms.Consider first the following particular case: G is a subgroup and at the

same time a smooth submanifold of GL(n, R). Such a group is usually calleda matrix Lie group. Actually this particular case is almost a general one:every Lie group is locally isomorphic to a matrix Lie group. It meansthat there exists a diffeomorphism between some neighborhoods of the unitscompatible with the group law. Two locally isomorphic Lie groups haveisomorphic covering groups.

We introduce convenient notation which allows us to treat general Liegroups as matrix groups. We call this matrix notation.

Let v be a tangent vector to G at a point x E G. The left and right shiftsby an element g E G are smooth maps of G to itself. The correspondingderivative maps send the space TxG to T9 G and T 9G. We denote theimage of v under these maps by g v and v g, respectively.

Note that for a matrix Lie group G C GL(n, 1R) the tangent space T2Gcan be identified with a subspace in Mat,, (R) and the expressions g v andv g can be understood literally as products of matrices.

269

270 Appendix III. Lie Groups and Homogeneous Manifolds

We shall also use matrix notation for covectors: for f E T* G we denoteby g f the covector in G defined by (g f , v) = (f , v g). In the sameway we f ET,G, vE 9_,XG.

For a matrix Lie group G C GL(n, R) the cotangent space TzG is iden-tified with a factor space of modulo TG-i using the pairing

(A, B) = tr (AB).

Here again the expressions g f and f g can be understood as products ofa matrix g and a class of matrices f E Indeed,

(g f, v) = tr (g f )v = tr f (vg) = (f, v g),(f g, v) = tr (f g)v = tr f (gv) = (f, g v).

1.2. Lie algebras.A Lie group is a rather complicated non-linear object. Fortunately, it can

be almost uniquely determined by a linear object, the so-called Lie algebra.By definition, a Lie algebra is a vector space over some field K (in our

book we shall use only real and complex fields) endowed with a bilinear map9X9-p9.

This map is called the commutator and is usually denoted by brackets[X, Y]. By definition, it satisfies the following conditions:

Antisymmetry: [X, Y] [Y, X] (or, equivalently, [X, X] = 0);

Jacobi identity:

(1) 0 [X, [Y, Z11 [X, [Y, Z11 + [Y, [Z, X11 + [Z, [X, Y11 = 0.

Here and below the sign 0 is used for the summation over cyclic permu-tations of arguments.

The definition of a Lie algebra, especially the Jacobi identity, at firstsight looks rather cumbersome and far-fetched. We give two importantinterpretations of this identity here that are easier to memorize:

a) The map ad X : Y ' [X, Y] is a derivation of g, i.e. satisfies theLeibnitz rule:

(2) ad X ([Y, Z]) = [ad X (Y), Z] + [Y, ad X (Z)].

b) The map ad : X - ad X is a Lie algebra homomorphism of g toEnd g, i.e.

(3) ad[X,Y]=adXoadY-adYoadX.In fact, the definition of a Lie algebra is well justified. The main reason

will be given in the next section, but meanwhile we present an argumentwhich shows that Lie algebras are objects at least as natural as associativecommutative algebras.


Proposition 1. Let A be a real algebra with the multiplication law denotedby *.

a) Suppose that in A some non-trivial bilinear identity holds:

where a, 0 are some fixed real numbers and a, b denote arbitrary elementsof A. Then A is either commutative or anticommutative.

b) Assume further that A admits some non-trivial trilinear identity:

=0where A, p, v are fixed real numbers and a, b, c denote arbitrary elements ofA.

Then, in the commutative case, A is either an associative commutativealgebra or a non-associative commutative algebra where the identity x3 = 0is satisfied.

In the anticommutative case either A is a Lie algebra, or an algebrawhere the product of any four elements is zero.

Proof. a) By interchanging a and b in the first relation we get the systemof linear equations

1. /3- 0.

If the multiplication is identically zero, we have nothing to prove. Oth-erwise, the system has non-trivial solutions and its determinant must bezero. We get a = ±,3, hence our multiplication is either commutative oranticommutative.

b) Let us apply a similar argument to the trilinear relation. Namely,consider the system

v a(bc)+A b(ca)+p c(ab)0,1 it =0.

The determinant of this system is

A3+µ3+v3 -3Auv = (A+a+v)(A+Eµ+E2V)(A+E2p+EV)ZAt

where E = e 3 is a cubic root of unity.If a * (b * c) is not identically zero, this determinant must vanish. Note

that the last two factors of the determinant are complex conjugate. So wehave the following alternative:'

'This is related to the following representation-theoretic fact: the space R3 under the actionof the permutation group S3 splits into two irreducible subspaces: the one-dimensional spaceLl = {(A, A, A)} and the two-dimensional space L2 = Li = ((A, µ, v) I A + µ + v = 0).


Either (A + Ep + e2v) = (A + E2µ + Ev) = 0, hence A=IL= v and we getthe identity

(4) a*(b*c)+b*(c*a)+c*(a*b)-0,

or the trilinear relation holds for any A, p, v with A + p + v = 0 and we get

(5) a * (b * c) = b * (c * a) = c * (a * b).

In the commutative case (4) implies x3 = 0. Conversely, if the identityx3 - 0 holds in a commutative algebra A, then, putting x = as + Qb + rycand comparing coefficients for aQry, we get the relation (4).

The identity (5) in the commutative case gives the associativity law.In the anticommutative case (4) is the Jacobi identity, while (5) means

that the triple product a * (b * c) is totally antisymmetric.Let La (resp. Ra) denote the operator of left (resp. right) multiplication

by a in A. The antisymmetry of the product a*b and the total antisymmetryof the triple product a * (b * c) implies the relations:

R. = -La; LaLb = -LbLa; La*b = LbLa.

From these relations we deduce

LaLbLc = -LaLbLc = LcL0.b = LeLbLa = -LaLbLc,

hence LaLbLc =- 0 and the product of every four elements is zero.

An important class of Lie algebras is formed by matrix Lie algebrasthat are subspaces of Mat,,(K) closed with respect to the ordinary matrixcommutator

[X, Y] = XY - YX.

Both conditions above can be easily checked.Actually, this class is universal because of

Ado's Theorem. Any Lie algebra is isomorphic to a matrix Lie algebra.

The example below historically was the first appearance of a Lie algebrain mathematics.

We start with a definition. An algebra A is called a division algebraif any non-zero element of A is invertible. Associative non-commutativedivision algebras are also called skew fields.


Frobenius' Theorem. There are only three (up to isomorphism) asso-ciative division algebras over R: the real field R, the complex field C, and askewfield 1111 of quaternions. 0

The latter, very remarkable, algebra was discovered by W. Hamiltonin 1843 after many years of unsuccessful attempts to generalize complexnumbers and define an associative multiplication in 1R3. The crucial ideawas to switch from J3 to lR4 (and also to sacrifice the commutativity law).We describe this algebra below.

Hamilton represented a quaternion in the form q = xo + x where xo isan ordinary real number and x = (x1 i x2, x3) is a vector in 1R3. he calledthese constituents the scalar and vector parts of q. The product of twoscalars and the multiplication of a vector by a scalar are as usual. So, oneneeds only to define a product of two vectors. This product x y splitsinto a scalar and a vector part, which were called by Hamilton the scalarproduct (x, y) and the vector product x x y (these terms and notationare still presently used).

The explicit formulae are:

(x,y) + xxy,

(x, Y) = x1111+x2y2 + x31!3, x x y = det

where i, j, k are standard basic vectors in &t3.

via

i j kx1 x2 x3

Y1 112 113

The algebra 1111 can be conveniently realized as a subalgebra of Mat2(C)

q = xo + x1 i + x2j + x3k --> (xo+ix3 x1 + ix2-x1 + ix2 xo - ix3

It can also be defined as an associative algebra spanned by one real unit 1and three imaginary units i, j, k satisfying the relations2

(H)

Example 1. The 3-dimensional Euclidean vector space R3 endowedwith vector multiplication is a real Lie algebra. Q

Exercise 1. a) Check that the operation of vector multiplication satis-fies all the axioms of a Lie algebra.

The legend is that these very relations were carved by Hamilton on the railings of a bridgehe usually crossed during his mathematical walks.


b) * Show that the Jacobi identity implies the following well-known factfrom Euclidean geometry: for any triangle ABC the three altitudes areconcurrent.3

Hint. Consider points and lines in the Euclidean plane R2 in terms ofhomogeneous coordinates on P2(R): a point (x, y) E R2 corresponds to (x :y : 1) E P2(R), and the line ax+by+c = 0 corresponds to (a : b : c) E P2 (R).

Note that the point loo = (0 : 0 : 1) does not correspond to any line inR2. It is related to the "infinite line" P2(R)\R2. The solution to b) followsfrom the statements:

(i) A point x = (xa : xi : x2) belongs to a line a = (ao : al : a2) if

3

(a, x) aixi = 0.i=1

(ii) The line a passing through points x and y is given by a = x x y.(iii) The intersection point x of two lines a and b is given by x = a x b.(iv) The line p passing through a point x and perpendicular to a line a

is given by p = x x a where &=a-loo a'IO°)

(v) Three lines a, b, c are concurrent if their mixed product

as al a2

(a, b, c) := (a x b, c) = det ba bI b2

Co CI C2

vanishes (in other words, the vectors a, b, c are linearly dependent).(vi) 0x,g,z (x, y, a)(z, b, c) = (x, y, z)(a, b, c) where the sign c5x,y,: denotes

the sum over cyclic permutations of three vectors x, y, z.

1.3. Five definitions of the functor Lie: G w g.To any Lie group G one can canonically define the Lie algebra g =

Lie (G). In more rigorous terms, there is a functor from the category ofLie groups (objects are Lie groups, morphisms are smooth group homomor-phisms) to the category of Lie algebras (objects are Lie algebras, morphismsare Lie algebra homomorphisms).

This functor can be defined in many equivalent ways. We give fivedifferent constructions here, each of which has its own advantage.

3This fact is mentioned in the recent paper [Arl] which is of independent interest. It isworthwhile to mention that the proof indicated becomes even simpler in the case of sphericalgeometry: the line through the point x perpendicular to the line a is given by x x a (cf. (iv)below).

§1. Lie groups and Lie aIgebras 275

All definitions will be illustrated on the example G = Aff(l, R), the Liegroup of all affine transformations of the real line Rl with parameter t.

To define the functor Lie we have to associate to any Lie group G someLie algebra g = Lie (G) and to any smooth group homomorphism 0 : G -H some Lie algebra homomorphism ep : g - h.

As a vector space, we define g as the tangent space TeG to the group Gat the unit element e. The map cp is just the derivative of 6 at e. It remainsto endow g with the Lie algebra structure, i.e. define the commutator. Wedo it below in five different ways.

1. Let us write the multiplication law on G in the coordinate form.It is given by n functions of 2n variables expressing the coordinates {zk}of the product in terms of coordinates {x`}, {yJ} of the factors: zk =fk(xl,...,xn; yl,...,yn).

Assume that our local coordinate system is centered at the unit element.Then we have f k(xl, ... , xn; 0'...,0) = xk, fk(0,... , 0; y1,. .. , yn) = yk.We conclude that the Taylor decomposition of f k at the origin has the form4

(6) fk(xl, ... , xn; y1, ... , yn) = xk + yk + b?x:y' + higher order terms.

Now we introduce the quantities

(7) 4 := b - bV .7t

and consider them as structure constants of the desired Lie algebra g.This means that the commutators of basic vectors X1, ..., Xn in g are

given by the formula

(8) (Xt, Xil =

Let us see how this construction works for the example G = Aff(1, R).The general affine transformation of Rl has the form

¢a,b: tHat+b where aER\f0}, bER.

Hence, the whole group can be covered by one (non-connected) chart withcoordinates (a, b). To put the origin at the unit element, we choose anothercoordinate system: xI = a - 1, x2 = b, and define the transformation 0r1,xzas t " (x' + 1)t + x2.

4Recall that we use the Einstein notation (see Remark I in Appendix II): if an index appearstwice, once as a lower index and once as an upper one, then the summation over this index isunderstood.


The composition of transformations 0xi x2 and lpy,,y2 has the form lpsI Z2with

z1 = x1 + y1 + x1y1, z2 = x2 + y2 + x1y2.

We see that the only non-zero coefficients b are bi 1 = bit = 1. Hence,the only non-zero structure constants are c 2 = -c21 = 1.

The basic commutation relation is [X1, X2] = X2-2. Let X, Y be any two tangent vectors to G at e. Define their commu-

tator [X, Y] as follows. Let 0(r) and 91'(r) be any smooth curves in G withthe properties

0(0) ='(0) = e; (0) = X, '(0) = Y.

Then the curve

(9)

is 1-smooth for T > 0 and we put [X, Y] = (0).In the example let X1 = 81 :_ a , X2 = 6912 := . As representatives

of these vectors we can choose the curves

0(r) = Vr,o: t' (1 + r)t, V1(r) ='0.r: t i t +,r.

Then the transformation (5) is t ' (1 + -'Ar-) (- + vf) = t + r, i.e.I+ VFcoincides with V)(r). We come again to the commutation relation [X1, X2] _X2.

3. For g E G consider the map A(g) : G - G : h '-+ g h g-1. Itis a so-called inner automorphism of G. In particular, it preserves themultiplication law and fixes the neutral element e.

Put Adg := A(g).(e), the derivative of at the point e. Thisis a linear transformation of the space g = TQ(G). Moreover, the chainrule for derivatives implies that Ad g1 Ad g2 = Ad 9192. Thus, the mapg 1-+ Ad g is a linear representation of G in the space g, called the adjointrepresentation.

Now, Ad is a smooth map from G to the group Aut g of all automor-phisms of the vector space g. Let us denote by ad the derivative of this mapat e. It is a map from g = TG to End g = TidAut g. So, for any X E 9 thequantity ad X := Ad. (e)(X) is a linear operator in g.

Finally, we put [X, Y] _ (ad X)(Y).In the example let g = Vx1,x2, h = 1/iyl,y2. Then the transformation

A(g) sends h to ghg-1: t (1 + yl)t + (1 + xl)y2 - xtyl (check it!).


Hence, in coordinates (yl, y2) on G the map A(g) is a linear transfor-mation:

11 0A(g) _ -x2 1--x1

Therefore, its derivative Ad g is given by the same formula, if we identifythe neighborhood of the unity in G with a neighborhood of the origin in gusing the coordinate system (yl, y2).

Put as above X,=B,=8, i=1,2. Then we get:

8Ad g (0 0), X2 _ 8Ad g0 0).ad X1 =

8x1 0 8x2 -1 0We come once again to the commutation relation [X1, X2] = X2.

4. To any vector X E g there corresponds the unique left-invariant vectorfield k E VectG, which takes the value X at e. Since the Lie bracket is anatural operation, it commutes with all diffeomorphisms and, in particular,with group translations. Therefore, the Lie bracket of two left-invariantvector fields is also left-invariant. So, [X, YJ has the form Z for some Z E 9.We then define [X, Y] as Z.

Warning. If we use the right-invariant vector fields X E Vect G thattake the value X at e instead of left-invariant fields X, we get a differentdefinition of the commutator, the negative of the previous one. The reasonis that the left-invariant vector fields are generators of right shifts on G andthe right-invariant vector fields generate left shifts (see below). So, the lattergive a representation of g while the former give an antirepresentation.

To get the same Lie algebra structure we have to associate to X E g theright-invariant field -X.

On the group G = Aff(1, R) with coordinates (a, b) the general vectorfield has the form v = A(a, b)OQ + B(a, b)86. The left shift by the element0a,,s sends the point (a, b) to the point (a', b') = (aa, ab +,6) - We have

da' = ada, db' = adb, and8 _18 a

=_8

8a' =a

8a' ab'a

8b'

Hence, the above field v goes to

Q)V' = A(a', Y)8., + B(a',

y),* =A(aa,

8a +B(aa, ab +

aab +Q)

a 8b.

Therefore, the field v is left-invariant iff

A(aa, ab + (3) = aA(a, b), B(aa, ab +,6) = aB(a, b).


The general solution to these equations is A(a, b) = cla, B(a, b) = c2a.We get

X 1 = aOa, k2 = abb.

Note that the same result can be obtain using the following principle:

a left-invariant vector field is a generator of a right shift :

exp X (g) = g exp X.

We leave it to the reader to derive the following formula for right-invariant fields on Aff(1, ]R) (which are generators of left shifts):

X 1 = aOa. + bOb, X2 = cob.

It is clear that [X1, X2] = X2, while [Xl, X2] = -X2.5. Consider any matrix (or operator) realization 7r of G by operators on

R". Then 7r(G) is a submanifold in Matn(]lt). Let g be the tangent vectorspaces to ir(G) at the point 1n. Define the commutator in g by the formula[X, Y] = XY - YX.

In our example we can use the following matrix realization of the groupG = Aff(1, R):

{x H ax + b}

Indeed, the equality

a

1)(1)

- (ax1b)

{0

shows that the standard linear action of the matrix ( a 1) on column

vectors (x) E R2 defines a standard affine action on the line y = 1.

The tangent space g = TTG is the space of matrices of the form

X= (0 0), a,,(QE

with the natural basis

X1 = 0 0) , X2 = (0 0).

SDo not mix the linear subspace g C with the geometric tangent space that is theaffine manifold 1a + g C Mat,,(R).

§ 1. Lie groups and Lie algebras 279

Here we get once more the commutation relation [X1, X21 = X2.

Let us discuss the five definitions above. First, we mention their positivefeatures.

The first and last definitions are the most convenient for practical com-putations (see examples below).

The second is more geometric and shows the relation between the Liegroup commutator and the Lie algebra commutator: the latter is a limit ofthe former.

The third definition is the most conceptual (it contains no computationsat all) and therefore is the most convenient in theoretical questions.

The fourth definition introduces a useful notion of left- (or right-) invari-ant vector fields on G. They are particular cases of left- (or right-) invariantdifferential operators on G which play an important role in harmonic anal-ysis on Lie groups.

On the other hand, the first definition has a serious disadvantage: itdepends on the choice of local coordinates. Consider this dependence inmore detail.

Note that under a linear transformation of local coordinates, the quan-tity b; . behaves like a tensor of type (2, 1). If, however, we make a non-lineartransformation xk xk + a x'xJ + , then b k.. changes in a more compli-cated affine way, namely

b k '--+ b - 2a x`yJ.

The remarkable fact is that the quantities ck still behave like coordinates ofa tensor. (In particular, under the above transformation they do not changeat all.)

Hence, a bilinear operation [x, y1 k = cijx`yJ is correctly defined on thetangent space TeG.

There is another argument in favor of ck comparing with bk;. Let{x1, ... , x"} be a local coordinate system. Call such a system symmetricif xk(g-1) = -xk(g) The following lemma shows the existence of symmetriccoordinate systems and describes their useful properties.

Lemma 1. Let {xi, ... , X' j be any local coordinate system centered at e.Introduce the functions xk(g) :_ ='F(s)-2"`(s-') Then

a) The functions x"} form a symmetric coordinate system.

b) The Jacobi matrix ex, 11 is equal to 1n at the origin.

c) In any symmetric local coordinate system the quantities bki are anti-symmetric with respect to i, j and therefore coincide with 2ck.


d) The group law in symmetric local coordinates looks like

(10) f (x; y) = x+y+ 2 [x, y]+Ti (x, x, y)+T2(x, y, y)+ terms of order > 4

where [x, y]k =`,x`y' and Ti, i = 1, 2, are trilinear vector-valued formssatisfying the conditions:

(i) Ti(x, y, z) = T1 (Y' x, z), T2(x, y, z) = T2(x, z, y),(ii) Ti(x, x, x) =T2(x, x, x).

Proof. a) is clear, provided we know that {yk(g)} is a coordinate system.b) In any coordinate system centered at e we have xk(g-1) = -xk(g)+

higher order terms. So, II a=; II(e) = 1. It also proves that is indeeda local coordinate system.

c) Note that a symmetric local coordinate system is necessarily centeredat e, so equation (6) holds. Substituting y = -x and using the propertyf (x; -x) = 0 we conclude that bk,x'xj - 0. Hence, the coefficients 6 areantisymmetric in i, j.

d) Consider the term of third order in the Taylor decomposition off (x; y). It can be written in the form >3o T1 where degy T1 = i.

Since f (x; 0) = x, f (0; y) = y, we have To = T3 = 0. It follows that (10)holds together with condition (i). Again using the property f (x; -x) = 0,we obtain T, (x, x, -x) + T2(x, -x, -x) = 0, which implies (ii).

Now we are ready to check the Jacobi identity for the first definition ofLie (G).6 The ultimate reason for the Jacobi identity is the associativity ofthe group law. In terms of coordinates the associativity property has theform:

f(f(x; y); z) = f(x; f(y; z))We assume that our coordinate system is symmetric and compare the tri-linear terms in the Taylor decompositions of the left- and right-hand sides.We get

4[[x, y], z] + 2T1(x, y, z) =

4[x, [y, z]] + 2T2(x, y, z).

Put commutators on one side, trilinear forms T1i T2 on the other, and takethe sum over cyclic permutations of x, y, z (denoted by 0). We obtain

0 [[x, y], z] = 4 U Ti(x, y, z) - 4 0 T2(x, y, z).

Using equations (i) and (ii) above we conclude that the right-hand side isactually zero and we get the Jacobi identity.

6Note that this check is very easy for the last definition and more involved for the others.

§ I. Lie groups and Lie algebras 281

Theorem 1. All five definitions of the commutator in g = Lie (G) = TeGgiven above lead to isomorphic Lie algebras.

Proof. The simplest way to establish the isomorphism of all five variantsof g is to compare definitions 2-5 with definition 1 using the appropriatelocal coordinate system. Let us do this.

2. e--* 1. Choose two vectors X, Y E g and consider two curves inG, which in some symmetric local coordinate system are given by O(T) _TX, t'(T) TY. Then

= fX+fY+2r[X,Y]- fX-2r[Y,X]- fY+o(T)= T[X, Y] + O(T).

3. 1. Suppose g E G has coordinates eX and h E G has coordinates6Y in some symmetric coordinate system. Then ghg' 1 has coordinate JY +jeb[X, Y] + higher order terms. To compute Ad g := A(g),(e) we have totake the term that is linear in 8. So, we get

Ad g Y = e[X, Y] + higher order terms.

Further, to compute ad Y := Ad#(e) we must take the term which is linearin e. So, ad X Y = [X, Y] and we are done.

4. ' 1. Let use the fact quoted above: all left-invariant vector fieldsare generators of right shifts and vice versa. The right shift on an elementg with coordinate 8Y is

X - X + SY + S[X, Y] + higher order terms.

Therefore, the corresponding vector field is given by

1vy(x) = (1c + ak + higher order terms.

From this we get [vy, vz] (0) _ [Y, Z]. Hence, [vy, vZ] = vIy,Zl.

5. --# 1. Let G C GL(n, R) be a matrix group, and let

g := T1G C T1GL(n, R) =

be its tangent space.We shall use the functions exp X and log g, which are defined for the

matrix arguments X and g by the series:

k9)k (defined for Jig- 111 < 1).k>0 k>0


We claim that these functions map g to G and some open part of G, definedby the inequality fig - 111 < e, to g, respectively.

Indeed, if X E 9, then there exists a smooth curve {g(t), t E 1[P} C G suchthat g(t) = 1+tX +o(t). Consider the sequence gn, = g(n) = + n +o (n)The element gn belongs to G and we have

lim gn = lim exp (n log gn) = exp (limn(+o())) = exp X.n-oo n n

Since G is closed, exp X E G.

Conversely, let U. C G be a neighborhood of the unit defined by jjg -111 < n . We temporarily assume that for any n E N there exists gn E Unsuch that log gn V g and come to a contradiction.

Let us split Matn(R) in the form Matn(R) = g ® V where V is anysubspace complementary to g. The map ep(X (D Y) = exp X exp Y, X E9, Y E V, has the Jacobian 1 at (0 ® 0). Hence, it establishes a diffeo-morphism between a neighborhood of 0 E Matn(1R) and a neighborhood of1 E GL(n, R).

We conclude that any matrix g E GL(n, R) that is sufficiently close to1 can be uniquely written in the form g = exp X exp Y, X E g, Y E V.

In particular, for the sequence gn above we eventually get

gn=expXn - eXpYn, XnE9,YnEV

Note that gn := exp Yn = exp(-Xn)gn E G, Yn 0 0 and IlYnli - 0 whenn --+ oo. We can choose a subsequence {Ynk } and integers Mk so thatMk Ynk -+ Y E V \O fork -' oo. Then Y = lim Mk log gnk E g and at thesame time Y E V\0, a contradiction.

Hence, there exists a neighborhood of unit Un for which log Un C g. Thisallows us to introduce the "logarithmic" coordinate system x(g) = log g E gin Un. Traditionally, this system is called exponential. It is evidentlysymmetric and in this system we have

f (x; y) = log (ex ey)

= log I 1 + x + 2 x2 + y + xy + 2 y2 + higher order terms 1

x + y +1

2(xy - yx) + higher order terms.

Hence, [x, y] = xy - yx.

Consider again the adjoint representation Ad introduced in the thirddefinition of g. When G is a matrix group, the adjoint representation is

§ 1. Lie groups and Lie algebras 283

simply the matrix conjugation:

(11) XEg, gEG.

The same formula holds in the general case if we accept the matrixnotation introduced in Section 1.1.

The proof of the Jacobi identity, or, more precisely, of its interpretations(2) and (3) above, can be derived from the following very useful relationbetween Lie groups and Lie algebras.

Theorem 2. There exists a unique exponential map exp : g -+ G thathas the following properties:

a) For any X E g the curve gx(t) = exp tX is a 1-parameter subgroupin G satisfying

9x(t)9x(s) = gx(t + s), 9x(0) = e. 9x(0) = X.

b) For any Lie group homomorphism : G -+ H the following diagramis commutative:

G H

c) If G is connected and simply connected, then for any Lie algebrahomomorphism 0 : g - h there exists a unique Lie group homomorphism4 :G,H such that

Proof. a) Let X E g, and let X be the left-invariant vector field on G withX(e) = X. We define the curve gx(t) as the unique solution to the equation

9x(t) = X(9x(t))

with the initial condition gx(O) = e. The two curves with parameter s,

Oi(s) = gx(t+s) and 02(8) = 9x(t)9x(s),

both satisfy the equation d (s) = X (46(s)) and the initial condition 0(0) _gx(t). Hence, they coincide and we get gx(t)gx(s) = gx(t + s).

b) The image under 1 of a 1-parameter subgroup {9x(t)} is a 1-parame-ter subgroup h(t) in H with h(0) = Y := 4.(e)X. Hence, (D(gx(t)) = hy(t)and tD(exp X) = exp Y = exp 44(e)X.


The proof of c) is more involved and we omit it. Note only that fornon-simply connected G the statement can be wrong (consider the caseG=H=T'). O

In particular, we have the following important formula, relating Ad andad:

(12) Ad (exp X) = ead X.

For matrix groups the exponential map is given by the usual formula foran exponent:

X"expX=eXn>O

n.

and (12) gives the following identity:

eX Ye-X = Y + IX, Y] + 2 IX, IX, YJ) +6

IX, IX, IX, YJ]J + .. .

The remarkable discovery of Sophus Lie, the founder of Lie group theory, wasthat a Lie group can be almost uniquely reconstructed from its Lie algebra.To formulate the precise result, we introduce the following definition.

Two Lie groups Gl and G2 are called locally isomorphic if there existsa diffeomorphism cp between two neighborhoods of unity Ui C Gi, i =1, 2, which is compatible with the multiplication laws: p(xy) = cp(x)yo(y)whenever both sides make sense.

Theorem 3. a) For any real Lie algebra g there exists a Lie group G suchthat Lie (G) = g and this group is unique up to local isomorphism.

b) Among all connected Lie groups G such that Lie(G) = g there existsexactly one (up to isomorphism) simply connected Lie group G.

c) Let C be the center of G. Every connected Lie group 69' with Lie(G) _g is isomorphic to G/I' where r is a discrete subgroup of G contained in C.

Scheme of the proof. Sophus Lie proved a) using the partial differentialequations for a 1-parameter subgroup in G. One can also use the Ado the-orem to prove the existence of G. The uniqueness and claim b) follow fromTheorem 2 if we observe that any Lie group G admits a simply connectedcovering group G and the natural projection p : G -+ G is a local isomor-phism, so that I' = ker p is a discrete normal subgroup. Finally, c) followsfrom

Lemma 2. Let G be a connected Lie group, and let r be a normal discretesubgroup. Then I' is contained in the center of G.


Proof of the lemma. Consider the map Ay : G -' r : g t-' gyg 1. Thismap is continuous, hence the image is a connected set. But r is discreteand the connected subsets are just points. Therefore A-(G) = Ay (e) _ {y}.Hence, gyg 1= y for all g E G and y E I'. o

Example 2. Let g = R" be the n-dimensional Lie algebra with thetrivial (zero) commutator. Then the corresponding simply connected Liegroup G is just RI with the group law given by vector addition. This groupis abelian, hence coincides with its center.

The classical Kronecker Theorem claims that every discrete subgroup1 in G has the form

where Vt, v2i ... , vk are linearly independent vectors.

Therefore, there are exactly n + 1 non-isomorphic connected Lie groupsG for which Lie(G) = Rn, namely 'Il'k x Rn-k, 0:5 k < n. Q

A very important fact is the following hereditary property of Lie groups.

Theorem 4. a) Any closed subgroup H in a Lie group G is a smooth sub-manifold and a Lie subgroup in G.

b) For any closed subgroup H in a Lie group G the coset space G/Hadmits the unique structure of a smooth manifold compatible with the groupaction. 0

The non-formal meaning of this theorem is that practically all reason-able groups are Lie groups. This result should not be overestimated. Forexample, all discrete groups are 0-dimensional Lie groups. But it does notgive us any useful information.

On the contrary, for a connected group G the fact that G is a Lie groupis very important and allows us to reduce many geometric, analytic, ortopological problems about G to pure algebraic questions about g.

In conclusion we formulate one more useful observation here.

Lemma 3. Let G be a Lie group, and let H be an open subgroup. Then His also closed in G.

Proof of the lemma. All cosets gH are open in G. Therefore, the setU90H gH is open. Hence, its complement H is closed. 0

Corollary. A connected Lie group G is generated by any neighborhood Uof the unit element.


Proof of the corollary. First, replacing U, if necessary, by a smallerneighborhood V = U f1 U-1, we can assume that U = U-1. Then thesubgroup H C G generated by U is the union of open sets

U':=U, etc.,

and, therefore, is open. By Lemma 3 it is also closed, hence coincides withG.

1.4. Universal enveloping algebras.A Lie algebra is a more algebraic notion than a Lie group, but still

Lie algebras are not as clear and customary objects as associative algebras.Therefore, it is important to know that from the representation-theoreticpoint of view one can replace a Lie algebra g by a certain associative algebraU(g), which has exactly the same category of representations.

The formal definition of U(g) is better formulated in categorical lan-guage. Consider the category C(g) whose objects are linear maps of g intosome complex associative unital7 algebra A (its own for each object), satis-fying the condition

(13) W([X, Y]) = V(X)So(Y) - cp(Y)tp(X)

A morphism from (gyp : g A) to (il) : g - B) is by definition a morphisma : A B of unital associative algebras such that the following diagram iscommutative:

A0' 0 B.

Theorem 5. The category C(g) has a universal (initial) object that is de-noted by (i : g --+ U(g)). The algebra U(g) is called the universal envelop-ing algebra for g; the morphism i : g -+ U(g) is called the canonicalembedding of g into U(g).

This definition, in spite of its abstractness, is rather convenient. Forexample, it implies immediately that to any representation it of g in a vec-tor space V there corresponds a representation of U(g) in the same space.Moreover, the categories of g-modules and U(g)-modules are canonicallyisomorphic. (Check it!)

7A unital algebra is an algebra with a unit. In the category of unital algebras morphisms areso-called unital homomorphisms that send units to units.


On the other hand, the definition above is not constructive. Therefore,we give two other definitions of U(g), which are more "down-to-earth". Onecan use these definitions to prove Theorem 5.

First, choose a basis X1, ..., X,, in g and consider the unital associativealgebra A(g) over C generated by X1, ..., X with relations

(14) XiX3 - XjXi - [Xi, Xi] = 0.

It means, by definition, that A(g) is a quotient algebra of the tensor algebraT(g) by the two-sided ideal generated by the terms occurring in (14).

Second, let G be any connected Lie group with Lie (G) = g. Consider thealgebra B(g) of all differential operators on G that are invariant under leftshifts on G. The Lie algebra g is embedded in B(g) via the left-invariantvector fields (= first order differential operators without constant term):X H X (see Appendix 11.2. 1).

Exercise 2. Show that the algebras A(g) and B(g) defined above areisomorphic to U(g).

Hint. Use the universal property of U(g) to define the maps a : U(g) -bA(g) and Q : U(g) - B(g) and prove that these maps are surjective.

To prove that a and 3 are injective, use the famous

Poincar&Birkhoff-Witt Theorem. Let X1, ..., X be any basis in g.Then the monomials Mk := i(Xi)kl .. k := (k1,..., kn) E Z+,form a basis of the vector space U(g).

So, any choice of a basis in g allows us to identify the space U(g) withthe space S(g) of ordinary polynomials in X1, ... , X. Namely,

CkMk «--+ E ckXk where Xk := Xi ` Xn".kEZ+ kEZ+

A more natural and more convenient identification of U(g) and S(g) isgiven by the so-called symmetrization map. This map sym : S(g)U(g) is defined by the formula

(15) sym(P) = P(o,, ..., 0an)ea1t(X1)+...+Qni(X")

The name "symmetrization map" comes from the following property of sym.

Exercise 3. Show that for any elements Y1, ..., Ym from g we have

sym (y1y2 ... Ym) = ii i(Ys(1))i(Ys(2)) ... i(ys(m))sES,,,


In particular, sym satisfies the equation

(16) sym(Yk) = (i(Y))k for all Y E g, k E Z+,

and is uniquely determined by it. We refer to Chapter 2, Section 1.1.2, forthe worked example of this map.

Recall that there is a natural G-action on S(g) and on U(g) coming fromthe adjoint representation. The important advantage of sym is that it is anintertwiner for this G-action. In matrix notation this property has the form

(17) sym(P(g , X , g-1)) = g , sym(P(X)) , g-1

where the elements of U(g) are interpreted as differential operators on G.The proof follows immediately from (16), if we take into account that

the space Sk(g) of homogeneous polynomials of degree k is spanned by{Yk, Y E g}.

Let Z(g) denote the center of U(g), and let Y(g) be the algebra of G-invariant elements of S(g) that can also be considered as polynomial func-tions on g`, constant along coadjoint orbits.

Gelfand-Harish-Chandra Theorem. The map sym defines a bijectionof Y(g) onto Z(g)-

Proof. From (17) we see that sym(Y) is the set of G-invariant elements ofU(g). Since G is connected, the G-invariance is equivalent to g-invariance.Therefore, sym(Y) consists of elements A satisfying [X, A) = 0 for all X E g.But this is exactly the center of U(g). 0

Warning. Both Y(g) and Z(g) are commutative algebras. Moreover,we shall see later that for any Lie algebra g they are isomorphic. Butthe map sym in general is not an algebra homomorphism. It becomes anisomorphism after the appropriate correction (see the modified Rule 7 inChapter 4). 06

2. Review of the set of Lie algebrasHere we try to give the reader an overview of the totality of Lie algebras.

2.1. Sources of Lie algebras.The main source and raison d'etre of real Lie algebras is the theory of

Lie groups (see Section 1.3). There are, however, at least two other sourcesof pure algebraic nature.


Lemma 4. a) Let A be an arbitrary (not necessarily associative) algebra,and let Der A denote the space of all derivations of A, i.e. linear operatorsD : A A satisfying the Leibnitz rule

(18) D(ab) = D(a)b + aD(b).

Then Der A is a Lie algebra with respect to the commutator

(19) [D1,D2]=D1oD2-D2oD1.

b) Let A be an associative algebra; introduce the commutator in A by

(20) [a, b] = ab - ba.

Then the space A with this operation is a Lie algebra. Sometimes, it isdenoted by ALie or Lie(A).

The proof is a check of the Jacobi identity by a direct computation whichis the same in both cases.

One can deduce a more conceptual proof from the fact that the Liealgebras defined in Lemma 4 are related to certain Lie groups, possiblyinfinite-dimensional.

Remark 1. Both constructions of Lie algebras given by Lemma 4 admita natural generalization. To formulate it, we have to assume that A is a Z2-graded algebra, consider the graded derivations, and correct the definitionof the commutator according to the following sign rule:

If a formula contains a product ab of two homogeneous elements, theneach time these elements occur in the inverse order ba, the sign (-1)dleg a'deg b

must be inserted.

So, equations (2), (3), (22). (23), (24) take the form

(2') ad X ([Y, Z]) = [ad X (Y), Z] + ad X (Z)].

(3') ad[X,YJ = adX oadY - y oad X.

(18') D(ab) = D(a)b+ (-1)degn-degaaD(b),

(19') [D1, D2] = D1D2 - (-1)degDI.degD2D2D1.

(20') [a, b] = ab -

The new object arising in this way is called a Lie superalgebra. Manyimportant Lie algebras can be considered as even parts of some superalge-bras. This approach turns out to be very useful in representation theoryand other applications.


For readers who do not want to be overloaded by superterminology, wegive the description of Lie superalgebras in terms of ordinary Lie algebrasand their modules. Namely, a Lie superalgebra is the collection of the fol-lowing data:

1. An ordinary Lie algebra go.

2. A go-module g1.

3. A symmetric pairing of go-modules: 91 ® 91 - go.This data should satisfy the condition:

(*) foranyXEg1.

In fact, points 1, 2, 3 and equation (*) correspond to four differentvariants of the super-Jacobi identity which can involve 0, 1, 2, or 3 oddelements.

Example 3. Let go = s[(2, R), 91 =1R2.The action of go on 91 is the usual multiplication of a matrix and a

column vector. The pairing is

[x, y] = (xyt + yxt)J2(-xlY2 - x2y1 -2x2y2

=2x1y1 x1y2 + x2y1)

Condition (*) holds because [x, x]x = 2xxtJ2x = 0.This Lie superalgebra is denoted by osp(1 12, R) and is the simplest

representative of a series of superalgebras osp(k I 2n, R) whose even partsare direct sums so(k, 1R)®sp(2n, 1W), while the odd parts are tensor products1Rk (9 R2n. 0

Two particular cases of Lemma 4 deserve more detailed discussion.1. Let M be a smooth manifold. Put A = A(M) in Lemma 4 a). As

we have seen in Appendix 11. 2. 1, derivations of .A(M) correspond to smoothvector fields on M. So, we obtain a Lie algebra structure on the infinite-dimensional space Vect M. The commutator in this case is the Lie bracketof vector fields described in Appendix 11.2.3.

In terms of local coordinates this operation has the form

[v, w]` = v149jw1 - w39jvi.

One of the main results of the original Sophus Lie theory can be formulatedin modern terms as the

Sophus Lie Theorem. Every n-dimensional real Lie algebra 9 can be re-alized as a subalgebra of Vect M for some manifold M of dimension n. 0


So, this example has a universal character.2. Put A = Mat, (R) in Lemma 4 b). The Lie (A) in this case is usually

denoted by g[(n, R). As a vector space it coincides with Mat,,(lR), but thebasic operation is the commutator (20) instead of matrix multiplication inMat a(la).

This example is also universal as we saw before (see Ado's Theorem).

Before going further, let us introduce some terminology. The collectionof all Lie algebras over a given field K forms a category CA(K) wheremorphisms are Lie algebra homomorphisms, i.e. linear maps ir : 91 - 92that preserve the brackets:

(21) [r(X), ir(Y)] = 7r([X, Y]).

Homomorphisms of a real Lie algebra g to gl(n, R) (resp. gl(n, C)) arecalled real (resp. complex) linear representations of g. We can identifyR" (resp. C') with any real (resp. complex) vector space V. Then we usethe notation gl(V) and speak of representations of g in the space V, whichis called a representation space or g-module.

The collection of all g-modules forms a category 9-Mod where mor-phisms are linear maps that commute with the g-action. They are calledintertwining operators or simply intertwiners.

We shall consider in this book not only finite-dimensional but also infinite-dimensional representations of Lie algebras.

2.2. The variety of structure constants.To define a Lie algebra structure on a vector space V one usually chooses

a basis X1, ..., X,, and defines the structure constants ckj by

[Xe, Xj] = c Xk.

These constants satisfy two sets of equations: the linear equations

(22) c. k? _ -4 (Antisymmetry);

and the quadratic equations:

(23) 0 c?k = 0 (Jacobi identity).ijk

Thus, the collection of all real Lie algebras of dimension n with a fixed basiscan be viewed as the set A (1R) of real points of the affine algebraic manifoldA,a defined by the equations (22), (23).


If we change the basis, the structure constants will change according tothe standard action of GL(n, IR) on tensors of type (2, 1). So, two pointsof An(R) correspond to isomorphic Lie algebras if they belong to the sameGL(n, R)-orbit.

We see that the collection Ln(IR) of isomorphism classes of n-dimensionalreal Lie algebras is just the set An(R)/GL(n, IR) of GL(n, R)-orbits inA,, (R).

Note also that the orbit corresponding to a Lie algebra g, as a homo-geneous space, has the form GL(n, IR)/Aut g where Aut g is the group ofautomorphisms of the Lie algebra g. So, Lie algebras with a bigger auto-morphism group correspond to smaller orbits.

A deformation of an n-dimensional Lie algebra g is a curve 1(t), 0 <t < e, in Ln(IR) that is a projection l(t) = p(a(t)) of a smooth curve a(t) inAn(R) such that 1(0) represents the class of g.

Two deformations 11(t) = p(ar(t)) and 12(t) = p(ar(t)) are equivalentif there is a smooth function ¢ mapping (0, el) to [0, e2) with 4(0) = 0 anda smooth curve g(t) in GL(n, IR) such that ar(t) = g(t) a2Wt)) in someneighborhood of 0.

If an orbit 0 is open in An, we say that the corresponding Lie algebra gis rigid. In this case all deformations of g are trivial, i.e. equivalent to theconstant deformation lo(t) - 1(0).

In other words, g is rigid if a small perturbation of structure constantsleads to an isomorphic Lie algebra.

Proposition 2. All semisimple Lie algebras are rigid.

The converse is not true; e.g. the only non-commutative 2-dimensionalLie algebra aff(1, IR) is rigid but solvable.

If an orbit 0 has non-empty boundary, then this boundary is a unionof smaller orbits. The Lie algebras corresponding to these orbits are calledcontractions of the initial Lie algebra. In particular, the abelian Lie algebraIRn is a contraction of any n-dimensional Lie algebra.

To illustrate the introduced notions, we consider in more detail the setsAn(R) and Ln(lR) for small n.

We can use the linear equations (22) to simplify the system and reduceit to a system of n2(n-6)(n-2) quadratic equations in n2

2n-1unknowns. But

the full description of all solutions is still very difficult for large n.A complete classification of Lie algebras has been obtained up until now

only for n < 8 (using a completely different approach).On the other hand, for small n it is not difficult to solve the system (22),

(23) directly and thus describe all Lie algebras of small dimensions.


Exercise 4. Show that:a) There is only one 1-dimensional real Lie algebra: the real line R with

zero commutator.

b) There are exactly two non-isomorphic 2-dimensional real Lie algebras:the trivial Lie algebra R2 with zero commutator and the rigid Lie algebraaff(1, 1R) with the commutation relation [X, Y] = Y.

Hint. For n < 2 the quadratic equations (23) hold automatically. rConsider now 3-dimensional real Lie algebras. For n = 3 the system

(22), (23) admits a simple geometric description. Let us replace the tensorckj of type (2, 1) by a tensor density bk! := c, e'j' where E'' i is the standard3-vector in 1R3.

This new quantity bkt splits into symmetric and antisymmetric parts:

el = 8kt + akt, skl = slk, akl = -atk

which are transformed separately under the action of GL(3. 1R).Further, we replace the antisymmetric tensor akl by the covector v;

Etktakl. Using the identities

EijkEjk! = 2ok. EiJ3Eskl = 6ka - &'1

we can reconstruct the initial tensor c from sk' and vi as

Cij = Zsk1E=jl + 4 (6 v{ - bkvj).

It turns out that the system (22), (23) in terms of new quantities s and vtakes a very simple form

(24) Sk1vt = 0.

We see from (24) that the manifold A3 is reducible and splits into twoirreducible components:

1) the linear component defined by v = 0 (in the initial terms ckk = 0);2) the non-linear component given by conditions: det s = 0, v E ker s.Both components are 6-dimensional affine varieties; their intersection is

non-empty and has dimension 5.

The action of g E GL(3, R) on A3(1R) in new coordinates (v, s) has theform:(25)1v'-' (gt)-'v, s - gsgt det g- .


In other words, vt transforms as a covector (row vector) and s as a productof a quadratic form on (1R3)` by a volume form on R3.

The first component consists of quadratic forms s = s`-? on R3 on whichthe group GL(3, II2) acts according to the second part of (25). The only in-variant of this action is the signature8 (n+, no, n_) considered up to equiv-alence (a, b, c) - (c, b, a).

Thus, we get the following list:

Lie algebra

su(2) so(3)

s[(2) = so(2,1)

so(2) x 1f82

so(1,1) x R2

Heisenberg algebra h

trivial Lie algebra J3

signature dimension of the orbit

(3, 0, 0) ' (0, 0, 3) 6

(2, 0, 1) , (1, 0, 2) 6

(2, 1, 0) (0, 1, 2) 5

(1, 1, 1) 5

(1, 2, 0) " (0, 2, 1) 4

(0, 3, 0) 0

We see that the first component of L3(R) consists of six points, two ofwhich are open. They correspond to the rigid Lie algebras su(2) so(3)and sl(2) so(2, 1).

Consider the Lie algebras corresponding to the second component. Inthis case we have a non-zero vector v = {v,,,} such that skmv,n = 0. Besides,akmvm = 2ekmpvpv,n = 0. Therefore, bkmvm = 0 and 0.

This means that [g, g] C vl. So, our Lie algebra has a 2-dimensionalideal. It turns out that all such Lie algebras have the form of a semidirectproduct JR K J2. This means that in the appropriate basis the commutationrelations have the form

[X, Y] = aY + [3Z, [X, Z] = yY + 6Z, [Y, Z] = 0.

The matrices

A=C7 and A`=Cy, a/of

define isomorphic Lie algebras g and g` iff

(26) A' = c BAB-1 for some c # 0 and B E GL(2, 1R).

8For a quadratic form a on R° the numbers n+, no. and n_ denote respectively the numberof positive, zero, and negative coefficients c; in the canonical representation s(x) ctx?.The triple (n+, no, n..) is called the signature of s.


Let A, It be the eigenvalues of A; then A' has eigenvalues (cA, c p). Also,we cannot distinguish the pairs (A, p) and (µ, A). Thus, the quantity

zf (A) = A p det A)

is the simplest invariant of the transformations (26). It is a rational functionof four variables a, 3, 'y, 6 that is correctly defined when (A, p) 0 (0, 0) (i.e.outside the cone given by tr A = det A = 0).

Warning. A rational function f of n variables can be written in theform

Qwhere P and Q are relatively prime polynomials. We consider this

expression as a map from C' to P'(C) = C U {oo}. For n = 1 this mapis everywhere defined and continuous. For n _> 2 the map f is defined andcontinuous only outside the so-called indeterminancy set I(f) of codimension2 given by the equations P = Q = 0.

To express f (A) in terms of structure constants, let us consider thebilinear forms s and a, which are defined by tensors ak1 and ski on g. Ina generic point they have a 1-dimensional kernel spanned by the vector v.The induced forms on the quotient space g/Rv are non-degenerate and theratio of their discriminants equals

r (A)(A+µ)2= 2+f(A)

A - µ 2 - f(A)

So, the second component admits a non-trivial GL(3, R)-invariant ratio-nal function f (A). Therefore, it splits into a 1-parameter family of invariant5-dimensional levels f (A) = c and a singular 4-dimensional subset A= µ= 0where f (A) is not defined. Most of these levels are single GL(3, R)-orbits.The levels f (A) = ±2 and the singular set split into two orbits each.

We collect this information in the following table:

c = f (A) number of orbits

-oo<c<-2 1

c = -2 2

-2<c<2 1

c=2 2

c>2 1

C = 00 1

not defined 2

eigenvalues of ad X

real, of different signs;

A = -µ 0 0, real or pure imaginary;complex conjugate;

real, equal, non-zero;

real, different, of the same sign;one zero eigenvalue;

two zero eigenvalues.


2.3. Types of Lie algebras.The direct study of equations (22), (23) is already rather difficult for

n = 4.There is another approach to the classification problem for Lie algebras.

It is based on the study of their inner structure. For the reader's conveniencewe recall below some standard definitions.

Definition 1. A Lie algebra g is an extension of a Lie algebra gl by a Liealgebra 92 if the following exact sequence exists:

(27) 0-+02 9P+91-+0.

Here all maps are Lie algebra homomorphisms and the exactness meansthat the image of each arrow coincides with the kernel of the next one.

In other words, (27) means that g contains an ideal isomorphic to 92and the quotient Lie algebra 9/92 is isomorphic to 91.

The extension is called central if i(92) is contained in the center of g.The extension is called trivial if the map p in (27) admits a section, i.e.

a homomorphism s : 91 -+ g such that p o s = Id. In this case we also saythat 9 is a semidirect product of gl and 92. It is denoted by 91 IK g2.

We now can introduce several important types of Lie algebras:1) Commutative, or abelian Lie algebras - those with zero commuta-

tor.

2) The class of solvable Lie algebras - the minimal collection of Liealgebras that contains all abelian Lie algebras and is closed under extensions.

3) The class of nilpotent Lie algebras - the minimal collection of Liealgebras that contains all abelian Lie algebras and is closed under centralextensions.

4) The class of semisimple Lie algebras - the minimal collection ofLie algebras that contains all non-abelian simple9 Lie algebras and is closedunder extensions.

For your information we formulate several facts here from real Lie algebratheory.

Levi's Theorem. Any Lie algebra g has a unique maximal solvable ideal r,the corresponding quotient Lie algebra s = g/r is semisimple, and g = s a r.0

9A Lie algebra is called simple if it has no proper ideals, i.e. different from {o} and fromthe whole Lie algebra.


Cartan's Theorem. Any semisimple Lie algebra is isomorphic to a directsum of non-abelian simple Lie algebras. 0

So, the classification of all Lie algebras is reduced to three problems:

A) Describe all simple Lie algebras.

B) Describe all solvable Lie algebras.

C) Describe all semidirect products gl x 92 where gl is semisimple and92 is solvable.

At the present time only problem A) is completely solved (see the nextsections), while B) and C) are regarded as hopeless.

3. Semisimple Lie algebrasThe class of semisimple Lie algebras is the most interesting for many appli-cations and therefore has been the most thoroughly studied. The structureof these Lie algebras and their classification are related to the notion of aroot system.

We include the general facts about root systems in the next sections.We hope that this allows our readers not only to get the general impressionabout this theory but also use the basic results in their research. For detailedproofs and further information we refer to the books [Bou], [FH], [Hu],[MPR], and [OV].

3.1. Abstract root systems.We discuss here the notion and properties of a remarkable geometric

object: a root system in Euclidean space. It appears in surprisingly manyquite different domains of mathematics (see, e.g., [H2SV]). Our main goalis to list basic facts and explain how to use them. So, the detailed proofsare given only if they are not too involved and help in understanding. Werefer to [Bou], [FHJ, and (OV] for further information.

Definition 2. A finite set R C RI is called a root system if it satisfiestwo conditions:

R1. Zoo EZ for all a,$ER;

R2. Q-2: aER for alla,QER.The elements of R are called roots.

Besides these main conditions, some additional properties are often re-quired that define special kinds of root systems. We formulate these re-quirements here together with the name of the corresponding kind of rootsystem.

§3. Semisimple Lie algebras 299

R3a. The set R spans the whole R" (Non-degenerate root system).R3b. The set R cannot be represented in the form R1 U R2 with R11 R2

(Indecomposable root system).

R3c. If a E R, then 2a 0 R (Reduced root system).R3d. All vectors from R have the same length (Simply-laced root sys-

tem)-Let

us comment on the main definition and the additional requirements.1. The geometric meaning of Rl: the angles between two roots can be

only from the following list:

0,7r W n it 21r 37r 57r

6' 4' 3' 2' 3 ' 4 ' 6 '1r,

and for any two non-perpendicular roots the ratio of the squares of theirlengths can be only 1, 2, or 3, depending on the angle between them. Allpossible configurations of two non-perpendicular roots are shown in Figure2.

Figure 2

2. Let MQ C R" be the hyperplane orthogonal to a E R. It is called amirror corresponding to a and the reflection with respect to this mirror isdenoted by sa. The condition R2 means that the set R is symmetric withrespect to all mirrors, i.e. is invariant under all reflections s,,, a E R.

The group generated by reflections se,, a E R, is called the Weyl groupcorresponding to the root system R. It is a finite subgroup of O(n, R).

Note that conditions RI and R2 are invariant under similarities (i.e.rotations and dilations) of R'. We shall not distinguish between two rootsystems that are similar to one another.

3a. If this condition is not satisfied, we can simply replace R" by asmaller space Rm spanned by R. The number m is called the rank of theroot system R.

3b. If R1 U R2 and R11 R2, then both subsets R1 and R2 are themselvesroot systems and we can study them separately.

3c. We propose the following

Exercise 5. a) Let a E R and ka E R for some k E R. Then k =±1, ±2, or ±'.


b) Show that all possibilities above are indeed realized. (Hint: considerroot systems of rank 1.) 6

It is worthwhile to mention that for any n E Z+ there is only one inde-composable non-reduced root system of rank n. It is denoted by BC" andconsists of vectors ±e2 ± ej, 1 < i 96 j < n, and ±ek and ± 2ek, 1 < k < n.

3d. The simply-laced root systems are also called ADE-systems (becauseof their classification described below). They appear in a larger number ofclassification problems than general root systems.

Now we introduce some basic definitions for general root systems.I0

The complement to the union of all mirrors in R" splits into several0

connected components C1, called open Weyl chambers; their closures C;are called simply Weyl chambers.

We say that a vector A E R is regular if it lies in an open Weyl chamberand singular if it belongs to at least one mirror.

A linear order in Rn is an order relation that is compatible with thestructure of a real vector space, i.e. such that the sum of positive vectors ispositive as well as a positive multiple of a positive vector.

Exercise 6. Show that any linear order relation is the lexicographicalone with respect to an appropriate (not necessarily orthogonal) basis in R".

Hint. Show that for any linear order in R" there exists a hyperplanesuch that its complement splits into open half-spaces, one of which consistsof positive vectors and the other of negative ones. Then apply induction. 46

Every linear order in R' induces an order relation in the root systemR. We denote by R+ (resp. R_) the set of positive (resp. negative) roots.Clearly, there are finitely many order relations on R. To formulate a moreprecise statement, recall that for any cone V C II8" one can define the dualcone V' as the set of vectors v' satisfying the inequality (v', v) > 0 for allvEV.

Proposition 3. For any linear order in lf8" the convex cone generated byR+ is exactly the dual cone to one of the Weyl chambers.

Later on we shall fix an order relation and denote by C+ the positiveWeyl chamber defined by

(28) C+= {AE1W1 (A, a)>0for all aER+}.We call a root a E R+ decomposable if it can be written in the forma = f3 + -y where both summands are also from R+. Otherwise, we call a asimple root.

10Some of these definitions we already used in Appendix 11.2.2 for a special root system inconnection with finite-dimensional representations of GL(n, R).

§3. Semisiznple Lie algebras 301

Lemma 5. Any root a E R has a unique decomposition a = Ek=1 ckakwhere all ak are simple roots. Moreover, the coefficients ck are either allfrom Z+ or all from Z_.

Proof. It is clear from the very definition of simple roots that any positiveroot is a linear combination of simple roots with non-negative integer co-efficients. Hence, any negative root is a linear combination of simple rootswith non-positive coefficients.

It remains to prove the uniqueness, i.e. linear independence of simpleroots. We derive it from the following geometric property: two simple rootsa, /3 never form an acute angle (or, in algebraic form: the scalar product(a. /3) is never positive).

Assume the contrary. Then the number k = 2Q Q is a positive integerand if we assume that (a, a) this number can only be 1. Thus,sQf3 = /3 - a. If sQ/3 E R+, then 1 = s,,,3 + a is not simple; if sQ/3 E R_,then a = /3 - sa/3 is not simple. Contradiction.

We return to the proof of the lemma. Suppose there is a non-triviallinear relation between simple roots: >k ckak = 0. Put the terms withpositive coefficients ci on one side and the terms with negative coefficientscj on the other. We get a relation of the form

v := E alai = > bjaj =: wiEI jEJ

with positive coefficients ai, bj.

Since the relation is non-trivial, we have v = w 0 0, hence Iv12 = Iw12 >0. On the other hand,

(v, w) = E a.,bj(ai, aj) < 0,iEI,jEJ

which is impossible. 0

We denote by IT the set {ai, ... , an,} of simple roots. Observe that C+is bounded by mirrors M1 i ... , M, corresponding to simple roots.

The whole root system R can be reconstructed from the system H ofsimple roots. The proof of this and many other facts (including Proposition3) is based on the following fundamental fact.

Proposition 4. The group W acts simply transitively on the set of Weylchambers.

The proof is based on the following lemma, which is of independentinterest.


O

Lemma 6. Let A E C+, and let it E R" be an arbitrary vector. Let W(p)denote the W-orbit of A. Then W(p) has a unique common point with C+,which is the nearest point to A of W (,u).

Sketch of the proof. Assume for simplicity that p itself is the nearestpoint to A among all {w p, w E W1. If p 0 C+, it is separated from Aby a mirror MO, ,6 E II. But in this case the point spp is strictly closer toA than p. Conversely, if p E C+, one can show that for any w E W thedifference p - w p is a linear combination of simple roots with non-negativecoefficients. It follows that p is the only nearest point to A. D

Corollary 1. The group W acts simply transitively on the set of all linearorder relations on R. So, the number of order relations on R and the numberof Weyl chambers are both equal to the order of the Weyl group.

Corollary 2. The stabilizer in W of any vector A E R" is generated byreflections in the mirrors that contain A. In particular, the stabilizer of aregular vector ,\ E R" is the trivial subgroup {e}.

To describe the system IT = {al, ... , an) of simple roots, it is convenientto introduce the so-called Cartan matrix A E Mat"(Z) with entries

(29) Aij = (a,,:), 1<i,j:5 n.

The famous result by Dynkin claims that all the information about a givenroot system R is contained in its Cartan matrix A.

Since W acts on R" by orthogonal transformations, Proposition 4 impliesthat the Cartan matrix does not depend on the choice of the order relation(hence on the choice of II).

There exists a simple graphic way to present the information encoded ina Cartan matrix A. Introduce the Dynkin graph rA with vertices labelledby simple roots, or just by numbers 1, 2, ..., n. The two different verticesi and j are joined by nb,j = Az,j Aj,= edges. If ja=i > 1021, we add an arrowdirected from i to j.

The same graph without arrows is sometimes called a Dynkin diagram.The following properties of a Cartan matrix A show that it can be re-

constructed from the graph I'A.

Proposition 5. a) The diagonal elements of A are all equal to 2.b) The of diagonal elements of A are non-positive and satisfy the con-

ditionAi,j=0 Aj,1=0.


c) All principal minors of A are positive. In particular, the quantitynij = Ai,J A9,i can take only four values: 0, 1, 2, or 3.

Thus, we have a chain of objects that define each other up to naturalisomorphism:

R(root system) i- lI(set of simple roots) +-+ A(Cartan matrix)

FA(Dynkin graph)

Example 4. Here we list all Cartan matrices of size 2 x 2 and thecorresponding graphs, and we also draw the reduced root systems of rank 2:

17A: 0 0 0-0 0===> 0 0.4:-- 0 000 0 0

Figure 3. Reduced root systems of rank 2.

0Example 5. The system A. Let {ei}o<i<n be the standard basis in

Rn+1 Put

R={ai?:=ei-ej I 0<i96 j <n},

the set with 2 (2) = n2 - n elements. This root system is degenerate of rankn. We leave it to the reader to check that it is indecomposable, reduced,and simply laced.

Let x 0 , . . . , xn be the standard coordinates in JEtn+1. The mirrors Mi3 are

given by the equation xi = x3. They split Rn+1 into (n+ 1)! Weyl chambers.If we choose the standard lexicographical order in Rn+1, then R+ = {aij Ii < j} and the positive Weyl chamber C+ is defined by inequalities xo >xl > ... > xn.

The Weyl group is generated by permutations xi <-+ x-1 and coincideswith the symmetric group Sn+1


Lemma 7. The system of simple roots for An is II = {ak := ak-1,k I 1 <k < n}.

Proof. If j > i + 1, then aij = a;,i+1 + ai+1,j. So, a positive root atj canbe simple only if j = i + 1. On the other hand, the number of simple rootsis at least n, the rank of the system. Hence, all roots ak are simple. 0 0

Example 6. The system D. Let R C Rn, n > 2, be the set {±ei ± eji 0 j}. It contains 4 (2) = 2n2 - 2n vectors. This is a non-degenerate rootsystem of rank n. It is also indecomposable, reduced, and simply-laced.The mirrors for this system are given by the equations xi = fxj. The Weylgroup consists of all permutations of coordinates and all changes of an evennumber of their signs. As an abstract group, it is a semidirect product.S DC

Znn 2

With respect to the standard ordering, the positive Weyl chamber isgiven by inequalities x1 > , , , > xn-1 > Ixnl

Exercise 7. Show that the system of simple roots for Dn is

II={ek-ek+1 I 1 <k<n-1}U{en_1+en}.

Hint. Follow the proof of Lemma 6 and use the relation

(ei + ej) _ (ei - ej+1) + (ej + ej+1) for i < j < n.

*0Example 7. The system E8. This is the most complicated simple root

system. It has many realizations from which we choose only two:1. Let R C 1R8 be the set of 4 (2) + 27 = 112+128 = 240 vectors:

8 8

{fe,±e.I Ui=1 i=1

2. Consider 2 (2) = 72 vectors of the form ei - ej, 0 < i # j < 8, and2 (9) = 168 vectors of the form ±(ei + ej + ek) with different i, j, k. Theprojections of these 240 vectors on the hyperplane >t=o xk = 0 form a rootsystem, isomorphic to system 1.

These two realizations show that E8 contains D8 and As as subsystems.Exercise 8.' Show that Es is a reduced indecomposable simply-laced

root system of rank 8. Find the simple roots.


Answer: In the first realization:

ak = ek+1-ek+2, 1 < k < 6, a7 = e7+e8, as = 12(e1 -e2-...-e7+e8).

In the second realization:

ak=ek-ek+1, 1 <k<7, as= 3(eo+e1+...+e8)-e1-e2-e3.

40All root systems are classified. To formulate the result, we observe that if

the Dynkin graph r of R is not connected, then the system R is a direct sumof its orthogonal subsystems corresponding to the connected components ofF. So, it is enough to classify all connected Dynkin graphs corresponding toindecomposable root systems. Also, keeping in mind the application to Liegroups, we restrict ourselves by reduced systems.

Theorem 6. The connected Dynkin graphs corresponding to reduced rootsystems form four infinite series and five isolated examples. They are drawnin Figure 4. O

An : o-o-...-o, n>1, Bn : o-o- -o-o : o, n>2,0

1Cn: n>3, Dn: o-o-...-o-o, n>4,0

En: n=6,7,8.F4: o-oho-o, G2: 000.

Figure 4. Dynkin graphs of indecomposable reduced root systems.

Exercise 9. Compute the determinants of the Cartan matrices for theseries An, Bn, Cn, Dn, E.

Hint. Show that detn satisfies the recurrent relation

detn+1 = 2 detn - detn_ 1,

hence is a linear function of n. Also use the isomorphisms Al - Bl - C1,B2 = C2, A3 D3, E5 D5, E4 = A4, E3 A2 + Al.

Answer: In the self-explanatory notation we have an = n + 1, b _cn = 2, do = 4, en = 9 - n. 46

Now we discuss some properties of root systems related to the Weylgroup. Define the length of w E W by the formula:

(30) 1(w) = # (w(R+) n R-).

In words: 1(w) is the number of positive roots a that w transforms to anegative root.


Proposition 6. The number 1(w) is equal to the minimal length of thedecomposition

w=silsiz...sik

into the product of canonical generators. The decomposition of minimallength is called reduced.

Proof. Let A be any interior point of the positive Weyl chamber C+. Con-sider a path joining A with wA. This path intersects some mirrors. Choosethe path with a minimal number of intersections. Suppose this number is m

0 0 0 0

and the path goes through the open Weyl chambers C+ = CO, C1,..., C.-

Choose representatives Ai E Ci so that Ai_1 and Ai are symmetric with re-00 0

spect to the mirror separating Ci_1 and Ci. Then all points A = A0, Al, ... ,Am = w(A) have the form Aj = wj (A).

Moreover, since w3A and wj_1A belong to adjacent Weyl chambers, thesame is true for A and w,-lwj_1A. But we already observed that C+ isbounded by mirrors corresponding to simple roots. Therefore w.,-. Iwj_ 1 = sipand wj = wj_1sip. We get the decomposition wj = si,si2 sip.

Thus, to any path joining A with w A and intersecting m mirrors therecorresponds the reduced decomposition of w into a product of m generators.We proved that the length of a reduced decomposition of w E W is equal tothe number of mirrors separating A and w A for any regular vector A.

Now we use the following

0Lemma 8. For A E C+ the inequality (a, A) > 0 is equivalent to a E R.

0

Proof of the lemma. By definition, C+ consists of points A satisfying(a, A) > 0 for all a E R+.

Conversely, if (a, A) > 0, then a E R+, since otherwise a E R_ and(a, A) is negative. 0

Suppose now that a mirror MQ separates A and w A. Replacing, ifnecessary, the root a by -a, we can assume that (a, A) < 0 but (w-1 a, A) _(a, w A) > 0. From Lemma 8 we conclude that a E R_, w-1 a E R+.Hence, w sends the positive root w-1 a to the negative root a.

Conversely, if w sends a positive root a to a negative one, then the mirrorMw.a separates A and w A. Therefore, the number of mirrors, separating Aand w A, is exactly the length of to. 0

Corollary. The generators si have length 1, i.e. only one positive root ahas the property sia E R_. (Certainly, this is the simple root ai.)


To each root system of rank n (or rather to the corresponding Weylgroup W) one can associate the series of integers el < e2 < < en thatare called exponents and have many remarkable properties." We mentionsome of them here.

Proposition 7. a) The distribution of elements of W according to theirlength is given by the following generating functions:

_ n 1 - tei+171

(31) E tI(w =H 1t =fl(1+t+...+te').

wEW i=1 i=1

b) The algebra of W-invariant polynomials on ][fin is freely generated byhomogeneous polynomials P,, ..., P such that deg Pi = e; + 1.

c) The algebra of W-invariant elements in the Grassmann algebra A(Rn)is freely generated by homogeneous elements Ql, ... , Qn such that deg Qi =lei+1.

d) The exponents satisfy the following relations:

1) f 1 (ei + 1) = IWI; 2) E 1 ei = IR+I; 3) ei + en-i+l = hwhere h is the order of the so-called Coxeter element C = s182 sn EW.12

To a root system R there correspond two important lattices (i.e. dis-crete subgroups) in Rn: the so-called root lattice freely generated by simpleroots:(32) Q=Z.a1+...+7L.anand the weight lattice

(33) P= JLERnI 2(A'a)EZforallaErI(a, a)The weight lattice P is freely generated by fundamental weights w1, ... ,wn defined by

2(w, , ai) -() (ai, ai)

J

In terms of the Cartan matrix the relation between simple roots and funda-mental weights can be written in the form

(35) aj = wiAia or, symbolically, a = w A.

In particular, (39) implies that Q is a sublattice in P. Moreover, it has afinite index in P, i.e. the quotient group P/Q is finite and #(P/Q) = det A.

I I There is a more general notion of generalized exponents introduced by B. Kostant. Seealso (Ki13( and references therein.

121n fact, this product depends on the choice of 11 and on the numeration of simple roots,but all these elements belong to the same conjugacy class in W, hence have the same order.


3.2. Lie algebra s1(2, C).This Lie algebra enters as a building block in all other complex semisim-

ple Lie algebras. Therefore, it is important to know in detail the structureof this Lie algebra and of the corresponding Lie group SL(2, C), as well astheir representation theory. We recall here the basic facts of this theory.

Proposition 8. a) Any finite-dimensional representation of sl(2, C) is adirect, sum of irreducible representations.

b) For any integer n > 0 there is exactly one equivalence class 7r,t ofirreducible representations with dim rn = n + 1.

In particular, ro is the trivial representation ro(X) = 0, 7rl is the defin-ing (tautological) representation 7rl (X) = X, and 7r2 is the adjoint represen-tation 7r2(X) = ad X.

c) The representation Tr,t is equivalent to the n-th symmetric power ofr1

d) There are the following isomorphisms:

rnin(nt.n)

rtrt 0 rn = ® rm+n-2s;s=0

n

A2(rn) = ® r2n-2s+1.9=1

s=0

Proof. We start with the explicit construction of irreducible representa-tions. Observe that the group G = SL(2, C) acts from the right on thespace C2 of row vectors

(x, y) '-' (x, y) 1 a ) = (ax + 7y, ;3x + by).

Let V be the space of polynomials in the variables x, y. Then we havea representation it of the group C in V:

(r(g)P) (x, y) = P(ax +'ry, /3x + by) for g = ( a E G.

Let us compute the corresponding action of g = sl(2, C). Assume that X E 9

has the form X = I a a I . Then exp rX = (1 rc a 1 T b a i + ONand we get \ ///

tS2(rn) =E) r2n-2s*,

r.(X) = - -ir(exp rX) I T=o= a(x8x - ye) + byex + cx8y.


The representations (7r, V) of G and (7r., V) of g are infinite-dimensionalbut they split into the sum of finite-dimensional representations since V isthe direct sum of invariant subspaces V. consisting of homogeneous polyno-mials of degree n.

It is clear that (7rn, Vn) = Sn (7r1 i V1) because Vn = Sn (Vi) . We showthat all these representations are irreducible.

Let {E, F, H} be the canonical basis of sl(2, C) (see Example 8 in Sec-tion 3.4 below). Then the formula above for 7r. implies

irn(E) = xay, 7rn(F) = yax, 7rn(H) = xax - yOy.

Choose in Vn the natural monomial basis

Vk=xn-kyk, 0<k<n.

The representation 7rn is given in this basis by the formulae:

(36) 7rn(E)vk = kvk-1, 7rn(F)vk = (n - k)vk+1, 7rn(H)vk = (n - 2k)vk.

Exercise 10. Prove that all 7rn are irreducible.Hint. Show that starting from any non-zero polynomial P E Vn and

acting several times by 7rn(E) we obtain the monomial cxn with c # 0.Then, acting by 7rn(F), we get all other monomials. 4

Thus, we have constructed a family {7rn}, 0 < n < oo, of irreduciblerepresentations of g. They are pairwise non-equivalent, since they havedifferent dimensions. The next step is to prove that any finite-dimensionalirreducible representation of g is equivalent to some of the representationsirn

Let (p, W) be any finite-dimensional representation of g. Consider aneigenvector w E W for the operator p(H), and let A E C be the correspond-ing eigenvalue. From the commutation relations (43) we obtain:

p(H)p(E)w = p(E)p(H)w + [p(H), p(E)]w= Ap(E)w + 2p(E)w = (A + 2)p(E)w.

In other words, the vector p(E)w, if non-zero, is also an eigenvector for p(H)with the eigenvalue A + 2. It follows that vectors w, p(E)w, p(E)2w, ... arelinearly independent until they are zero because they are eigenvectors forp(H) with different eigenvalues. Since dim V < oo, we have p(E)kw = 0for some k _> 1. Choose the minimal k for which this is true and denotep(E)k-1w by wp. It is an eigenvector for p(H) and we denote the corre-sponding eigenvalue A + 2k - 2 by Ao.


Now, introduce the vectors wk := p(F)kwo. The same argument showsthat all wk are eigenvectors for p(H) with eigenvalues (Ao - 2k). Hence,vectors wo, are independent until they are zero. Let w;, be the lastnon-zero vector in this sequence.

Exercise 11. Prove that the linear span W' of wo, ..., w;, is a g-invariant subspace in W.

Hint. The invariance of W' with respect to p(F) and p(H) follows fromthe very construction. Show by induction on k and using (43) that

p(E)wk = Ckwk_1 with ck = k(Ao - k + 1).

If (p, W) is irreducible, we conclude that W' coincides with W. Thespectrum of p(H) consists of eigenvalues (.\o - 2k), 0 < k < n. On the otherhand, tr p(H) = tr [p(E), p(F)] = 0. It follows that ao = n = dim V - 1.We come to the following final formula:

p(E)wk = k(n-k+1)wk_1, p(F)wk = wk+1+ p(H)wk = (n-2k)wk.Comparing this with (36), we see that the map w'k '- nik ,vk establishesthe equivalence between (p, W) and (7rn, Vn).

It remains to show that any finite-dimensional representation of g is adirect sum of irreducible representations. There are two ways to do this.

The first, algebraic, way uses the quadratic Casimir element 0 in theuniversal enveloping algebra U(g). We note that a dual basis to E, F, Hin g with respect to the Killing form is 117, 1E, a H. Hence, 0 =

4 4

s (H2 + 2EF + 2FE).An easy computation shows that 7rn(p) = n 82 , 1v,,. Therefore, the

spectral decomposition of 7rn(0) in V gives us the decomposition of V intoisotypic components for g.

The second, analytic, way uses the equivalence of representation cate-gories for g, for its compact real form su(2, C) and for the correspondingcompact group SU(2, C) (see Chapter 5). The latter category is evidentlysemisimple: any object is a direct sum of simple (indecomposable) objects.0

3.3. Root system related to (g, h).We start with the formulation of basic facts and main structure theorems

for semisimple Lie algebras.For any Lie algebra g the following bilinear form ( , )K was first defined

by W. Killing and extensively used by Elie Cartan; we call it the Killingform:

(37) (X, Y)K := tr (ad X ad Y).It is clearly invariant under all automorphisms of 9.


For solvable Lie algebras this form is practically of little use because itvanishes on any nilpotent ideal and, in particular, on the derived Lie algebra[g, g]. On the contrary, for semisimple Lie algebras it is a very importanttool.

Theorem 7 (E. Cartan). The Killing form on a Lie algebra g is non-degenerate if 9 is semisimple.

Indeed, if g has a non-zero abelian ideal a, then it is in the kernel of theKilling form. Conversely, if g has no abelian ideals, then it is a direct sumof non-abelian simple ideals, hence is semisimple.

So, it remains to show that a simple Lie algebra has non-degenerateKilling form. It follows from the fact (which we do not prove here) that thekernel of the Killing form is a nilpotent ideal in g. 0

For any semisimple Lie algebra g we fix the G-invariant form ( , ) ong`, which is dual to the Killing form on g.

The main structure theorem for semisimple Lie algebras is the following.

Proposition 9. Let g be a complex semisimple Lie algebra.a) There exists a canonical decomposition of g into a direct sum of

three vector subspaces, which are subalgebras, but not ideals in g:

(38) 9=n-0 4®n+

such that for any linear finite-dimensional representation (7r, V) of 9, af-ter suitable choice of a basis in V, the elements of these subspaces go re-spectively to lower triangular, diagonal, and upper triangular matrices ins((m, C), m = dim V.

b) The decomposition (38) is unique up to an inner automorphism of g.c) The subalgebra h is called a Cartan subalgebra and is character-

ized by the property: it is a maximal abelian subalgebra consisting of ad-semisimple elements.13 0

Applying Proposition 9 to the adjoint representation (ad, g) of g, we seethat all operators ad H, H E h, are simultaneously diagonalizable. It followsthat

9 = ® 9aaEA

where A is a finite subset in b` and go consists of elements X E g satisfying

(39) (H, XJ = a(H) X for all H E h.

13An element X is called ad-semisimple if the operator ad X in an appropriate basis is writtenas a diagonal matrix.


It is clear that go = h (since 4 is a maximal abelian subalgebra).

Exercise 12. Show that

C ga+Q fora +E A,a) [ga' gQ] { = 0 fora +,13 A.

b) (ga,g0)K=0ifa+00.

Hint. Use the relation (39) and the Jacobi identity.It follows that the Killing form establishes the duality between ga and

g-a and, in particular, is non-degenerate on h = go.Let us denote by R(g, ll) or simply by R the set of non-zero elements in

A. Denote by V the real vector subspace in tl" spanned by It.The relation between complex semisimple Lie groups on one hand and

abstract root systems on the other hand can be formulated as follows

Theorem 8. a) The bilinear form is positive definite on V and R(g, f})is a reduced non-degenerate root system in the Euclidean space V.

b) Any abstract reduced non-degenerate root system is obtained from acertain pair (9, ll).

c) Two complex semisimple Lie algebras are isomorphic if they haveisomorphic root systems (i.e. isomorphic Dynkin diagrams).

Sketch of the proof. The proof is heavily based on the representationtheory of the simple Lie algebra sl(2, C) (see the previous section).

For any root a we choose a non-zero vector X. E ga. Ftom the invarianceof the Killing form we conclude that

([X0, X_a], H)K = (Xa, [X-a, H])K = a(H)(X0, X-a)K

Since the Killing form is non-degenerate, we can assume that (Xa, X_a)K =1. Then the element a := [Xa, X_a] has the property a, H)K = a(H) forall H E Il. In other words, a is exactly the element of fl that corresponds toa E h` under the isomorphism h t, induced by the Killing form.

The element a is sometimes called the dual root to a. The collectionof all dual roots forms a root system which is denoted by R'' and is calledthe dual root system.

Let g(a) denote the 3-dimensional subalgebra in g spanned by Xa, X_a,and a. It is isomorphic to sl(2, C) with the canonical basis E = Xa, H =Ha, F = X_a where Ha (a

QLet us study the adjoint action of g(a) in g. Choose a root,3 and considerthe so-called a-string of ;Q, i.e. the set of roots of the form {,0 + ka, -p <

§3. Semisimple Lie algebras

k < q} such that the root vectors Xk E 00+k« satisfy the relations

[Xa, Xk]ckXk+1 for k < q,

0 fork=q,r for k > -p,

(X-«, Xk] - {l 0 for k = -p,

313

with some non-zero constants ck, c'k.

It is clear that the subspace S(3) C g spanned by elements {Xk}_p<k<qis an irreducible 9(a)-module. From the results of the previous section itfollows that for some n E Z+ we have q + p = n and

[H«, X,3+k«] = (2k + p - q)X,3+k«

Since H« = aa

, we conclude that

2(a, )3) _ (p - q)(a, a) or2(a, Q)

a)= p - q E Z.

(a,

So, we have verified the first axiom of root systems and have also given arepresentation-theoretic interpretation of the integers A,,.,6

To check the second axiom, we consider again the Lie subalgebra g(a)and define the element s« E exp g(a) c G by

s« = exp 2 (X. - X_«)-

The corresponding inner automorphism of 9 preserves all subspaces of typeS(/3) and interchanges the extreme vectors X-P and Xq. Since the roots,l3-pa and 13+qa are symmetric with respect to the mirror M«, we concludethat the corresponding automorphism of 1)' is just the reflection s«.

As a byproduct we get the realization of the abstract Weyl group as thegroup W = NN;(h)/H of automorphisms of h and of 4`. In Chapter 5 wesaw that W is also the group of automorphisms of full flag manifolds.

The proofs of statements b) and c) are more involved and we omit them.We refer the reader to [Bou], [FH], [GG], [OV]. 0

The decomposition (42) defines the additional structure on R(g, h): thesplitting of R into two disjoint subsets R+ and R_ = -R+ defined by therelations

n+ = ® 9«, n- = ® 9«.«ER+ «ER_


The roots from R+ (resp. R_) are called positive roots (resp. negativeroots).

We denote by r the dimension of of (i.e. the cardinality of Rt) andby n =: rk g the dimension of l). These numbers are related by the equality2r + n = dim g.

Let al, ..., a,, be the simple roots. Let Hi = ay, 1 < i < n, be thedual roots. They are characterized by the property

(40) aj (Hi) = Ai,j or wj(Hi) = bid for all j = 1, ..., n.

The second important result is

Proposition 10. a) For any a E R the subspace g,,, is 1-dimensional, henceis spanned by a single element X, which is called a root vector.

b) The root vectors Xa can be normalized so that the following relationshold:

Na,o Xa+,a when 0 # a +,3 E R,(41) (Xa, X,g] = 0 when 0 34 a +,3 R,

H,, when a +# = 0

where Na j are non-zero integers satisfying N_a,_p = -Na,0 = No,,,. 0

Corollary. a) The real span of elements Xa, X_a, Ha for all a E R is areal form of g. It is called the normal or split real form and ie denoted by9n

b) The real span of elements x°-x_° iH Xa+X_a2 2, 2i for all a E R is

also a real form of g. It is called the compact real form and is denoted by9c.

Informally speaking, this corollary tells us that every complex semisim-ple Lie algebra g is in a sense "built" from r copies of sl(2, C), while itsnormal (resp. compact) real form is constructed from r copies of sl(2, R)(resp. su(2, C)).

Therefore, a good understanding of these three Lie algebras is very essen-tial for the whole theory. In particular, the simplest proofs of Propositions6-9 can be obtained by considering the restrictions of different representa-tions (ir, V) of g to subalgebras g(a), a E R.

Now we are in a position to formulate the classification theorem forsimple Lie algebras. As usual, we start with the complex case.

Theorem 9. Every complex semisimple Lie algebra g is a direct sum ofsimple Lie algebras. There are four infinite series of complex simple Lie


algebras, which are called classical simple Lie algebras:Ar, 2.1 sl(n + 1, C), n > 1; B. so (2n + 1, C), n > 2,Cn sp(2n, C), n > 3, D ^_' so(2n, C), n > 4,

and five isolated examples: G2, F4, En, n = 6, 7, 8, which are called excep-tional simple Lie algebras. The corresponding root systems are describedin Chapter 5. 0

The index n here denotes the rank of g which was defined above as thedimension of a Cartan subalgebra . It is also equal to the codimension ofa generic adjoint orbit.14

All classical simple Lie algebras have explicit matrix realizations:

sl(n + 1, C) -- all traceless complex matrices of order n + 1;so(n, C) - all complex antisymmetric matrices X of order n.

sp(2n, C) - all complex matrices X of order 2n which satisfy theequation XtJ2n + J2nX = 0 (or the equivalent condition: S := J2nX issymmetric).

Remark 2. The classical Lie algebras A,,, Bn, C, Dn are defined forall natural numbers n. The restrictions on n in Theorem 9 are made toavoid the appearance of non-simple or isomorphic Lie algebras. O

Exercise 13. Prove the following isomorphisms:a) B1 Cl Al; b) C2 ^_- B2; c) D1 Cl; d) D2 Al ® A1;

e) D3 ^_- As.

Hint. Compare the corresponding root systems. See also Chapter 5. 4We will not give the description of exceptional Lie algebras here. The

interested reader can get some information about them in Chapter 5.Here we discuss only some basic notions which are needed to understand

the main ideas. We hope that this will allow our readers to not only getthe general impression about this theory, but to also use the basic resultsin their research. For proofs and further information we refer to the books[Bou], [FHJ, [Hu], and [OV].

3.4. Real forms.For a real Lie algebra (g, we define its complexification (gc,

as a complex vector space gc = g®RC with bilinear operation extendedfrom .J by complex linearity. Simply speaking, passing from g to gCmeans that we keep the same structure constants but allow complex linearcombinations of basic vectors.

"If we replace "adjoint" by "coadjoint". we get the definition of rank for an arbitrary Lie

algebra. See Chapter 1 for details.


Now let g be a complex Lie algebra. We say that a real Lie algebra gois a real form of g if g is isomorphic to (go)c as a complex Lie algebra.

Remark 3. A given complex Lie algebra g can have no real form at allor have several different real forms (non-isomorphic as real Lie algebras).

Let 0 c A,(C) be the GL(n, C)-orbit corresponding to g (see the pre-vious section). The existence of a real form means that 0 has non-emptyintersection with In other words, for an appropriate choice of basisall structure constants are real.

The general theory of real algebraic groups says that 0 f1 splitsinto a finite number of GL(n, R)-orbits. It means that for any complex Liealgebra g there exist only finitely many real forms (up to isomorphism). Q

Example 8. Let g = sl(2, C) be the 3-dimensional complex Lie algebraof traceless 2 x 2 matrices. It has two remarkable bases with real structureconstants:

1st canonical basis: X = 2 (01 0 )' Y 2

(00 )' Z

Z (0 of )i

with commutation relations`

l

(42) [X,Y]=Z, [Y,Z]=X, [Z,X]=Y,

and

(0 1)2nd canonical basis: E =0 0

with commutation relations J

0 )F =(?

)H=(0 -1

(43) [E, F] = H, [H, E] = 2E, [H, F] = -2F.

Exercise 14. Check that the two bases above define different real Liealgebras.

Hint. Consider the signature of the Killing bilinear form in g:

(44) (X, Y) := tr (ad X ad Y).

4We see that g has at least two different real forms: one isomorphic to

su(2) and the other isomorphic to sf(2, R). Actually, s[(2, C) has no otherreal forms.

The description of real simple Lie algebras is based on the following fact.

Theorem 10. a) Every complex simple Lie algebra remains simple whenconsidered as a real Lie algebra.


b) Every complex simple Lie algebra has a finite number (> 2) of realforms that are simple real Lie algebras.

c) Every real simple Lie algebra is obtained from a complex simple Liealgebra by the procedures described in a) or b). 0

We use the following standard notation to list all the real forms of clas-sical simple complex Lie algebras:

sl(n, R) - the set of all traceless real matrices of order n;

sl(n, H) - the set of all quaternionic matrices X of order n thatsatisfy the condition Re tr X = 0;

sp(2n, R) - the set of all real matrices X of order 2n that satisfythe equation XtJ2n + J2nX = 0 (or the equivalent condition: S := J2nX issymmetric);

so*(2n) (denoted also as u*(n)) - the set of all quaternionic matricesX of order n that satisfy the equation a(X)`J2n + J2na(X) = 0 (the equiv-alent condition: S := J2na(X) is Hermitian); here a denotes the embeddingof Matn(H) into Mat2n(C);

so(p, q, R) - the set of all real matrices X of order n = p + q thatsatisfy the equation XtIp,g + Ip,qX = 0;

su(p, q, C) - the set of all complex traceless matrices X of ordern = p + q that satisfy the equation X *Ip,q + Ip,gX = 0;

su(p, q, H) (denoted also as sp(p, q)) the set of all quaternionicmatrices X of order n = p+q that satisfy the equations X'Ip,q + Ip,qX = 0,RetrX=0.

In the last three cases when p = n, q = 0 the shorter notation so(n),su(n), sp(n) is used.

The list of real forms.An : sl(n + 1, R); su(p, q, C), p,.5 q, p + q = n + 1; sl(n , H) for

n odd.Bn: so(p,q,R),p:5q,p+q=2n+1.Cn : sp(2n, R); sp(p, q) su(p, q, H), p < q, p + q = n.Dn : so(p, q, IR), p < q, p + q = 2n; so`(n) for n even.Recall that among all the real forms of a given complex simple Lie alge-

bra g there are two of special interest (cf. Corollary in Section 3.3).The compact form & is characterized by the following equivalent prop-

erties:

a) it admits a matrix realization as a subalgebra of su(m);


b) the quadratic form Q(X) = tr (ad X)' is negative definite.The normal form g,, is also characterized by two equivalent properties:

a) it admits a real canonical decomposition, i.e. a real matrix realizationsuch that

(38') 9 =n_®(1®n+

where the three subspaces are intersections of g with lower triangular, diag-onal, and upper triangular subalgebras;

b) the quadratic form Q(X) = tr (ad X)2 has the signature (NN, N2n" )

where N = dim g, n = rk g := dim h.Exercise 15. Locate compact and normal forms in the above list of

real forms.Hint. Compute the signature of the quadratic form Q(X) = tr (adX)2.

46

Exercise 16. Establish the following isomorphisms of the real formscorresponding to the isomorphisms from Exercise 13.

a) su(2, C) so(3, R) su(1, H), s((2, IR) so(2, 1, R) ~ sp(2, R).b) sp(4, R) so(3, 2, R), su(2, H) - so(5, 1K), su(1, 1, H) - so(4, 1, K).c) so(2, 1K) so(1, 1, 1K) ^--K'.

d) so(4, 1K) = su(2) e su(2), so(3, 1, K) - s((2, C), so(2, 2, 1K)su(1, 1, C), so* (4) - so(3, K) (D s((2, 1K).

e) so(6, 1K) ^_- su(4, C), so(5, 1, H) ^' s((2, H), so(4, 2, 1K) su(2, 2, C),so(3, 3, 1K) s((4, H), so*(6) - su(3, 1, C).

There is one more isomorphism, obtained from the extra symmetry ofthe Dynkin graph D4 : so`(8) = so(6, 2, H). 4

4. Homogeneous manifolds

4.1. G-sets.Here we recall some known facts mainly to establish convenient termi-

nology to be used later.Let G be a group. The set X is called a left G-set if a map

GxX -+ X : (g,

is given which satisfies the condition

91(92x)=(9192)'x, ex=x.

§4. Homogeneous manifolds 319

This means that we have a group homomorphism G - Aut X that associatesto g E G the transformation x - g x. In this case we say also that G actson X from the left. The homomorphism G - Ant X is called faithful ifit has the trivial kernel: no element except e acts as identity. We call suchan action effective.

Sometimes the notion of a right G-set or right action is used. It isdefined as a map

XxG -,X:that satisfies the condition

=x (9192),

This means that the map G - Aut X that sends g E G to the transformationx'-+ x - g is an antihomomorphism (i.e. reverses the order of factors).

Often the set X in question has an additional structure preserved by thegroup action; e.g., X can be a vector space or a smooth manifold. Then weshall call it a G-space or G-manifold, etc.

The collection of all left (or right) G-sets forms a category Q-Sets. Ob-jects of this category are G-sets and morphisms are so-called G-equivariantmaps 0: X --' Y for which the following diagram is commutative:

X -Y91 19

X 0 ; Y.

Example 9. The usual matrix multiplication defines a left action ofGL(n, R) on the space R" of column vectors and a right action of the samegroup on the space (R")* of row vectors:

vi-,Av, f i-+fA, vER", f E (R")*, A E GL(n, R).

0Example 10. For any category K the set MorK(A, B) is a left G-space

for G = Aut A and a right G-space for G = Aut B. Q

Remark 4. In fact it is easy to switch from left action to right action(and vice versa) using the antiautomorphism g H g-1 of the group G.Namely, on any left G-space X we can canonically define a right action bythe rule

=9-1 X.It follows that the categories of left and right G-spaces are equivalent.


Later on we often omit "right" and "left" and speak just on G-sets, keep-ing in mind that each of them can be considered in both ways as explainedabove. In any formula it will be clear from the notation which kind of actionwe have in mind. V

We introduce (or recall) some more definitions and notation.(i) There is a natural operation of a direct product in the category

of G-sets: it is the usual direct product of sets endowed by the diagonalaction of G:

g.(x,y)=(g.x,g.y)(ii) For any G-set X we shall denote by X/G or XG the set of G-orbits

in X.(iii) We denote by XG the set of fixed points for G, i.e. those points

x E X for which forall9EG.(iv) For any two G-sets X and Y their fibered product over G is

defined as (X x Y)/G and denoted by X x Y.G

Here it is convenient to assume that X is a right G-set and Y is a leftG-set. Then the set X x Y is a quotient of X x Y with respect to the

Gequivalence relation (x g, y) ti (x, g y). If we denote the class of (x, y)by x x y, then the equivalence above takes the form of an associativity law:

(45) x y=x x

Note that in general there is no group action on X x Y. But in the caseG

when X or Y is endowed by an action of another group H that commuteswith a G-action there is a natural H-action on the product space X x Y.

GIn particular, this is the case when G is abelian and any G-set can also beconsidered as a (G x G)-set.

Let H be a subgroup of G. Then the categories c-Sets and N-Sets arerelated by two functors:

The restriction functor resH : c-Sets -i f-Sets is defined in theobvious way: any G-set is automatically an H-set.

The dual induction functor indH : f-Sets 9-Sets is defined onthe objects as

(46) indH X = G x X.H

Here G is considered as a left G-set and a right H-set. So the product is aleft G-set. If z/i : Y1 - Y2 is an H-equivariant map, we define indG O by the


formula

(47) x y) = g x (y)

We highly recommend that the reader verify that indH is indeed a func-tor. The most important part of this assignment is to formulate in "down-to-earth" terms what you have to verify.

The two functors resH and indH are related by

Theorem 11 (Reciprocity Principle). For any X E Ob(Cg-Sets) and anyY E Ob(7-I-Sets) there is a natural bijection:

(48) Morg_Set, (indH Y, X) = Mor%_set, (Y, res H X).

Proof. Let 0 E Morx_set, (Y, res H X) - This means that 0 is a map from Yto X that is H-equivariant: -O(h y) = h 0(y).

On the other hand, an element 4) E Morg_set, (indH Y, X) is a G-equi-variant map from G

HY to X, i.e. 4)(glg

Hy) = gl 4)(g x y). We leave it

to the reader to check that the formulae

(49) 0(y) := 4)(e x y), 4)(g x y) := g . 0(y)H H

establish the required bijection 0 «w» 4). O

Remark 5. Sometimes indH is called a left adjoint functor to resHbecause the equality (48) looks like the well-known formula for the adjointoperator in a space with an inner product:

(A*x, y) = (x, Ay).

In our case the role of a (set-valued) inner product in a category C is playedby the bifunctor

C x C -Set : X,Y i Morc(X, Y).O

Example 11. a) If X is the group H acting on itself by right shifts,then indH X is the group G acting on itself by left shifts.

b) If X is a one-point set with trivial action of H, then indH X is theset G/H of left H-cosets in G (see below). Q

This example shows the geometric meaning of the induction functor insome special cases. We shall come back to it in the next section.

A G-set X is called homogeneous if for any two points x1i x2 of Xthere is an element g E G that sends xl to x2. This is equivalent to thestatement: X/G is a one-point set. In this case we also say that G actstransitively on X.


The following simple facts are frequently used.

Lemma 9. There are natural one-to-one correspondencesa) between homogeneous G-sets with a marked point and subgroups of G;

b) between homogeneous G-sets and conjugacy classes of subgroups of G.

Proof. a) Let X be a homogeneous G-set with a marked point xo E X. Wecan associate to this data the subgroup H = Stab(xo) C G, which is calledthe stabilizer of xo and consists of those g E G that fix the point xo.

Conversely, to any subgroup H C G we can define the set X = G/H ofleft H-cosets in G with the marked point xo = H.

(Recall that left H-cosets are precisely H-orbits in G if we consider thegroup G as a right H-space; so, the general H-coset is xH = {xh I h E H}.)

We leave it to the reader to verify that the correspondences (X, xO)Stab(xo) and H w (G/H, H) are reciprocal.

b) Let X be a homogeneous G-space with no marked point. We ob-serve that all subgroups Stab(x), X E X, belong to the same conjugacyclass, which we denote by C. This follows from the relation Stab(g x) _g Stab(x) g-1.

Conversely, if a conjugacy class C of subgroups in G is given, we canchoose a representative H E C and define the homogeneous G-set X = G/H.It remains to check two things:

1) that different choices of H E C produce isomorphic homogeneousG-sets;

2) the correspondences X w C and C w X constructed above are

reciprocal.

We leave the checking to the reader. 0

Let X = G/H be a homogeneous G-set. We want to describe the groupAut(X) of all automorphisms of X (as an object of the category of G-sets).

Define the normalizer of H in G as NG(H) = {g E G I gHg-1 = H}.It is clear that it is the maximal subgroup in G that has H as a normalsubgroup.

Theorem 12. The group Aut(X) is isomorphic to NG(H)/H.

Proof. Let 0 E Aut(X). Then 0(H) = gH for some g E G. Since 0 is anautomorphism of X, we get O(g1H) = g1gH for any 91 E G. In particular,for gl = h E H we get gH = 0(hH) = hgH. Therefore HgH = gH andg E NG(H). Conversely, for any g E NG(H) the map glH '-+ g1gH = g1 Hgis an automorphism of X.


It remains to observe that two elements gl, g2 of Nc(H) define the sameautomorphism of X iff g2 E gi H.

4.2. G-manifolds.The main statements of the previous section remain true for Lie groups

and homogeneous manifolds if we agree to consider only closed subgroupsH C G.

In particular, it is true for Lemma 9 and Theorems 11 and 12. Forexample, we have

Lemma 10. Let G be a Lie group. There is a natural one-to-one corre-spondence

a) between homogeneous G-manifolds with a marked point and closedsubgroups H C G;

b) between homogeneous G-manifolds and conjugacy classes of closedsubgroups H C G.

The proof is a combination of Lemma 9 and Theorem 4 above.Example 12. Consider the 2-dimensional sphere S2 C R3. It is a

homogeneous space with respect to the group G = SO(3, R). The stabilizerof the north pole is the subgroup H = SO(2, IR) naturally embedded inSO(3, R) as a subgroup of matrices of the form

fcos t - sin t 0h(t) = sin t cos t 0

0 0 1

The normalizer of H in G consists of two components: No = H and N1,which is the set of matrices of the form

- cost sin t 0n(t) = sin t cos t 0

0 0 -1

So, Aut(S2) consists of two points: the trivial automorphism, correspondingto No, and the antipodal map, corresponding to N1. 0

Example 13.* The previous example has a far-reaching generalization.Let K be a compact connected Lie group, and let T C K be a maximalconnected commutative subgroup of K usually called a maximal torus.It is well known that all such subgroups form a single conjugacy class. Wedefine the flag manifold to be the homogeneous space F := KIT.

This is one of the most beautiful examples of homogeneous manifolds.It is widely used in representation theory of semisimple groups, in topology,and in geometry.


The group 14" = Aut F plays an important role. It is called the Weylgroup associated to K. It turns out that F has a very rich geometricstructure; e.g. it admits several K-invariant complex and almost complexstructures. The group G of holomorphic transformations of F is a complexLie group G in which K is a maximal compact subgroup. The stabilizer ofa point F E F in G is a maximal solvable subgroup B C G. The manifoldF also admits a Kihler structure and is often used as a basic example inmany geometric theories.

More details about this example are discussed in Chapter 5.Now we discuss the topology of homogeneous manifolds. Since a homoge-

neous manifold M = G/H is determined by the pair (G, H), all topologicalproblems about this manifold can in principle be reduced to pure algebraicquestions.

Here we list some facts about homotopy groups and related homologyand cohomology groups.

Let X = G/H be a homogeneous manifold. Then we have the followingexact sequence of homotopy groups (see Appendix 1.2.3):

(50)... -, 7rn(H) - 7r,, (G) - irn(X) - irn-1(H) --....... - irl(H) r1(G) - iri(X) iro(H) - iro(G) 7ro(X) -- {1}.

Later on we shall usually assume that the group G in question is connectedand simply connected. Then 7ro(G) = irl(G) = {1}. It is known that in thiscase we also have 7r2(G) = {1}. So, (50) implies the following isomorphisms:

(51) 7ro(X) = {1}, 7rk(X) ^_' irk-1(H), k = 1,2.

We also recall that some homology and cohomology groups can easily berecovered from homotopy groups.

For example, for X connected the homology group Hl (X) := H1(X, Z)is just the abelianization of the homotopy group iri(X):

(52) H1(X) irl (X)/fir1(X ), iri (X )].

For X simply connected we also have

(53) H2(X) := H2(X, Z) 7r2(X).

If the homology groups Hi(X) have no torsion (for instance, isomorphicto Zk for some k), then the cohomology group with real coefficients has theform

(54) H2(X, R) = Hom(Hi(X), ][t) = lRbi(x)

where bi (X) is the i-th Betti number of X.


For future use we formulate one corollary of these results as

Proposition 11. Let G be a connected and simply connected Lie groupacting on some smooth manifold M. A G-orbit St C M is simply connectedif the stabilizer GF of a point F E Il is connected. If this is the case, wehave a natural isomorphism:

(55) H2(l, R) H1(GF, 1) ^_' Hom (7r1(GF), R).

Sketch of the proof. We argue in terms of differential forms. A class c EH2(f , R) can be represented by a closed differential form o. Let p : G -+ Ilbe the canonical projection: p (g) = g F. Then the form p* . is closed,hence exact, because H2(G, R) = Hom(7r2(G), R) = 0.

Therefore p*a = dO for some 1-form 0 on G. The restriction on GF givesus a 1-form 00 := 0I GE . This form is closed since dOo = dOIGF. = p"o(CF. =p`(aJp(GF)) and p(GF) is a point F. The class [9o] represents the image of cin H1(GF, IIt) L' Hom (7r1(GF), R).

Conversely, let X : 7r1(H) -+ R be a homomorphism. Since 7r1(GF) ^-'7r2(1) (see (55)), we get a homomorphism from 7r2(1l) to R. But for a simplyconnected manifold ft the homotopy group 7r201) coincides with H2(Sl).Thus we get a homomorphism from H2(cl) to R, which is an element ofH2(f), R).

It remains to check that the two constructed maps are reciprocal. Thisis essentially the Stokes theorem: fD dO = fOD 9 where D is a 2-dimensionalfilm in G with 8D a 1-cycle in GF.

Example 14. Let G = SU(2), H = U(1), and X = P'(C) ^_' S2. SinceSU(2) ^_' S3 and U(1) S' as smooth manifolds, we get from (55):

7ro(S2) = 7r1(S2) = {1}, 7r2(S2) ?' 7r1(S1) ... Z.

Proposition 11 implies in this case that H2(S2, IIt) H1(S', R) ^-' IR

4.3. Geometric objects on homogeneous manifolds.Consider the following data.

(i) A smooth manifold M with the action of some group G on it; wedenote the action by m p-+ g - in.

Usually G is a Lie group, but in some examples it is convenient to use aninfinite-dimensional group in the role of G, such as Diff(M), the group of alldiffeomorphisms of M, or a subgroup preserving some geometric structureon M.


(ii) A fiber bundle F -+ E 4 M, where F is a smooth manifold, possiblywith some additional structure.

(iii) An action of the group G on the total space E, denoted by x --+ g xand compatible with the G-action on the base M. This means that for anyg E G the following diagram is commutative:

E S EP1

MIP

When all three conditions (i), (ii), (iii) are satisfied, we shall say that thebundle F - E -+ M is a G-bundle. If moreover the action of G on M istransitive, we say that F --+ E 4 M is a homogeneous G-bundle.

We have seen in Appendix 11.2.2 that geometric objects on a manifoldM can be viewed as sections of some natural bundle on M, i.e. a bundleF'-+ E p M such that the action of the group Diff(M) can be lifted fromM to E. So, a geometric object on M is just a section of a Diff(M)-bundle.

Note that if M is connected, the group Diff(M) acts transitively on M:for any two points ml, m2 E M there exists a diffeomorphism 0 E Diff(M)such that O(ml) = m2.15 So, any Diff(M)-bundle on a connected manifoldis homogeneous.

Let mo be a marked point on M, let F = p I (mo) be the fiber over mo,and denote by H= Stab(mo) the stabilizer of mo in G. The main fact abouthomogeneous G-bundles is that such a bundle is completely determined bythe H-set F.

Indeed, consider the map

GxF --+E: (g, x)

Since for any x E F we have (gh) x = g (h x), this map can be factoredthrough the fibered product G x F defined in the previous section. Moreover,

the preimage in G x F of the point g x is exactly the equivalence classg x x. Thus, the total set E is identified with the fibered product G x F.

H H

Using the notation of Section 4.1, we can also write

(56) E= ind H F.

'5To see this, consider a smooth path joining ml and nag and a tiny tube along this path.The question for a general manifold reduces to the same question for a cylinder, which is rathereasy.


This gives the geometric interpretation of the functor ind H.Denote by IF(E) the set of all smooth sections of E. For any G-bundle

E we define the action of G on IF(E) by

(57) (g s)(m) = 9.8(g-1 m).

In the case of a trivial bundle this is the standard action of the group G onthe space of functions on the G-set B : (g f)(m) = f(g-1 m).

A section s : M -* E is called G-invariant (or simply invariant ifthere is no doubt about G) if g s = s, or s(g m) = g s(m) for all g E G.

Lemma 11. Let F -p E - M be a homogeneous G-bundle, and let H C Gbe the stabilizer of the point m E M. There is a natural bijection betweenG-invariant sections of E and H-invariant elements in F = p-1(m).

Proof. Indeed, let s be a G-invariant section. Then its value at m is anH-invariant element of F since h s(m) = s(h m) = s(m). Conversely, ifx E F is an H-invariant element, then the formula s(g m) = g x definescorrectly a section of E since the last expression depends only on the cosetgH. This section is G-invariant because (gi s)(g m) = gl s(gi 1 g m) _919119' x=s(g.m).

Note that if G is a Lie group and the action of G on M is smooth andtransitive, then any invariant section is automatically smooth.

Remark 6. Actually, the construction of a G-invariant section s : M -+E from an H-invariant element x E F is nothing but the application of thefunctor ind H to the morphism {m} -+ F : m H x in the category of H-sets.

G

We give several useful applications of Lemma 11 here.

Example 15. We show the pure algebraic procedure to find out whetherthe connected homogeneous manifold M = G/H is orientable. We can andwill assume that the group G is connected.

Consider the fiber bundle 1±11 -' E --P+ M for which the fiber F overany point m E M consists of two points ±1 and the transition function ¢Q Qacts as

q5Q,p(e) = sign(det JQ,Q) e

where JQ,p = I x, I is the Jacobi matrix.

Further, we can define the G-action on E by

g-(m,E)=(gm,detg.,(m)E).


It should be clear for the reader (check if it is really so!) that the followingstatements are equivalent:

a) M is orientable;b) the bundle E admits a continuous section s : M -> E:c) the bundle E admits a G-invariant section s : M --> E;d) E is trivial.Indeed, a section s is continuous iff it is G-invariant. (Actually, both

properties mean that s is locally constant in any trivialization of E.)But according to Lemma 11, a G-invariant section exists if there is an

H-invariant element of the fiber, i.e. when det h.(m) > 0 for any h E H.Note that the action of h. on Tm M is simply the quotient action of Ad h onthe quotient space g/h. So, we come to

Criterion 1. Let G be a connected Lie group. Then the homogeneous man-ifold M = G/H is orientable if

det(Adah)>0 forallhEH.det(Adh h)

In particular, this is certainly true if H is connected.

As an illustration, consider the case M = P"(R), G = SO(n + 1, R),H = O(n, R). The element h E H has the form

h=(0 detA)' AEO(n,R).

The action of h.(m) on TmM = 0/b R" is given by

h.(m)

Therefore, deth.(m) = (det A)"+' and P" (R) is orientable iff n is odd.Quiz. The statement above is false for n = 0. Where i. the gap in our

argument? Q

Example 16. We want to find out when a G-invariant measure existson a homogeneous space M = G/H. It turns out that this question canbe solved in the same way as in the previous example. Here the auxiliarybundle E has the fiber R >0 and the transition function

O..o(p) =I detJa,5 I p.

We leave the details to the reader and formulate the answer.


Criterion 2. Let G be any Lie group. Then the homogeneous manifoldM = G/H admits a G-invariant measure if

59 det(Ade h) _ 1 all h E H.( ) det(Adh h)

for

For example, this is true if H is either a compact, or a connected semisimple,or a nilpotent subgroup in G.

Note that if instead of the invariant measure, we were looking for theinvariant differential form of top degree (the volume form) on M, then con-dition (59) will be replaced by the similar but stronger condition

(60)det(Ada h) = 1 for all h E H.det(Adh h)

As an illustration, we can cite the following example. The Mobius bandM can be viewed as a quotient of R2 by the action of the group Z. Namely,the element k E Z sends (x, y) E IR2 to (x + k, (-1)k y). In this casecondition (59) is satisfied, while (60) is not. Therefore, there exists an R2-invariant measure on M, but there is no 1R2-invariant 2-form.

Example 17. Let us find out when the homogeneous manifold M =G/H admits a G-invariant almost complex structure. Recall that an almostcomplex structure on M is just a complex structure on fibers of the tangentbundle TM. Analytically it is given by the smooth family of operatorsJ(m) E End(T,,,M), m E M (in other words, by a smooth tensor field oftype (1, 1) on M), satisfying j2 = -1.

According to Lemma 11, the G-invariant almost complex structures onM are in bijection with the H-invariant operators J on g/b satisfying J2 =-1, hence with H-invariant complex structures on T,,,a M.

The analogous question about G-invariant complex structures on M =G/H is more delicate but also can be solved in pure algebraic terms. Werecommend that the reader compare the result below with the discussion ofNijenhuis brackets in Appendix 11.2.3.

Theorem 13. Let G be a connected Lie group, and let H be a closed sub-group of G. Assume that the homogeneous manifold M = G/H possesses aG-invariant almost complex structure J. Denote by P the i-eigenspace forJ in the complexification THM ^_' gC/(C, and let p be its preimage in gc.Then J is integrable ie° the complex vector space p is a Lie subalgebra ingC.

Not only G-invariant sections but all sections of a homogeneous G-bundleadmit a simple analytic description. Namely, to a section s : M E we


associate its representing function fg : G -p F, given by

(61) f9(g) = g-1 s(9 mo) = (g-1 . s)(mo)

Lemma 12. a) An F-valued function f on G is a representing functionfor a section of the homogeneous G-bundle F -+ E -* M if it satisfies thecovariance condition:

(62) f(9h) = h-1 of (g).

b) If the covariance condition is satisfied, the section s represented by fis given by

(63) s(g m) = 9 f(9)

c) The G-action on sections in terms of representing functions has theform:

(64) fg.s(91) = fs(9-191)

O

Remark 7. Equations (62) and (64) would look slightly better if weconsider M as a right G-set but keep the left action on E. Namely, theytake the form

(62') f (h9) = h f (9)'(64') f9.s(91) = f-(919)-

Of course, we have to change the definitions of a representing function andof the G-action on r(E) in an appropriate way. Instead of (57) and (61) wewould have

(57') (9 . s)(m) = g s(m . g),

(61') fs(9) = s(mo - g) g-1.

G

Example 18. Vector fields on a Lie group. A Lie group G can beconsidered as a homogeneous G-space in two different ways: as a left G-space with respect to left shifts x H gx and as a right G-space with respectto right shifts x - xg. Moreover, it can be considered as a (G x G)-spacewith respect to the left action (gl, g2) x = glxg2 1.


Therefore, the sections of homogeneous vector bundles over G admitthree different descriptions. We give these descriptions for the case of thetangent bundle TG.

A smooth section of TG, i.e. a vector field v on G, can be described asa smooth g-valued function on G in two different ways:

foef t(9) = 9-1 .V(g), fvght(9)

= v(9) . g-1.

It is also described as a smooth g-valued function on G x G:

Fv(91, 92) = 91 1 . v(9192')

- 92

satisfying the condition

Fv(919, 929) =Adg-'

F,(91, 92).

These three descriptions are related by:

Fv(91, 92) = Adgi1 fright(9192 1) = Ad92 1 fief (9192').

The action of G x G on vector fields in terms of representing functions lookslike:

f left -(g) = Ad92 fief t(gl 192), f right-1(9) = Ad g1 fvaght(91192),

1

Fg1.v.g. 1(9i, 92) = Fv(919i, 92,92)

In particular, for the left-invariant vector field X we have

f left(g) = X,fright(9) = AdgX, F'X(91, 92) = Adg2'X.

For the right-invariant vector field k we have:

fXft(9) = Adg-1X,fright (9) ° X, Fg(91, g2) = Ad g1 X.

0In conclusion we collect some facts about integration on Lie groups.On any Lie group G there exist unique (up to a scalar factor) left-

invariant and right-invariant differential forms of top degree. We have de-noted them dig and dg respectively. Put 0(g) = det Ad g.

Proposition 12. The following relations hold:

di(h9) = dig; di(9h) = i(h)-1d19; di(9-1) = c d,-g;(65)

dr(hg) = i(h)dr9 d,-(9h) = dr9; dr(9-1) = c 1. d19.


Proof. We prove only the first three relations; the proof of the others isquite analogous. The first relation is just the definition of the left-invariance.

To prove the second, we remark that the form dt(g) under the innerautolnorphism A(h-1) : g - h-1gh goes to another left-invariant form,hence is simply multiplied by a scalar. The direct computation at the point eshows that this scalar is actually det Ad(h)-' = 0(h)-1. Therefore dt(gh) _di(h-'gh) = 0(h)-ldlg

The third relation follows from the observation that the form dt(g-1) isright-invariant: dt((gh)-1) = dt(h-1g-1) = dt(g-1). 0

Appendix IV

Elements of FunctionalAnalysis

1. Infinite-dimensional vector spaces

In this appendix we collect some definitions and results from functionalanalysis that are used in representation theory. Most of them have rathersimple formulations and the reader acquainted with them can read the maintext without difficulties. The proofs are often omitted, except in those caseswhen the proof is not difficult and/or helps to understand the matter andmemorize the formulation.

1.1. Banach spaces.In representation theory we often deal with infinite-dimensional vector

spaces. The pure algebraic approach to such spaces is rather fruitless. Amuch more interesting and powerful theory arises when we combine algebraicmethods with analysis and topology.

One of the main objects of this theory is the notion of Banach space,which is the combination of two structures: a real or complex vector spaceand a complete metric space. Here we recall some basic facts from the theoryof Banach spaces and the linear operators on them.

A Banach space is a vector space V over the field K = R or C endowedwith a distance function d : V x V --+ 1[t+ such that

a) (V, d) is a complete metric space (see Appendix 1.1.2);b) the distance is compatible with the linear operations:

(i) d(v + a, w + a) = d(v, w) (Translation invariance)(ii) d(cv, cw) = Icl d(v, w) for any c E K (Homogeneity)

333

334 Appendix IV. Elements of Functional Analysis

These two distance properties allow us to replace this function of twovariables by a function of one variable, the so-called norm of the vector vdenoted by Ilvll and defined by

Iivll := d(v, 0).

The initial distance function can be reconstructed from the norm d(x, y) =lix - yll. The homogeneity of distance is equivalent to the homogeneity ofnorm:

IIcxII = lei - IIxII

The triangle inequality for the distance is equivalent to the following basicproperty of the norm:

IIx + yl1 <_ IIxII + Ilyll (Semiadditivity).

Example 1. The space C(X, K) of continuous K-valued functions ona compact topological space X is a Banach space with respect to the norm

(1) Ilfll = max f(x)I.

Note the remarkable universal property of this space: any separable Banachspace (i.e. possessing a countable dense subset) over K is isomorphic to aclosed subspace in C([0, 1], K). 0

Example 2. The space LP(X, p, K) of measurable K-valued functionson a measure space (X, µ) is a Banach space with respect to the norm

If (x)IPdu(x)) 1 < P:5 x.(2) Of lip = (fxFor p = oo the norm is defined as

(2') IIf1Ioo = li m Ilf lip = essXup If (x)I.00

0Example 3. The space Ck(M) of k-smooth K-valued functions on a

smooth compact manifold M. In this case there is no preferable norm, butthe corresponding class of convergent sequences is easy to describe.

A sequence If,,) converges to f in Ck(M) if for any chart U C M witha coordinate system x1, ... , x"' and for any multi-index p = (pi, ..., Pm)with I pI < k the sequence ow f,,(x) = di...... dl m" f" (x) converges to Op f (x)uniformly on every compact subset of U.

§1. Infinite-dimensional vector spaces 335

We omit the explicit definition of a norm and the proof of completeness(see, e.g., (KG] or (Y]). 0

Warning. The space C°°(M), where the class of convergent sequencesis defined in the same way, but using all multi-indices p, is not a Banachspace. More precisely, there is no norm in CO°(M) with the above givenclass of convergent sequences.

However, the topological space C°°(M) is metrizable: one can defineon C°O(M) a metric such that it will be a complete metric space and theconvergence defined by the metric will be exactly the convergence describedabove. This metric, however, is not homogeneous.

The space C°°(M) for a non-compact manifold M is still more sophisti-cated: it is non-metrizable (see loco cit.). 6

1.2. Operators in Banach spaces.A linear operator A from one Banach space Vl to another Banach space

V2 is called bounded if it has a finite norm, given by

(3) IIAII:= sup IIAxIIyz

O,XEVi IIxhly.

It is well known that A is bounded if it is continuous.The collection of all bounded operators from Vl to V2 is denoted

Horn (V1, V2). It is a Banach space with respect to the operator norm (3).In particular, V* := Hom(V, K) is a Banach space. It is called the

dual Banach space to V and its elements are called linear continuousfunctionals (or, for short, functionals) on V.

Example 4. It is known that LY(X, u, K)* ^_' Lq(X, p, K) for 1 _< p <oo where p and q are related by the equalities

Pi+q-1=1 pq=p+q (p-1)(q-1)=1.

More precisely, any functional F E LP(X, M, K)* has the form

F9(f) = Jx f (x)g(x)dµ(x) for some g E LQ(X, jz, K)

and II FgII L,(x,, ,K) =11911 Lq(x,µ,K) 0The class of all Banach spaces forms a category 13 (morphisms are

bounded linear operators). The passage from V to V* can be extended to acontravariant functor from 13 to itself. In particular, to any A E Hom(Vj, V2)there corresponds the dual or adjoint operator A* E Hom(VZ , Vi*) defined


by

(A*f. x) Ax).

One can prove that IIA*II = IIAII.

In the finite-dimensional situation the Banach spaces V and V* playdual roles: the second dual V** :_ (V*)* is canonically isomorphic to V.

This is no longer true in the infinite-dimensional case. In general wehave only the map: V - V**. Indeed, to any v E V we can associate thelinear functional F on V* : (Fe, f) := (f. v). It is known that this map isactually an isometric embedding of V into V**.

Those Banach spaces for which the embedding V V** is an isomor-phism are called reflexive; e.g., the spaces Lp(X, p. K) are reflexive for1 <p<oc.

1.3. Vector integrals.In Appendix 11.2.4 we defined the integral of a density over a smooth

manifold.

In representation theory we need the notion of an integral not only forreal or complex-valued densities but also for densities w with values in agiven real or complex vector space V. When V is finite-dimensional, we canintroduce a basis {v1, ... , v } and define the integral IN w(m) as a vectorin V whose k-th coordinate is the integral f Al wk(nt) where wk(m.) is thek-th coordinate of w(m). Of course, we have to check that this definitiondoes not depend on the choice of a basis. We leave this to the reader.

For an infinite-dimensional space V the definition of an integral is moredelicate. Let V be a Banach space and w a V-valued density on a manifoldAl. Suppose that for any f E V* the scalar density (f. w(m)) is integrableover Al (e.g., it is so when w is weakly continuous and has a compact sup-port). Then we can define the weak integral w- f fJ w(m) as a vector in V**such that

(4) (wJ w(m), f) =1 (f, w(m)).

If we assume that w is strongly continuous (i.e. loc.3lly w = f (x)d"xwhere f is a strongly continuous vector function) and has a compact support,then the strong integral s-fa, w(m) is defined. It is a vector in V which isthe limit of Riemannian integral sums constructed exactly as in the scalarcase (see Appendix 11.2.4):

s-J/w(m)=limS(f;{AI;},{a,}) =lion f(xi)IvolI(Ali).

§1. Infinite-dimensional vector spaces 337

Since the integrand is strongly continuous and has a compact support. theintegral sums form a Cauchy sequence, hence have a limit when the diameterof the partition M =11 NI1 tends to zero.

Exercise 1. Prove that when the strong integral s- fMf w(m) exists. thenthe weak integral w- fM w(m) also exists, takes its value in V. and coincideswith the strong integral. 46

1.4. Hilbert spaces.The infinite-dimensional real and complex Hilbert spaces are the most

direct analogues of the ordinary Euclidean space RN and its complex versionCN

They are the very particular cases of Banach spaces for which the non-linear object, the norm, can be expressed in terms of a linear object: thescalar (or inner) product.

By definition, a scalar product in a complex vector space V is definedas a sesquilinearl map: V x V -. C, denoted by (x, y), with the followingproperties:

(i) (x, y) _ (y, x) (Hermitian symmetry)(ii) (x, x) > 0 and (x. x) = 0 x = 0 (Positivity)The scalar product in a real vector space is defined in the same way,

but its properties are slightly simpler. Namely, the scalar product (x. y) isreal and bilinear in x and y. The symmetry condition (i) is just (x. y) _(y, x)

A Banach space V is called a Hilbert space if V admits a scalar productsuch that the norm is related to the scalar product by the equation

(5) 11xII_ (x,x) forallxEV.

It is known that this is the case if and only if the norm satisfies theso-called parallelogram identity:

lix + y1l2 + lix - y1l2= 211x112 + 211y112.

Geometrically it means that for any parallelogram the sum of squares of thetwo diagonals equals the sum of squares of the four sides.

Remark 1. This definition also includes finite-dimensional Euclideanspaces R" and C". Initially, the term Hilbert space was reserved for thespace introduced by Hilbert. In modern terms it can be defined as "com-plex infinite-dimensional separable Hilbert space" or "complex Hilbert space

1That is, complex-linear, in the first argument and antilinear. or conjugate-linear. in thesecond one. In Russian mathematical literature a more expressive term "?-linear" is used.


of countable Hilbert dimension" (see below). We prefer the more generaldefinition above. Q

For a vector v in a Hilbert space V the norm is usually denoted by Ivland called the length of v. In a real vector space one can also define theangle 0 between two vectors v and to by the.equality

(6) COSO=I(vl v''

Iww)i , 0<0<7r.

Two vectors v and to are called orthogonal if (v, w) = 0. This definitionmakes sense in complex Hilbert spaces in contrast with the notion of anangle.

For a finite system {vi } of pairwise orthogonal vectors the followingequality holds (Pythagorean Theorem):

2

(7)Vi

= Iv,l2

The system {va}aEA of vectors in a Hilbert space V is called orthonor-mal if

(va, vv) = bao-For any such system and for any vector v E V the quantities ca = (v, va)are called coefficients of va with respect to the system {va}aEA

Proposition 1. The following Bessel inequality holds:

(8) IC-1, < 1W.aEA

Proof. By definition, the sum on the left-hand side of inequality (8) isSUPAoCA E EA0 IcaI2 where AO runs over all finite subsets of A. Therefore,it is enough to prove the inequality for a finite set A0.

In this case we write v = v' + v" where v' = EweAU Cova, and observethat v and v' have the same coefficients with respect to the system {va}aEAoIt follows that v" is orthogonal to v' and from the Pythagorean Theorem wehave

(9) Ivl2 = lu I2 +IcaI2

+ Iv"12 > IcaI2,aEAo aEAo

which proves the Bessel inequality. 0

An orthonormal system {va}aEA in V is called complete if there is nonon-zero vector v E V that is orthogonal to all va, a E A. In this case thesystem forms a Hilbert basis in V in the following sense.

§2. Operators in Hilbert spaces 339

Proposition 2. For any v E V the series EQEA cavq converges to v.In more detail, for any e > 0 there is a finite subset A(c) C A such that

v - cQ v.

QEAI

< C for any Al D A(e).

Proof. The proof follows from the analogue of (9). 0

Warning. A Hilbert basis is not a basis in the algebraic sense : thefinite linear combinations of basic vectors do not exhaust the whole space.4

It is known that every Hilbert space V has a Hilbert basis and that allthese bases in a given space V have the same cardinality. This cardinalityis usually called the Hilbert dimension and is denoted by dimh V. TheHilbert dimension dimh V coincides with the algebraic dimension dim Vwhen the latter is finite.

All Hilbert spaces of given Hilbert dimension are isomorphic, i.e. thereexists a linear isometric operator from one space to another.

We shall deal mostly with separable Hilbert spaces that contain acountable dense subset. They have finite or countable Hilbert dimension(the latter sometimes is denoted by No).

The dual space V* to a Hilbert space V can be naturally identified withthe complex conjugate space2 V. Namely, any linear functional F on V hasthe form F = Fw where

(10) FF(v) _ (v, w), w E V.

It is convenient to denote Fw by U,. The correspondence w +-* F.,,, = w isantilinear and isometric.

2. Operators in Hilbert spacesIn this section we describe different types of operators in Hilbert space. Themost important fact is that some of these operators share the main propertyof numbers: they can serve as arguments for functions of one variable (seeRemark 2 below). This fact is used in the construction of the mathematicalmodel of quantum mechanics (see Section 3).

2By definition, V is the same real vector space as V. but the multiplication by a complexnumber a in V is defined as the multiplication by the complex conjugate number X in V.


2.1. Types of bounded operators.Let, as before, Hom(Vi, V2) denote the set of all bounded operators from

one Hilbert space Vl to another Hilbert space V2. This is a Banach spacewith respect to the operator norm.

The above identification of V' with Vi produces the antilinear map:

Hom(Vi, V2) - Hom(V2, Vi): A A'

where the adjoint operator A' is defined by

(11) (Avi, v2)V2 = (vl, A'v2)v, for all v; E V;.

If Vl = V2 = V, we simply write End V instead of Hom(V, V). Thisis an associative algebra with the anti-involution A +--+ A'. The algebraicproperties of this anti-involution are:(12)(A+ B)' = A'+ 13', (,\A)' = L41, (AB)' = B'A', (A')' = A.

It is also known that 11A511 = 11AII.

Using the anti-involution we can define several important classes of op-erators in a Hilbert space V.

Hermitian or self-adjoint operators S : S' = S;anti-Hermitian or skew-Hermitian operators A : A' = -A;normal operators N : NN' = N'N;unitary operators U : UU' = U'U = 1;orthoprojectors P : P' = P = P2.Remark 2. The normal operators can be considered as a far-reaching

generalization of complex numbers. More precisely, Hermitian operatorsare analogues of real numbers, anti-Hermitian operators correspond to pureimaginary numbers, and unitary operators correspond to numbers of abso-lute value 1.

In this approach the theory of unitary representations appears as thenatural generalization of the theory of characters for abelian groups. G

2.2. Hilbert-Schmidt and trace class operators.Let V be a Hilbert space, and let V' V be its dual. Consider the

tensor product E = V ® V'. This space can be identified with the subspaceEndo(V) c End(V) consisting of all operators of finite rank. Namely, to avector vi 0 v2 E V ® V' we associate the operator A,,, ®z2 E Horn (V, W)defined by

Avi®u2(v3) = (v3, v2) - v,.


There is no distinguished norm in V 0 V*. But still some norms are morenatural than others. We consider only so-called cross-norms in E thatsatisfy the conditions:

1. I{v1(9 V211EIv1I'IV21 for vi EV,V2EV,

2. Ilvi ®v211E = Iv1I ' Iv21 for V1 E V, V2 E V.

There are three remarkable cross-norms in E:a) the operator norm II ' II inherited from End V;

b) the Hilbert cross-norm II ' 112 given by

IIAII2 = tr(A*A) JAv=12

(13) iEI

where {vi}iEI is any orthonormal basis in V;

c) the so-called trace norm II ' II 1 There are three equivalent definitionsof this norm:(14)

IJAII1 =A->kAk

E IIAkii, rkAk = 1,

IIAIII = E sk(A) where sk(A) are the eigenvalues of Al J:= VA''-A,

k

IIAIII = sup E 1(Avk,wk)1 where {vk} and {wk} are two bases in V{vk},{wk} k

and supremum is taken over all pairs of bases.

Proposition 3. a) For any cross-norm p and any A E End0V we have:

(15) NAII < p(A) S IIAIII

b) There exists a unique cross-norm on E satisfying the parallelogramidentity, namely the Hilbert cross-norm II ' 112-

c) If A and B are operators of finite rank in V, then so is C = BA and

(16) IJCII1 5 JJA1121113112-

0

The completion of E with respect to the operator norm coincides withthe closure of Endo V in End V, which consists of all compact operatorsin V.

The completion E2 of E with respect to the Hilbert cross-norm con-sists of so-called Hilbert-Schmidt operators A in V characterized by theproperty:

IIAII2=trA'A<oo.


The scalar product in E2 is given by

(17) (A, B) = tr B'A = (Avi, Bvi).iEI

The completion El of E with respect to the trace norm consists of all oper-ators of trace class, characterized by the condition

(18) IAviI < oo for any Hilbert basis {vi}iEI in V.iEI

It follows from (18) that for a trace class operator A the series E EI(Avi, vi)is absolutely convergent for any Hilbert basis {vi}iEI. It turns out that thesum of this series does not depend on the basis. This sum is called the traceof A.

Proposition 4. Let V = L2(X. µ), and let A be an integral operator

(x. y).f (y)dtt(y)(19) (Af) (x) = fX a

Then A is of Hilbert-Schmidt class if the kernel a(x, y) belongs toL2(X x X, p x p). Moreover,

(20) JJA112 = 11a112L2(xxx,µx,,)

Proof. The proof follows immediately from (13). 0

Proposition 5. Let M be a smooth compact manifold, let p be a measureon M given by a smooth density, and let A be an integral operator given byformula (19). Then A is of trace class and

(21) trA = ja(xi x)dp(x).

roof. Let us choose a finite atlas on the manifold M x M such that thePdiagonal OM C M x M is covered by charts of the form Ui x Ui and therest of M x M is covered by charts of the form U3 x Uk where U, n Uk = 0.We can also assume that all Ui are bounded domains in R" endowed with aunimodular coordinate system.

Using the partition of unity subordinated to {Ui}, we reduce the problemeither to the case when supp a C Ui x Ui or to the case supp a C U, X Uk,U,nUk=O.


In the second case the right-hand side of (21) is evidently zero. Thetrace of A is also zero, because for an appropriate basis in L2(M, p) allsummands (Avi, vi) are zeros.

In the first case we can consider Ui as a bounded domain in R", hence asa domain in T", the n-dimensional torus. So, we have reduced the problemto the case M = T". In this case it can be easily solved using Fourier series.Indeed, if we put vk(x) = e2Ri(k,) and a(x, y) _ Ek,tEzn ak,te2ai((ks)+(t.y))then

J(Avi, vi) = E ak,-k = f" a(x, x)dx.i kEZ^ T

11

Remark 3. There are two important extensions of formula (21).First, it remains true in the case of non-compact manifolds if the ker-

nel function a(x, y) has compact support or rapidly decays at infinity. Inparticular, it is true for M = R" and a E S(R2"), the Schwartz space.

Second, this formula is also true for a positive integral operator A with acontinuous kernel a(x, y) on an arbitrary manifold with a continuous volumeform dp. Moreover, in this case we have IIAIII = trA.

Note also that formula (21) for the trace of an operator is the directanalog of the formula for the trace of a matrix: tr A = EjAii = (Avi, vi).We replace the sum of diagonal elements of A by the integral of the kernelfunction a over the diagonal. V

2.3. Unbounded operators.The theory of bounded operators in a Hilbert space is rather deep and

beautiful, but it does not enclose many natural operators (for instance, dif-ferential operators). There is the remarkable theory of unbounded operatorsin a Hilbert space and we recall here some results from this theory.

By an unbounded operator in a Hilbert space V we understand apair (A, DA) where DA is a linear subspace (not necessarily closed) in Vand A : DA -+ V is a linear operator. The space DA is called the domainof definition for A.

Often people simply write A instead of the full notation (A, DA). Wealso follow this tradition, although sometimes it leads to misunderstandingsor even mistakes.

When the Hilbert space in question is L2(M, ia) where M is a smoothmanifold, p is a measure defined by a smooth density, and the operator Ais a differential operator with smooth coefficients, then there is a so-called


natural domain of definition for A. It consists of all regular distributionsf E L2(M, p) for which Af also is regular and belongs to L2(M, µ).

It is convenient to study an unbounded operator A via its graph rA:

(22) r'A={(v,Av)EV®V I VEDA}.

An operator A is called closed if its graph rA is a closed subspace in V V.We say that A is an extension of A if I'A C I'A. In this case DA C DA

and AIDA = A.

The adjoint operator (A*, DA-) is defined by

(23) FA. = I F-AL

where 1 denotes the orthogonal complement:

r'A={(x,y)EV®V I (x,v)+(y,w)=0forall(v,w)EI'A}

and I denotes "the rotation on a right angle" in V ® V:

AX, y) = (-y, x).

Warning. This definition makes sense only for operators A with a densedomain of definition. Otherwise, if 0 u 1 DA, the vector (u, 0) E V ® Vis orthogonal to rA. Hence, (0, u) E DA., which is impossible: a linearoperator cannot send 0 to a non-zero vector u.

Note, however, that there is a theory of linear relations generalizingthe notion of linear operators. Any vector subspace in VI ® V2 is by defi-nition a graph of a linear relation between Vl and V2. One can define theinverse relation between V2 and VI and also the composition of relationsA E Rel(Vi, V2) and B E Rel(V2, V3).

In the case of Hilbert space V the adjoint relation A* is defined for anyrelation A E Rel(V, V). 41

We leave to the reader the labor and pleasure of convincing yourself thatfor a bounded operator A the definitions (11) and (23) coincide.

Note also that any adjoint operator is closed since the orthogonal com-plement is always a closed space.

An unbounded operator A is called self-adjoint if it coincides with itsadjoint operator A*. Here the equality DA = DA- is also understood.

An operator A is called essentially self-adjoint if its closure A is aself-adjoint operator. (In this case q also coincides with A*.)

An operator A is called symmetric if A C A*. This means that(Av, w) = (v, Aw) for all v, w E DA.

§2. Operators in Hil bert spaces 345

Example 5. Let H = L2(R, IdxI), and let A be the operator i2iwith DA = A(R), the space of smooth complex-valued functions on R with

compact support. This operator is symmetric as one can easily see viaintegration by parts.

One can check that the adjoint operator A* is also i d but with biggerdomain of definition. Namely, DA- is exactly the natural domain of defini-tion. It consists of all functions f with generalized derivative f E L2(R, dx).In this case we also have A** = A*. So, A* is self-adjoint and A itself isessentially self-adjoint.

The situation changes if we consider the open interval (0, oo) or (0, 1)instead of the whole line. The operator A = i d with the domain DA =A(0, oo) or DA = A(0, 1) is still symmetric but not essentially self-adjoint.

In the case of a finite interval it has infinitely many self-adjoint exten-sions, while for the infinite interval there is no self-adjoint extension at all.0

Exercise 2. Show that any symmetric operator A with a dense domainof definition admits a closure.

Hint. Prove that A* is the closure of A using the following fact.

Lemma 1. Let S be a not necessarily closed subspace in V. Then (Sl)

coincides with the closure S. O

For future use we give here the simple

Criterion. A symmetric operator (A, DA) in V is essentially self-adjointif

Proof of the non-trivial part. Assume that ker(A' +i 1) = ker(A' - i 1)= 0. For any operator B the following equality holds:

(im B) ' = ker B'.

Therefore, both V_ := im and V+ := im are dense subspacesin V. Define the operator Uo : V_ -' V+ by (Av - iv) -. (Av + iv). Since Ais symmetric, we have the equality JAv ± ivl2 = JAvJ2 + Iv12. It follows thatUo preserves the length, consequently can be extended to a unitary operatorU in V. Hence, A* = i(U + 1)(U - 1)-' is self-adjoint.

2.4. Spectral theory of self-adjoint operators.It is well known that any Hermitian matrix A can be reduced to a diag-

onal form by a unitary conjugation: UAU-1 = D. Moreover, the diagonal


entries Si = Dii of D (the eigenvalues of A) are defined uniquely up topermutation.

So, the full set of invariants for a Hermitian operator in an n-dimensionalHilbert space is the collection of real numbers 61 < 62 < . . < bk withmultiplicities n1, n2, ..., nk subjected to the condition Ek ; ni = n.

Another way to encode this information is to associate to a Hermitianoperator A a collection of disjoint finite subsets X1, X2, ..., Xk of R whereX3 is the set of eigenvalues of A with multiplicity j. Clearly, the collection{X3} is subjected to the condition E,=1 j #X3 = n.

Let us consider in more detail the case when ni = 1 for all i. In this casewe say that the operator A has a simple spectrum.

Recall that a vector v E V is called a cyclic vector for an operatorA if the vectors v, Av, ..., An-1v are linearly independent, hence span thewhole space V = C".

Proposition 6. The Hermitian operator A in an n-dimensional space hasa simple spectrum if and only if it has a cyclic vector. 0

More generally, we say that a Hermitian operator A in an n-dimensionalspace has a homogeneous spectrum of multiplicity m if all its eigenval-ues have the same multiplicity m.

It is clear that such an operator is just a direct sum of m copies of anoperator with a simple spectrum. Of course, this can happen only if m is adivisor of n.

It is also clear that any operator can be uniquely written as a direct sumof operators with disjoint homogeneous spectra.

The situation seems to be quite different in the infinite-dimensional case:a Hermitian operator can have no eigenvectors at all (see examples below).The main achievement of the theory of unbounded operators is the spectraltheory of self-adjoint operators. It claims that any self-adjoint (not neces-sarily bounded) operator in a Hilbert space can be reduced by a unitaryconjugation to some canonical form.

Below we give the rigorous formulation, which is rather involved.We start with a reformulation of the above properties of Hermitian op-

erators in the finite-dimensional case. Let us consider a finite-dimensionalcomplex vector space V as the space L(X) of all complex valued functionson some finite set X with # X = dim V. We define the inner product inL(X) as

(fl, f2) = E f(x)J.xEX


The natural orthonormal basis in L(X) consists of functions 6, x E X :

I1 ifx=y,0 ifx54 Y.

Assume that an operator A has a real diagonal matrix in this basis.Then it is just the multiplication operator f -+ a f where a is a real-valuedfunction on X. We see that any Hermitian operator A in the appropriate"functional" realization of the initial Hilbert space is an operator of multi-plication by a real-valued function.

It turns out that this statement remains true in the infinite-dimensionalcase. More precisely, consider any Hilbert space of the form V = L2(X, µ, C)(i.e. the space of square-integrable complex-valued functions f on a set Xendowed with a measure p). Let A be the operator of multiplication by areal-valued measurable function a E L°°(X, p, Q. Then A is a Hermitianoperator in V with the norm

IIAIIv = IIaIIL-(x,µ,c)

Moreover, if a is any real measurable (not necessarily bounded) function onX, we can consider the unbounded operator A acting as multiplication bya with the domain

DA={f E V I of EV}.

It is not difficult to check that A is a self-adjoint operator in V. The spec-tral theorem below claims that it is the most general form of a self-adjointoperator in a Hilbert space.

Weak Form of the Spectral Theorem. Let A be a self-adjoint operatorin a complex Hilbert space V. Then there exist an isomorphism 0: V -,L2(X, µ, C) and a real measurable function a on X such that 0 o A o 0-1 isthe operator of multiplication by a. 0

This variant of the spectral theorem seems to be less informative thanthe initial finite-dimensional theorem about the reduction of a Hermitianmatrix to a diagonal form. But it is quite useful and sufficient for manypurposes. For instance, we derive from this theorem the existence of theso-called operator calculus, which allows us to treat Hermitian operatorsas real numbers and use them as arguments of functions.

Theorem 1. Let A be a self-adjoint operator in a Hilbert space V. There isa unique homomorphism 0 H q(A) from the algebra of complex-valued Borel


measurable bounded functions on R to the algebra End V with the properties:

(i) (A) _ O(A)';(ii) IIO(A)II < esssup I.0(x)I;

xER

(iii) if t/:(x) = x4i(x) is bounded, then tp(A) = Ab(A);(iv) if I0,:(x)I < C and On (x) 4i(x) for all x E R, then cn(A) 4)(A)

in the strong operator topology.

Proof. First, prove the existence. We can assume, due to the weak formof the Spectral Theorem, that V = LZ(X, u, C) and A is the operator ofmultiplication by a real-valued measurable function a on X. Then we candefine O(A) as the multiplication operator by co(a(x)). Properties (i)-(iv)can be easily verified.3 This proves the existence.

Now prove the uniqueness. Recall that the set of all Borel measurablefunctions can be obtained from a set of piecewise constant functions bypointwise limits. So, it is enough to check the uniqueness for functions e, ofthe form

ec(x) =f

1

0

for x < c,

for x> c.

From (i) and the homomorphism property we conclude that Ec := e,(A)are orthoprojectors in V. Moreover, the ranges VG of EE form an increasingfamily of subspaces Vc C V on which A is bounded from above:

(Av, v) < c(v, v) for all v E V.

The reader acquainted with abstract spectral theory will recognize in EEthe spectral function of A. The characteristic property of this function isthe following integral formula for the operator calculus:

fR(24) V(A) = V(x)dEE = s-n im F,

k(Et - E00

kEZ

for any bounded Borel measurable function gyp.

So, the uniqueness of the operator calculus is equivalent to the unique-ness of the spectral function of A. But the latter follows from (24). Indeed,if EE is another spectral function for A, then

ec(x)dE'' = E,.EE = er(A) = fR

03Property (iv) follows from the Lebesgue Dominant Convergence Thecrem.


The important consequence of the uniqueness of the operator calculusis the following

Theorem 2. If a unitary operator U commutes with A (i.e. U preservesDA and UA = AU on DA), then it commutes with any function of A.

Proof. Indeed, the correspondence 0 H U,b(A)U-1 satisfies conditions (I)-(iv), hence coincides with 0 4i(A).

We say that two unbounded self-adjoint operators Al and A2 commuteif their spectral functions commute:

(25) E.,(Ai)Ecz(A2) = Ec2(A2)EE,(Ai)

It is useful to know that any family {A,},EI of pairwise commutingself-adjoint operators can be simultaneously reduced to the form of multi-plication by a function.

In the finite-dimensional case it means that any family {A,},E1 of pair-wise commuting Hermitian matrices can be simultaneously reduced to thediagonal form.

There exist more precise versions of the Spectral Theorem that generalizethe notion of multiplicity for eigenvalues.

Inspired by the finite-dimensional experience, we say that a self-adjointoperator A in a Hilbert space V has a simple spectrum if there exists acyclic vector v E V for A, i.e. such that the linear span of v, Av, ... Anv, .. .is dense in V.

We say that A has a homogeneous spectrum of multiplicity m if itis similar to a direct sum of m copies of an operator with a simple spectrum.

For operators with a simple spectrum the following stronger form of theSpectral Theorem holds:

Theorem 3. Let A be a self-adjoint operator with a simple spectrum ina Hilbert space V. Then there exist a Borel measure p on R and an iso-morphism 0: V - L2(IIt, p, C) such that the operator 0 o A o 0-1 is themultiplication by x in the space L2(R, p, C). The measure p is defined bythe operator A up to equivalence.4

In the finite-dimensional case the measure U has finite support X, whichis exactly the set of eigenvalues of A and has the property # X = dim V.The space L2(R, A, C) is isomorphic to L(X) above.

4Recall that two measures s and v are equivalent if they have the same collection of setsof measure zero. In this can there exists a Walmost everywhere positive function p suchthat µ= p. v.


The general self-adjoint operator is a direct sum of operators with dis-joint homogeneous spectra. Note that in a separable Hilbert space the mul-tiplicity of the spectrum can take the values 1, 2, ... , oc.

Now we can formulate the

Strong Form of the Spectral Theorem. Let A be a self-adjoint operatorin a complex Hilbert space V. Then there exists a family {pk}1<k<oo ofpairwise disjoint Bored measures on R such that A is equivalent to the directsum ®1<k<oo Ak where Ak is the direct sum of k copies of the multiplicationoperator by x in L2(IR, pk, C).

All measures ,4k, 1 < k < oc, are defined by the operator A uniquely upto equivalence. O

Remark 4. Note that a convenient form of the operator Ak above is theoperator of multiplication by x in the space of vector-functions L2(R, pk, Wk)where Wk is any complex Hilbert space of Hilbert dimension k.

Recall that the disjointness of pk means that there exists a partitionof the real line R = U1<k<. Xk into a family of disjoint Borel subsets {Xk}such that pk(R\Xk) = 0. Replacing, if necessary, every pk by an equivalentmeasure, we can assume that p = E 1 <k<oo pk is a finite measure on R.

If we assume now that all Wk are subspaces of a single space W spannedby the first k basic vectors, then the initial Hilbert space V becomes a partof L2(R, p, W) consisting of functions satisfying

f(x) e Wk forxEXk.

The operator A will simply be multiplication by x as in Theorem 3.Thus, the collection of disjoint measures {pk} can be replaced by a pair

(p, m) where p is a measure, called the spectral measure of A, and m isthe multiplicity function for A defined by

m(x)=k forxEXk.

The pair (p, m) forms the spectral data for the operator A.

2.5. Decompositions of Hilbert spaces.In the category of Hilbert spaces, besides the usual notion of a direct sum

of a family of objects, there exists a more general operation: a continuoussum or direct integral of objects.

The idea is as follows. Let X be a set with a measure it and supposethat to any x E X a complex Hilbert space Vx is attached. Denote by Vthe direct sum ®xEX V. Consider the collection of V-valued functions f on


X satisfying the condition f(x) E V for all x E X. It is a complex vectorspace. We want to make it a Hilbert space by introducing the inner product

(26) (fl, 12) (fl (X), f2(x))i.;.dµ(x)

To make this definition rigorous, we have to formulate conditions onfunctions f that ensure the existence of the integral (26). The difficulty isthat our functions take values in a rather complicated space V (or, if youwish, take values in different Hilbert spaces Vi for different x).

There are several ways to bypass this difficulty. All of there are essen-tially equivalent and we choose the most direct way. Namely, we fix thefamily of Hilbert spaces

ViCV2C CV,2C CV.=V, dimhLk=k,

and the measurable decomposition

k=oo

X= U Xk.k=1

Now we consider the complex vector space V of functions f : X - Vsatisfying the following conditions:

(i) f is a measurable map from X to V (i.e. has measurable coefficientsin any Hilbert basis);

(ii) f (x) E Vk for X E Xk;

(iii) f has a finite norm:

(27) ilf 11= IL if (x)FV,.dµ(x) < x.

The space V is a Hilbert space with respect to the norm (27). It issometimes denoted as

V Vi dµ(x) where V2 = Vk for X E Xk,x

and is called a continuous sum or direct integral of V, x E X.An operator A in V is called decomposable if there exists a family of

operators A,, E End Vr such that

(28) (A f) (x) = Axf (x) for almost all x E X.

352 Appendix N. Elements of Functional Analysis

In this case we use the notation

(29) A _ IxAx dy(x)

and say that A is a continuous sum, or direct integral, of A. x E X.For any set S C End V we define the commutant of S as the set

(30) S!={AEEndV I AS=SAforallSES}.

A symmetric (i.e. stable under the anti-involution `) operator algebra A CEnd V with a unit is called a von Neumann algebra if it satisfies thecondition:

(31) X:= (A!) = A.

From this definition it follows that von Neumann algebras always appear inpairs (A, A).

Exercise 3. Show that A = A' if A is a maximal symmetric abeliansubalgebra in End V. 4

The remarkable fact is that the pure algebraic condition (30) is equiva-lent to some topological conditions.

von Neumann Theorem. For symmetric operator algebras A with a unitthe following conditions are equivalent:

a) A is a von Neumann algebra;b) A is weakly closed;

c) A is strongly closed. 11

A von Neumann algebra A of operators in the space V = jx Vxdµ(x) iscalled decomposable if there exists a family of von Neumann algebras Ax CEnd Vx such that A consists of all decomposable operators (28) satisfyingthe condition

(32) Ax E Ax for almost all x E X.

In this case we denote A by fX Axdp(x) and call it a direct integral ofalgebras Ax, X E X.

Among all decomposable algebras in V there is a minimal and a maximalone.

The minimal decomposable algebra V consists of all diagonalizableoperators A which have the form

(Af)(x) = A(x)f(x)


where A is a measurable essentially bounded complex-valued function onX. Thus, P. = C 1V= . The algebra D is abelian and, conversely, everyabelian von Neumann algebra can be realized as an algebra of diagonalizableoperators in a direct integral of Hilbert spaces.

The maximal decomposable algebra R consists of all decomposable op-erators. Here Rx = End V.

Theorem 4. Let A be a decomposable von Neumann algebra in the spaceV = fx V. dpt(x). Then A' is also decomposable and (A'),, = (Ax)l foralmost all x E X.

For the proof, see, e.g., [Di2]. 0We draw several useful corollaries from this important result.

Corollary 1. Since (C 1yx) = (EndVx) and (End Vx)! _ (C. 1v=), weget

D' = R, RI = D.

Corollary 2. Let A be a self-adjoint operator in a Hilbert space V withspectral data (µ, m) (see Remark 4). Then the algebra {A}' is isomorphic toa continuous sum fR Matmixl (C)dpe(x) where Mat,,. (C) denotes the algebraof all bounded operators in a Hilbert space of Hilbert dimension tdo.

In particular, the algebra {A}! is commutative iff the operator A has asimple spectrum.

2.6. Application to representation theory.Let (ir, H) be a unitary representation of a group G. We say that 7r is

a continuous sum (or direct integral) of representations 7rx, x E X, andwrite

(33)ir =I

irx dlu(x)

if there is an isomorphism a : H --* V = fx Vx dA(x) such that

(i) all operators i(g) := a-1 o r(g) o a are decomposable operators in V:

*(g) = J jF'x(9) dp(x)x

and

(ii) the correspondence g u- nx(g) for almost all x E X is a unitaryrepresentation of 0 equivalent to 1r..


Theorem 5 (Gelfand-Raikov, 1942; see [Ge]). Any unitary representationof a locally compact group in a separable Hilbert space can be written as adirect integral of irreducible representations.

Proof. Let (ir, H) be a unitary representation of a locally compact group G.Let A be the von Neumann algebra generated by all operators 7r(g), g E G.Choose a maximal symmetric abelian subalgebra D in the dual algebra A'.There exists a decomposition of H into a continuous sum fX Hxda(x) suchthat D is the algebra of diagonalizable operators. Then A C D' will bedecomposable: A = fX Axdp(x). In particular, all operators zr(g), g E G,are decomposable: 7r(g) = fX 7rx(g)dp(x).

For locally compact groups one can derive from this that almost all 1rxare unitary representations of G in H.

Now, since A! nD! = D, the only decomposable operators in A' are thosethat are in D. Hence, (Ax)e = Dx = C 1H= for almost all x E X. Thismeans that almost all representations 7rx are irreducible.

This result, as in the finite-dimensional situation, justifies the intensivestudy of irreducible representations. But for infinite-dimensional represen-tations of general groups the role of this result is diminished by the lack ofthe uniqueness of the decomposition in question. We discuss this problembelow.

The great achievement in representation theory made in the 1960's isthe dichotomy of all topological groups onto so-called type I and non-typeI groups.

In [Ki2] I suggested the term wild for the non-type I groups. Accord-ingly, groups of type I will be called tame.

Roughly speaking, the tame groups have a nice representation theorywhile the wild groups have all possible unpleasant features (see examples inChapter 4):

1. The topological space d violates even the weakest separation axiomTo. This means that there exist two distinct points in G such that anyneighborhood of one point contains the second.

2. The decomposition of a unitary representation into irreducible com-ponents can be non-unique. It can even happen that two different decom-positions have no unirreps in common.

3. There exist factor representations of types II and III in the sense ofvon Neumann (see [Di2] or [Ki2]).

4. There exist unirreps that possess no generalized characters. Namely,operators 7r(O), 0 E A(G), are never of trace class unless they vanish.

§3. Mathematical model of quantum mechanics 355

For tame topological groups the decomposition into irreducible cornpo-nents is unique in the following sense. Let G denote the set of all equivalenceclasses of unirreps of G. It is a topological space where the neighborhood ofthe class A is defined in Chapter 3, Section 4.5.

Proposition 7 (see [Di2]). Any unitary representation it of a tame groupC can be written in the form

,r = rn(A) - -iradp(.\)

where p is a Borcl measure on G, m is a Bored measurable function that takesvalues from { 1, 2, ... , oc}, and Ira is a unirrep of class A. The measure pis defined up to equivalence and the function m is defined uniquely p-almosteverywhere.

3. Mathematical model of quantum mechanicsIn this section we give a brief dictionary between quantum mechanics andmathematics. Note, however, that a completely adequate translation ofall notions is impossible; e.g. the physical term "quantization" has several(non-equivalent) mathematical translations.

We recommend to the reader to compare this dictionary with Appendix11.3.3 where we describe the mathematical model of classical mechanics.

Physical notions

phase space

state of the system

physical observable

the value of an observable A

in a given state i'

the energy of a system

equation of motion (for states)

Mathematical interpretations

projective space P(H) associatedwith a complex Hilbert space H

an element of P(H), usually repre-

sented by a unit vector V E H

a self-adjoint operator A on H

a random variable with

the mean value (A4, yp)

and distribution (ES(A) b, r;%)

a non-negative operator E on H

hyi

27ri= Eyi (Schrodinger equation)


equation of motion (for observables) A = 2 i [E, A] (Heisenberg

equation)

state where the given observable A an eigenvector of A

has a determined value a with the eigenvalue a

observables that can be commuting operators

simultaneously measured

These rules are more visual when the operator A has the pure pointspectrum. i.e. there is an orthonormal basis jOk}o<k<oo consisting of eigen-vectors, Aa'k = akll)k. In this case the random variable A in the state V)takes the values ak, 0 < k < oc, with probabilities pk = Since

Ikpk=1.

Appendix V

RepresentationTheory

1. Infinite-dimensional representations of Lie groupsFormally, we do not assume any preliminary knowledge about representationtheory. But certainly, some acquaintance with finite-dimensional represen-tations of finite groups will be useful.

In this appendix we collect some basic facts and notions from the theoryof infinite-dimensional representations of Lie groups that are used in themain part of the book.

1.1. Generalities on unitary representations.A unitary representation of a group G is a pair (ir, H) where H is

a Hilbert space and 7r : g* --: U(H) is a homomorphism of G to the groupU(H) of all unitary operators in H.

In other words, it : G -. U(H) is an operator-valued function on Gsatisfying the multiplicativity property:

71(9192) = lr(91)ir(92) for all gl,92 E G.

The scalar product in H is denoted by (x, y), or by (r, y)I1 if there areother Hilbert spaces under consideration.

If we choose an orthonormal basis {xQ}QEA in H, then 7r(g) is given bya unitary matrix (possibly of infinite size) with entries

irQ.3(9) = (ir(9)xQ, x3)

357

358 Appendix V. Representation Theory

More generally, for any two vectors x, y E H we call a matrix element ofit the complex-valued function on G:

(1) xx.y(g) = (7f(g)x, y).

When G is a topological group, in particular, a Lie group, it is natural toimpose some continuity conditions. For finite-dimensional representationsthere is only one way to do this, since the unitary group U(N) has only onenatural topology.

So. we simply require the continuity of all entries 7r,,,Q(g) in any givenbasis. It implies the continuity of all matrix elements (1) and also thecontinuity of the matrix-valued function 7r(g).

For infinite-dimensional representations the situation is more delicate.There are several different ways to introduce a topology in the group U(H).The most common are weak, strong, and uniform topologies.

Recall thatweak continuity of it means that for any two vectors x, y E H the

matrix element 7rx,y(g) is continuous;

strong continuity of it means that for any vector x E H the vector-valued function g 7r(g)x is continuous;

uniform continuity of it means that the operator-valued function g -7r(g) is norm-continuous.

In fact, all interesting infinite-dimensional representations of Lie groupsare strongly continuous. But very few of them are continuous in the normtopology (see Example 1 below).

On the other hand, for unitary representations the conditions of weakand strong continuity are equivalent as the following theorem shows.

Theorem 1. If a unitary representation (7r, H) of a topological group Gis weakly continuous at the unit point e E G, then it is strongly continuouseverywhere.

Proof. We have to show that for any g E G, any x E H, and any e > 0there is a neighborhood V of g such that

l7r(g')x - 7r(g)x(2 < e for any g' E V.

Since it is unitary, the left-hand side can be written as

I7r(g)x12 - 2 Re (7r(g')x, 7r(g)x) + I7r(9 )x12 = 2 Re ((1 - 7r(g-19 ))x, x)

§1. Infinite-dimensional representations of Lie groups 359

But we know that (7r, H) is weakly continuous at e. Therefore, there is aneighborhood W of e, such that

((7r(h) - 1)x, x) I < 2 for h E W.

So, it is enough to put V = gW.

Example 1. Let it be the natural representation of G = R in L2(IR, dt):

(7r(a)f)(t) = f(t + a).

We claim that it is strongly continuous. Indeed, every function f EL2(R, dt) can be approximated in the L2-norm by a continuous function fowith a compact support:

llf - f011t2(R,dt) < e, suppfo C [-R, R].

Further, since fo is continuous on [-R, RI, it is uniformly continuousand for some 8 > 0 we have I fo(t) - fo(s)I2 < zR as soon as it - sI < 6.Then, for la - bI < b we have

117r(a)f - 7r(b)f11 :5 117r(a)f - ir(a)foll + 117r(a)fo - 7r(b)foll

+ I17r(b)fo - 7r(b)f 11 < 3e.

Thus, it is strongly continuous.On the other hand, we claim that 117r(a) - 7r(b)11 = 2 for a # b.

The inequality llir(a) - 7r(b)ll < 2 is clear. To prove the converse, wedefine a function IN, N E N, by the formula

fN(t) =irtsin for ItI < NIb - al,

1. 0 for Itl > Nlb - al.

Then ir(a)fN is almost equal to -7r(b) fN. More precisely,

117r(a)fN112 =117r(b)fNll2 = Nlb-al, ll7r(a)fN-7r(b)fN112 = (4N-1)Ib-al.

Therefore, it is discontinuous in the norm topology. 0This simple example can be widely generalized. Namely, G can be any

Lie group and the operators 7r(g) can act on functions (or, more generally, onsections of vector bundles with connection) on smooth G-manifolds by dif-feomorphisms combined with multiplication by a smooth (operator-valued)function. So locally, using a trivialization, we can write this action in theform (22) (see Proposition 5 below).


All such representations are strongly continuous but discontinuous inthe norm topology.

From now on by a unitary representation of a Lie group G we meana continuous homomorphism 7r : G - U(H) where U(H) is endowed withthe strong operator topology.

These representations form a category Rep(G). The objects of Rep(G)are unitary representations of G; the morphisms from (7r, H) to (7r', H') arethe so-called intertwining operators (or intertwiners for short) A: H -H' such that for any g E G the following diagram is commutative:

H r(9) I H(2) Al IA

H' H'.

The set of all intertwiners forms a complex vector space denoted byI(7r, 7r'), or by HomG(H, H'). The dimension of this space is called theintertwining number and is denoted by i(7r, 7r').

Remark 1. The intertwining number is a very important and usefulnotion in representation theory. It plays the role of a peculiar inner productbetween two representations. The evident relations)

i(7rl ®7r2, 7r) = i(7r1, 7r)+i(7r2, ir), i(7r1, 72) = i(72, 71), i(7r, 7r) > 0

are analogues of linearity, symmetry, and positivity of an inner product.Moreover, the famous Schur Lemma (see below) claims that for a finite

group G the non-equivalent irreducible representations 7r1, 7r2, ... , 7r,1 forman analogue of an orthonormal basis in Rep G:

i(7rk, 7rj) = (kj.

This analogy can be made precise if we pass from representations to theircharacters:

X. (g) := trir(g)

For any finite group G we have

i(7rl, 72) = (g)

IAre they really evident for you? If not, try to prove them using the definitions. (Note thatthe definition of a direct sum is given below.)

§ 1. Infinite-dimensional representations of Lie groups 361

The same formula is true for compact groups if we replace the average overa finite group by integration over a compact group with respect to a nor-malized invariant measure. V

Two representations (ir, H) and (?r', H') are called unitarily equiva-lent if there is a unitary intertwiner U E i(ir, 7r'). In the appropriate bases.related by U, these representations are given by the same matrix-valuedfunction. Therefore, in many cases there is no reason to distinguish betweenit and ir'. For instance, all classification problems deal with equivalenceclasses of representations, not with representations themselves.

Proposition 1. Two unitary representations of a group G are unitarilyequivalent if they are equivalent objects of the category Rep(G).

Proof of the non-trivial part. Let (7r, H) and (7r', H') be equivalentobjects of Rep(G). Then there exists an invertible intertwiner A : H - H'.The adjoint operator A* : H' H is also an invertible intertwiner. Itfollows that A*A is a positive self-adjoint invertible element of i(7r, ir). Thisoperator has a unique positive self-adjoint square root R = A*A. But forany g E G the operator R' = 7r(g)-'Rir(g) is also a positive self-adjointsquare root from A*A, since

(R')2 = ir(g)-1R2ir(g) = u(g)-'A*Ai(g) = A*A.Hence, R' = R and R E i(7r, 7r). It follows that U := AR-1 E i(7r, 7r'). ButU* U = R-1 A* AR-1 = 1. We see that U is a unitary intertwiner and it isunitarily equivalent to ir'. 0

Let (ir, H) be a unitary representation of a group G. If H has a closedsubspace H, that is stable under all operators ir(g), g E G, then the orthog-onal complement H2 = Hi is also stable under all 7r(g).

Let 7rk(g) denote the restriction of ir(g) to Hk, k = 1, 2. It is clear that(Irk, Hk) is itself a unitary representation of G. It is called a subrepresen-tation of (7r, H).

In the appropriate basis the matrix of 7r(g) acquires the block-diagonalform:

it(g) _ (7ri (g)0

0 7r2(g)

In this case we say that (7r, H) is (equivalent to) a direct sum of (7rl, Hl)and (ir2r H2) and write it = 7r1 ® 7r2.

Exercise 1. Check that irl e ire is indeed a direct sum in the categoryRep(G).

(See Appendix II for the definition of a direct sum of objects in a cate-gory.) 46


Exercise 2. Find a non-trivial subrepresentation of the representationfrom Example 1.

Hint. Consider the Fourier transform of the space L2(iR, dt).A representation (7r, H) is called irreducible if it has no non-trivial

subrepresentations. In other words, any closed subspace in H that is stableunder all operators 7r(g), g E G, is {0} or H itself.

Note that H can have non-closed stable subspaces. They are necessarilydense in H provided they are non-zero.

Theorem 2. For a unitary representation (7r, H) of a group G the followingare equivalent:

1. (7r, H) is irreducible.

2. i(7r,7r)={A.1, AEC}.3. Let k7r be the representation

k tames k times

(7r

Then any subrepresentation of k7r has the form (7r(91, H®W), W C Ck.

Proof. We show that the first two statements follow from the third one fork = 1, 2 respectively.

31 1. Any closed stable subspace in H ti H ® C is H ® W where Wcan only be {0} or C. Hence, it is irreducible.

32 = 2. We apply statement 3 for k = 2 to the graph rA of an intertwinerA E i(7r, 7r). This graph is a closed subspace in H ® H, which is invariantunder operators 27r(g), g E G. Therefore, rA must have the form H ® Wwhere W is a subspace in C2. If W has dimension 0 or 2, H ® W cannot bea graph of an operator. If W is 1-dimensional and is spanned by a vector(o, 3), then H ® W is the graph of the scalar operator A 1, A = a. (Ifo = 0, H (9 W is not a graph.)

1 =o- 2. Let A E i(7r, 7r) be an intertwiner. Then A* is also an intertwiner.We can write A in the form A = Al + iA2 where Al = A 2A , A2 = A aA*are Hermitian intertwiners. So, it is enough to show that every Hermitianintertwiner A is a scalar operator.

Let ES(A) be the spectral function for A. Since it is unique, all projectorsES(A) are intertwiners. Therefore the subspaces He = EC(A)H are invariantunder the operators 7r(g). But it is irreducible, hence every HH is either {0}or H. Since ES(A) is increasing and left-continuous, we conclude that there

1exists A E R such that ES(A)

0 for c < \,It follows that A = A A. 1.1 fore>A.

§I. Infinite-dimensional representations of Lie groups 363

2 3. Let V be a closed subspace in H ® C'. The orthoprojector P toV belongs to i(kir, kn). In the appropriate basis in H ® C k the operator Pis given by a block-matrix of the form

Pii Pik

Pk l Pkk

where p,, E i(rr, n). From statement 2 we get pjj = atj . 1 and P = 1 ® Awhere A is an orthoprojector in Matk(C). Hence, V = H ® W with W =ACk. O

Remark 2. The representations (a, H), satisfying the first conditionof Theorem 2, are called topologically irreducible.

If the representation space H has no invariant subspaces (no matter,

closed or non), (7r, H) is called algebraically irreducible.The representation (n, H), satisfying the second condition of Theorem

2, is called operator irreducible.The representations (7r, H), satisfying the third condition of Theorem

2, are sometimes called k-irreducible. This notion makes sense also fork=oo.

For unitary representations oo-irreducibility is equivalent to k-irreducibi-lity for any finite k.

It is known that for non-unitary representations all the conditions ofTheorem 2 are non-equivalent. The simplest example is the group T,, (C) ofupper-triangular matrices acting in C". There are many invariant subspaces.but no intertwiners except scalar operators. C7

1.2. Unitary representations of Lie groups.Let (a, H) be a unitary representation of a Lie group G. The matrix

elements of (7r, H) are bounded and continuous, but not necessarily smoothfunctions on G.

Example 2. Keep the notation of Example 1 and put x = y = x10,1](t),the characteristic function of the unit interval. Then

0 if jai > 1,(a) = 1 - Jai if Jai < 1.

0A vector x E H is called smooth if the vector-function g '-+ 7r(g)x from

G to H is strongly infinitely differentiable.The set of all smooth vectors in H is denoted by H°°.


Theorem 3. The vector x E H is smooth if for any vector y E H thematrix element 7rx,y is a smooth function on G.

The claim results from the following more general statement.

Proposition 2. Let Al be a smooth manifold. let H be a Hilbert space, andlet f : M -+ H be a strongly continuous vector function. Assume that f isweakly smooth on M, i.e. for any y E H the scalar function m - (f (m), y)is smooth. Then f is strongly smooth on M.

Proof. Since the statement is local, it does not depend on the manifold (alln-dimensional manifolds are locally isomorphic). Therefore, we can assumethat M = T", the n-dimensional torus. We represent it as T" = R /Z".

We define the Fourier coefficients {ck}, k E Z", for f as vectors in Hgiven by

(3) ck = r e-Uikt f(t)d"t.

Here t = (ti, ..., tn), k = (ki, ..., kn), kt = E; k:ti, d"t. = Idttn...Adt"I,and the integral is defined as the strong limit of Riemann integral sums (seeAppendix 11.2.4).

For any y E H the quantities (ck, y) are Fourier coefficients of the scalarfunction t -i (f (t), y). Since f is weakly smooth, this function is smooth.Hence, for any N E N there is a constant c(N, y) such that

I(ck, y)I (1 + IkI)" < c(N, y).

But in a Hilbert space any weakly bounded set is strongly bounded, so forany N E N there exists a constant c(N) such that

Icki (1 + IkI)" < c(N).

It follows that f can be written as a strongly convergent series with rapidlydecaying coefficients:

f (t) _ ck e2,rikt

kEZn

Therefore, f is a strongly smooth function. 0

It is not clear a priori if a given representation (ir, H) of a Lie groupG has at least one non-zero smooth vector. The remarkable fact is thatthe space H°° of smooth vectors is always dense in H. Moreover, it is bigenough to reconstruct the representation (Tr, H) of a connected Lie group Gfrom the representation of the Lie algebra g in HO°.


The main result of this section is

Theorem 4. Let (ir, H) be a unitary representation of a Lie group G. Then1. The subspace H°O of smooth vectors is dense in H and stable with

respect to all operators rr(g), g E G.2. For any X E g = Lie (G) the operator

(4) A = -irr. (X) =: -idrr(exp tX) LOdt

with the domain H°' is essentially self-adjoint.3. If G is connected, the representation (rr, H) is completely determined

by the representation rt. of g in HO" defined by (4). In particular, for anyx E H°O we have

(5) rr(exp tX)x = e'tAx

where the right-hand side is defined as the solution to the ordinary differen-tial equation

x'(t) = iAx(t)

with the initial condition x(O) = x.

The proof needs some preparations.We recall that to any X E g there correspond two vector fields on G:

the left-invariant field Rx (infinitesimal right shift) and the right-invariantfield Lx (infinitesimal left shift) given by

(Rxf) (g) := dtf(gexp tX)I _ , (Lxf) (g) :=wt

f(exp -tX)gl _ .

Lo t-o

For any function ¢ E A(G) we define the operator rr(O) on H by theformula

(6) rr(¢) = JC ¢(g)7r(g)dig where dig is a left-invariant measure on G.

Lemma 1. The following relation holds:

(7) rr.(X)x(O) = ir(Lxo).


Proof. Using the left-invariance of d,g, we obtain

a+(X)n(O)x = j dt 7r(exp tX)¢(g)x(g)xdtg

(g ' -' exp(-tX)g) =f d /(exp(-tX)g)x(g)xdtg

t=o

t=o

= f (Lx4)(g)x(g)xdjg = 7r(Lxo)x.c

0

Exercise 3. Show that on H°° the following relation holds:

7r(0)7r.(X) = -7r((Rx +tradX)0).

Hint. Use the relation dt(gexp -tX) = A(exp -tX)dt(g).

Proof of Theorem 4. 1. From Lemma 1 it follows that for any 0 E A(G)and any x E H the vector ir(0)x is smooth. Therefore, we sometimes call7r(O) a smoothing operator. We have to prove that the linear span of theimages of all smoothing operators 7r(¢), 0 E A(G), is dense in H.

Let x be any vector from H. Since 7r is strongly continuous, for anye > 0 there exists a neighborhood U of e E G such that 1ir(g)x - xj < Efor all g E U. Take a function 0 E A(G) with a support in U such thatfu 0(g)djg = 1, and put y = 7r(0)x. Then y E H°O and

.O(g)(ir(g)x - x)dtg1 < E,x,.ly - xI = I fU

2. For X E g consider the operator A = -i7r.(X) _ tX)I1=owith H°° as the domain of definition. From the unitarity of ir(g) we derivethat A is symmetric: (Ax, y) = (x, Ay) for all x, y E H. To check that Ais essentially self-adjoint, we use the criterion from Appendix IV.2.3.

Suppose that y E ker (A' ± i 1) and consider the function

f(t) = (zr(exp tX)x, y), x E DA.

We have

f'(t) = (iAir(exp tX)x, y) = (7r(exp tX)x, -iA`y) = f(t).

Hence f (t) = cert. But f (t) is bounded, therefore c = 0 and y isorthogonal to DA. Since DA is dense in H, the vector y must be zero. So,ker (A' ± i 1) = 0 and the operator iA is essentially self-adjoint.


3. The relation (5) follows from the very definition of eitA. Now, wehave to reconstruct (7r, H) from (7r,, H°°). Recall that for any self-adjointoperator A the operator eitA can be defined as follows. In the appropriate re-alization of H in the form L2(X, t) the operator A is just the multiplicationby a real-valued function a(x). Then we define eitA as the multiplication byeita(x)

When A is the closure of A, this definition coincides with (4) on thesubspace DA = HO°. Therefore ir(exp tX) coincides with eitA on H°O andwith eitA on the whole space H. So, we know the operators ir(g) in aneighborhood of the unit element covered by the exponential map. But G,being connected, is generated by any neighborhood of the unit (see Corollaryto Lemma 3 in Appendix 111.1.3). 0

Remark 3. The idea to use smoothing operators was first used byI. M. Gelfand for 1-parametric groups. It was soon applied by Girding forgeneral Lie groups. Therefore, the subspace in H°O spanned by the vectorsof the form ir(O)v, v E H, is called the Gelfand-Girding space. Thisspace often coincides with HO° (see examples below) but I do not know if itis always true. Q

Remark 4. Theorem 4 gives a non-trivial and important result evenin the case of a finite-dimensional representation. In the finite-dimensionalcase the space H°° can be dense in H only if H°O = H.

So, any unitary finite-dimensional representation of a Lie group is auto-matically smooth. Actually, it is true for all continuous finite-dimensionalrepresentations without the unitarity restriction.

You can try to find an independent proof of this statement in the simplestcase G = R. It looks as follows:

Proposition 3. Let f be a complex-valued function of a real variable tsatisfying the conditions:

1) f is continuous;2) f (t + a) = f (t) f (a) for all a, t E R.

Then f is smooth.

One more corollary from Theorem 4 and its proof is

Stone's Theorem. Let U(H) be the group of all unitary operators in aHilbert space H. Any strongly continuous 1-parametric subgroup in U(H)has the form u(t) = eitA where A is a self-adjoint (not necessarily bounded)operator in H. 0


1.3. Infinitesimal characters.Let U(g) be the universal enveloping algebra for the Lie algebra g =

Lie(G), and let Z(g) be the center of U(g) (see Appendix 111.1.4). If (7r, H)is a unitary representation of a Lie group G, we have a representation of U(g)in H°O that extends the representation 7r. of g. We use the same notation7r. for this representation (sometimes we simply write 7r when it causes noconfusion).

Theorem 5. Assume that (7r, H) is an irreducible unitary representationof a Lie group G. Then for any A E Z(g) the operator 7r.(A) in H°O isscalar:

(8) 7r.(A) = I,(A) 1H--

The map I,, : Z(g) -+ C is an algebra homomorphism, called the infinites-imal character of (7r, H).

Proof. Consider the anti-involution ' of U(g) that sends an element X E gto -X and the product X1 ... Xk to (-1)kXk ... X1. The elements A E U(g)satisfying A' = A are called Hermitian.

Since the center Z(g) is stable under this involution, any element A EZ(g) can be written as A = Al + iA2 where A1, A2 are Hermitian elementsof Z(g). Indeed, it is enough to put Al = A +A'

, A2 = A ZA' . Thus, in theproof of the theorem we can assume that A is Hermitian.

In this case the operator 7r(A) with domain H°O is symmetric. Hence,it admits a closure 7r(A). The graph of 7r(A) is a closed subspace in H ® Hthat is invariant under all operators (27r)(g) = 7r(g) ®7r(g). We already haveseen in the proof of Theorem 2 that such operators must be scalar. 0

In practice, the operators 7r(g), g E G, are usually combinations of shiftsand multiplications by a function. Therefore, operators 7r.(A), A E U(g),are differential operators. The decomposition of (7r, H) into irreducible com-ponents implies the decomposition of H into common eigenspaces for alloperators 7r. (A), A E Z(g).

Theorem 5 admits an important generalization. Namely, let R be a G-invariant rational function on g`. In Chapter 1, Section 3.1, we showed thatsuch a function can be written as a ratio P/Q of two relatively invariantpolynomials of the same weight.

The argument that we used in the proof of Theorem 4 shows that theoperators 7r(sym(P)) and 7r(sym(Q)) are proportional. Hence, the infini-tesimal character I, is defined for the extension of Z(g) that consists of allrational functions on g' that are constant along coadjoint orbits.

§ 1. Infinite-dimensional representations of Lie groups 369

1.4. Generalized and distributional characters.In the theory of finite-dimensional representations the central role is

played by the notion of a character. Recall that a character of a represen-tation (7r, V) of a group G is the scalar function on G given by the formula

X. (g) = tr 7r (g)

For infinite-dimensional unitary representations this notion no longermakes sense: unitary operators are not in the trace class. Nevertheless,using the smoothing technique, we can define for some infinite-dimensionalrepresentations (ir, H) the character of ir as a distribution or as a generalizedfunction on G.

Namely, assume that for a certain class of test densities ,u on G theoperators

lr(p) = f ir(g)d,a(g)rare of trace class. Then we can define the generalized character Xn asthe generalized function on G, i.e. as a linear functional on the space of testdensities given by the formula

(9) (Xir, k) = tr7r(p).

For Lie groups it is natural to take as test densities the expressionsp = O(g)dlg where 0 E A(G) and di(g) is a left-invariant measure on G. (Itis clear that using the right-invariant measure d,.(g) we get the same classof test densities.)

Then the right-hand side of (9) can be viewed as a linear functional onA(G), i.e. a distribution on G. We call it the distributional character of7r and denote it by X,r, the same as the ordinary and generalized characters.

It turns out that for some types of Lie groups the generalized or distri-butional characters indeed exist for all irreducible unitary representations.

They also inherit some important properties of ordinary characters.

Theorem 6. The generalized (distributional) characters have the proper-ties:

a) they are invariant under inner automorphisms of G;2b) if 7rl is equivalent to in, then Xn, = X12c) if irl and ire are irreducible and X,,, = X12 0 0, then irl is equivalent

to 9r2.

2For distributional characters this property holds only for unimodular groups.


Proof. a) The action of an inner automorphism A(h) on a generalized func-tion or a distribution is defined as a dual to the action on test measures ortest functions. Therefore, the statement follows from the invariance of thetrace under conjugation: tr 7r(A(h)rb) = tr (ir(h)lr(cb)7r(h)-') = tr 7r(O).

b) If U is an intertwiner for Tr1, 7r2, then tr7r2(rd) = tr (Uri (.O)U-') _tr7ri(0).

c) For the proof, which is rather involved, we refer to [Di3]. Note thateven for finite groups the standard proof actually reconstructs from X, not7r itself but d it where d = dim ir. So, in our case we need the analog ofTheorem 2 for k = oo (cf. Remark 2). 0

1.5. Non-commutative Fourier transform.The notion of a Fourier transform is very useful for commutative groups

(especially for groups T, Z, R). It turns out that it can be extended to non-commutative groups too. This extension is less simple and less useful butstill can help in situations involving a non-commutative group of symmetries.

To define this transform for a given topological group G we have tochoose a representative ira for any equivalence class A E G. Let dg denote aleft-invariant measure on G, which is known to exist on any locally compactgroup.

For a function f E L' (G, dg) we define its Fourier transform f as anoperator-valued function f on G given by the formula

(g)xa(g)dg(10) f(A) = fG f

This transform shares with its commutative prototype the following basicproperty: it sends the convolution to the ordinary product-

(11) fl * f2 = fl f2Note that on a non-commutative group G the convolution is non-commuta-tive. Therefore it cannot be realized by the ordinary multiplication of scalarfunctions. But for the operator-valued functions the ordinary multiplicationis also non-commutative!

For tame groups G the function f can be reconstructed from its Fouriertransform via the formula

(12) f (g) = Jc tr (f (A)xa(g)-) dp(A)

where p is a special measure on G called the Plancherel measure, dualto the measure dg on G.

§2. Induced representations 371

It is known that for a compact group G with the normalized measure dgthe Plancherel measure is just the counting measure on G (which is a discreteset) multiplied by dim zra. For non-compact Lie groups the computation ofthe Plancherel measure is one of the deepest problems of harmonic analysis.

We finish this section with the non-commutative generalization of thePontryagin duality principle. Denote by L2(G, µ) the Hilbert space of opera-tor-valued functions 0 on d taking at a point A a value in End(VA) with thenorm

I012 = ftr (4(A)

More accurately, this is a continuous surn over (G, µ) of Hilbert spacesformed by Hilbert-Schmidt operators in Va (see Appendix IV.2.5).

Non-commutative Plancherel formula. Let G be a tame locally compactgroup. The non-commutative Fourier transform can be extended to a unitaryisomorphism between L2(G, dg) and L2(G, µ), i.e.

(13) f If(g)12dg = If1L2(c,d9) = IfIL2(O du) = ftr (Y(,\) '

2. Induced representationsWe start with a brief description of the theory of induced representationsfor finite groups. We use this "toy model" to introduce the basic notionsand notation that will be used later in the case of infinite-dimensional rep-resentations of Lie groups.

2.1. Induced representations of finite groups.Let G be a finite group, and let H C G be a subgroup. For any finite-

dimensional representation (p, W) of H we construct a representation of thegroup G, which is called the induced representation and is denoted byInd y(p, W). The space where the induced representation acts is denotedby L(G, H, p, W). It consists of W-valued functions 0 on G satisfying

(14) ¢(hg) = p(h)O(g) for all h E H, g E G.

The group G acts on this space by right shifts:

(15) ((Ind Hp)(g)(k) (9i) = '(gig).

Note that right and left shifts on G commute: therefore the operators (15)preserve the condition (14).


The geometric meaning of an induced representation is especially clearwhen (p, W) is the trivial representation (1, C). In this case the spaceL(G, H, 1, C) is just the space L(X) of complex-valued functions on theright coset space X = H\G. More precisely, to a function 0 E L(G, H, 1, C)we associate the function f E L(X) according to the rule f (x) = 4)(g) whereg is any representative of the coset x E X. From (10) we see that the value4)(g) does not depend on the choice of a representative.

The action of G in L(X) is just the geometric action by right shifts:

(16) ((Ind Hp)(9)f) (x) = f (x . g).

In the general case we interpret the elements 0 E L(G, H, p, W) as sectionsof some G-vector bundle E over X with the fiber W.

Recall that from the algebraic point of view the space of sections r(E, X)is a projective module over the algebra of functions L(X). In our case thespace L(G, H, p, W) carries a natural L(X)-module structure: (f 4))(g) _f (H9)O(9)

Actually, for finite groups the bundle in question is always trivial. In al-gebraic terms it means that L(G, H, p, W) is a free L(X)-module: L(X, W)^- L(X) ® W. To establish it, we choose for any x E X a representative s(x)of the coset x.

The map s : X - G is called a section of the natural projectionp: G - X: p(g)=Hg,ifpos=Id.

It is convenient to choose e as a representative of the coset H, i.e. puts(H) = e. Later we always assume this condition.

Using a section s we can identify the set G with the direct productH x X. Namely, any element g E G can be uniquely written in the form

(17) g = hs(x), h E H, x E X.

Indeed, from (17) we derive x = Hg, h = gs(Hg)-1.We can now rewrite the formula for the induced representation in another

form that is often more convenient. To this end we introduce the so-calledMaster equation:

(18) s(x)9 = h(x,9)s(y)

Here X E X and g E G are given, y = x g, and h(x, g) E H is determinedby the equation (18).

Proposition 4. a) The map h(x, g) from X x G to H satisfies the followingcocycle equation:

(19) h(x, 9192) = h(x, 9i)h(x . 9i, 92).


U) Let us identify ¢ E L(G, H. p. 144) with f E L(X, W), using thesection s : f (x) = ¢(s(x)). Then the induced representation (11) takes theform

(20) ((Ind Hp) (9)f) (x) = A(x, 9)f (x - g)

where the operator-valued function A(x. g) is defined by

(21) A(x, g) = p(h(x, g)).

Proof. a) is an immediate consequence of the Master equation applied to9 = 9192

To prove b) we use the relation between f and 0 and the Master equa-tion.

Note that (20) is a natural generalization of the geometric representation(16). The remarkable fact is that the converse is also true: all representationsof the form (20) are in fact induced representations.

Proposition 5. Let C be a finite group, let H C G be a subgroup, andlet X = H\G be the corresponding right cosec space. Assume that 7r is arepresentation of G in the space of vector functions L(X, 14') acting by theformula

(22) (ir(9)f) (x) = A(x, 9).f (x . g)

where A is some operator-valued function on X x G. Then there exists a rep-resentation (p, W) of H such that (7r, L(X, W)) is equivalent to Ind G(p, W).

Proof. From the multiplicative property 7r(9jg2) = n(91)7r(92) we deducethat A(x, g) satisfies the cocycle equation

(23) A(x, g1g2) = A(x, gi)A(x - gi. 92).

Lemma 2. All solutions to the cocycle equation (23) have the form

(24) A(x, g) = C(x)-'p(h(x, g))C(x . g)

where C is an invertible operator-valued function on X, h(x, g) E H isdefined by the Master equation, and p is a representation of H in W.


Proof. Put gl = s(x), x = xo := (H), and g2 = g in (23). We obtain

A(xo, s(x)g) = A(xo, s(x))A(x, g) or A(x, g) = B(s(x)i-IB(s(x)g)

where B(g) := A(xo, g).If we now put x = xo, gi = h, and g2 = g in (23), we get B(hg) _

B(h)B(g). Finally, putting C(x) = B(s(x)) and p(h) = B(h) we obtain thedesired result.

To complete the proof of Proposition 5 it is now enough to make thetransformation f (x) '-+ C(x) f (x) that sends A(x, g) to the desired form(21) (and the representation (22) to an equivalent representation).

Now we prove a very useful criterion: when a representation (7r, V) ofa finite group G can be written in the form (22). In other words, we wantto know when (7r, V) is equivalent to Ind H(p, W) for some representation(p, W) of the subgroup H C G. To formulate this criterion we need somenew notions and notation.

Let X = H\G be the set of right H-cosets in G. Denote by A(X) thecollection of all complex-valued functions on X. From the algebraic pointof view A(X) is a commutative associative algebra (with respect to theordinary multiplication) with an involution f N f (complex conjugation).

Let us define a *-representation (11, V) of A(X) as a map IT : A(X) --+End W, which is an algebra homomorphism with an additional property11(f) = II(f )* where * means Hermitian adjoint operator.

We call a *-representation (H, v) non-degenerate if 11(1) = 1. Here1 in the left-hand side is the constant function on X with value 1, and 1 inthe right-hand side denotes the unit operator in V.

Let us say that a *-representation (11. V) of A(X) and a. unitary repre-sentation (ir, V) of G in the same space are compatible if

(25) x(9)11(0)7r(9)-' = 11(R(9)O) where (R(9)b)(x) = O(x 9).

To any representation of the form (18) we can easily associate a *-represen-tation (11, V) of A(X) compatible with (7r, V) = (IndH(p,W), L(X, W)).Namely, put

(26) (11(O)f) (x) = O(x)f (x)

Then

(7r(9)11(c)f) (x) = A(x, 9)O(x . 9)f (x . g)

= 5(x - g)A(x, 9)f (x g) = (II(R(9)0)7r(9)f) (x)

and (25) is satisfied. Note that this representation 11 is non-degenerate.


Inducibility Criterion. A unitary representation (7r, V) of a finite groupG is induced from some representation (p, W) of a subgroup H if thereexists a non-degenerate. *-representation (II, V) of A(H\G) compatible with(7r, V).

Proof. We have seen above that the condition is necessary. We prove thatit is also sufficient. Let X = H\G. We introduce in A(X) the special basisconsisting of functions ba, a E X. where

Mx)(1 for x = a,

0 forx54 a.

Let Pa := II(ba). From the relations

ba = ba = ba 6a6b=0 for a5b, 1: 5a=1aEX

we obtain

PP =PR =Pa, PaPb=O for a#b, > Pa=1.aEX

The geometric meaning of these relations is the following.

The space V is a direct sum of orthogonal subspaces Va, a E X, and Pais the orthoprojector on Va.

The compatibility condition (25) applied to g E G and ba E A(X) gives:

7r(g)Pa7f(g)-' = Pa.g-1 or 7r(g)Va C Va.g-3.

Since 7r(g) is a unitary operator, we conclude that all subspaces Va have thesame dimension, say k. Fix a k-dimensional Hilbert space W and somehowidentify each Va with W. An element v E V can be considered as a W-valuedfunction f on X. Namely, the value f (a) is equal to Pav E Va W.

Finally, the compatibility condition implies that the value of the trans-formed function 7r(g) f at the point x E X depends only on the value of theinitial function f at Therefore, 7r has the form (22), hence, according toProposition 5, is induced from some representation p of the subgroup H. D

The representation-theoretic meaning of induced representations is re-vealed by

Proposition 6 (Frobenius Duality). Ind N defines a functor from Rep Hto Rep G which is dual to the restriction functor Res H : Rep G -> Rep Hin the following sense:(27)I (7r, Ind yp) I (Res H7r, P) for any (7r, V) E Rep G, (p, W) E Rep H.


Proof. We establish a one-to-one linear correspondence between the spacesI(ir, IndHp) and I(ResHir, p). Let A : V --+ L(X, W) be an intertwiner be-tween Or, V) and (Ind Hp, L(X, W)). Then for any v E V its image f = Avis a W-valued function on X. Denote by a(v) the value of this functionat the initial point xo = (H). We claim that the map a : v --# a(v) be-longs to HomH(V, W) = i(ResHir, p). Indeed, a(7r(h)v) = (Air(h)v)(xo) _(IndHp (h)Av)(xo) = p(h)Av(xo) = p(h)a(v).

Conversely, if a E HomH(V, W), we define the operator A : V -L(X, W) by Av(x) = a(ir(s(x))v). Then

A(ir(g)v)(x) = a(ir(s(x)g)v) = a(ir(h(x,g)s(x-g))v)

= p(h(x, g))a(ir(s(x - g))v) = p(h(x, g))Av(x - g)= (IndH(g)Av)(x).

Therefore, A is an intertwiner between (7r, V) and (Ind Hp, L(X, W)). Fromthe definitions of A and a it is evident that the correspondence a f--+ A isa bijection.

Actually, the Frobenius duality can be used as an alternative definition ofan induced representation. Indeed, let 7r1, ..., irk be all (equivalence classesof) irreducible representations of G. Then any 7r E Rep G is uniquely writtenas

mj EZ+.In particular, it is true for 7r = Ind

Hp. But from (27) we have

mj = i(Ind y(p),irj) = i(p, ResH(7rj)).

Therefore, all mj are uniquely determined by the equivalence class of p.Hence, the equivalence class of it is determined by (23).

Three useful corollaries of this observation are:

Proposition 7 (Induction by stages). If K C H C G are two subgroups,then

(28) Ind HG lnd K _ Ind K.

Proof. The proof follows from the evident relation Res K Res H ^ Res Ko.0

Proposition 8 (The structure of the regular representation). Let pInd{el 1 be the regular representation of G. Then

k

p=Ed; 7ri where d, =dim7ri.i=1

In particular, we get the Bernside formula Ek 1 d? = IGI.


Proposition 9 (Frobenius formula). The character of the induced repre-sentation ir = Ind H (p) is given by the formula

(29) X'(9) _ j Xp(xgX 1)zEG/H

where X. is extended from H to G by zero.

In conclusion we present the formula for the intertwining number for twoinduced representations.

Proposition 10 (Mackey Formula). Let H and K be two subgroups ofthe finite group G. Let (p, V) and (a, W) be representations of H and K,respectively. Then

(30) i(IndHp, IndGa) = dimL(G, H, K; p, a)

where L(G, H, K; p, a) is the space of Hom(V, W)-valued operator func-tions f on G satisfying

(31) f (kgh) = a(k) o f (g) o p(h).

0

In particular, for trivial representations p and a we obtain

(32) i (IndH

1, Ind G 1) = I K\G/HI (the number of double cosets).

Note that the last number has three group-theoretic interpretations:1) the number of K-orbits in X = G/H,2) the number of H-orbits in Y = K\G,3) the number of G-orbits in X x Y.In many cases this simple formula already allows us to construct all

irreducible representations of a given finite group G.

Example 3. Let G be the group of rotations of the solid cube. Thereare several homogeneous G-sets (the index shows the cardinality of the set):

the set X6 of faces;

the set Xs of vertices;the set X12 of edges;

the set X4 of big diagonals;

the set X2 of inscribed tetrahedrons;the set X1 - the center of the cube.


Using the formula (32) we can fill up the table of intertwining numbersbetween the geometric representations (7ri, L(Xi)), i = 1, 2, 4, 6, 8, 12:

i\j 1 2 4 6 8 12

1 1 1 1 1 1 1

2 1 2 1 1 2 1

4 1 1 2 1 2 2

6 1 1 1 3 2 3

8 1 2 2 2 4 4

12 1 1 2 3 4 7

Let us explain, for example, the equality i(7r12i 7r8) = 4. The stabilizer of anedge is a subgroup K2 of two elements in G. The set X8 of eight verticessplits into four K2-orbits. Another example: there are exactly two G-orbitsin X6 x X8i one consists of pairs (f, v) such that the face f contains thevertex v, another contains all other pairs. Therefore, i(7r6i 1r8) = 2. Thereader is advised to make an independent check of some other numbers inthe table.

Now we show how to describe the set G of all (equivalence classes of)irreducible representations of G by just contemplating the table above.

First, notice that 7r1 is a trivial 1-dimensional representation. We includeit in the set G under the name pl. We see from the table that it enters withmultiplicity one in every 7ri.

Further, since i(1r2, 7r2) = 2, ire splits into two irreducible components:pl and another 1-dimensional representation pi, non-equivalent to pl.

The second row of the table shows that this representation p' enters inthe decomposition of 7r8 with multiplicity 1 and does not occur in 74, 76, or7T12

From 474, 7r4) = 2 we conclude that IN is the sum of the "obligatory"pl and an irreducible 3-dimensional representation, which we denote p3.

Since i(7r6, 7r6) = 3, ir6 splits into three non-equivalent representations.The equality i(7r6i 7r4) = 2 shows that one of them is pi and another is p3.The remaining one we denote by p2.

The equality i(7r8, 1r8) = 4 means that 7r8 is either a sum of two equiv-alent irreducible components, or splits into four pairwise non-equivalent ir-reducible components. The first possibility contradicts i(ir8, irk) = 1. Fur-thermore, the equations i(7r8i 7r2) = i(7r8, 7r4) = 2 imply that 7r8 containsp and p3 with multiplicity 1. Together with the obligatory pl it gives thedimension 5. Therefore, the remaining component has dimension 3 and wedenote it by p'3'


The squares of dimensions of pl, p'1, p2, p3, p3 sum up to 24 = IGI.According to the Bernside formula (Proposition 8), we have found all therepresentations of G.

We leave it to the reader to find the decomposition of 7r12, to describegeometrically the intertwiners between L(Xi) and L(X3), and to locate theirreducible components in L(Xi). 0

2.2. Induced representations of Lie groups.Now we turn to the case of unitary representations (possibly infinite-

dimensional) of Lie groups. Let G be a Lie group, let H C G be a closedsubgroup, and let M = H\G be a smooth manifold of right H-cosets inG. Let (p, W) be a unitary representation of H in a Hilbert space W. Wewant to define the induced representation Ind

Hp, which must be a unitary

representation of G.We cannot use the formulae (14), (16) because in general there is no

G-invariant inner product in the space L(G, H, p, W) even in the simplestcase (p, W) = (1, C) when L(G, H, p, W) coincides with the function spaceL(M).

However, in this case we can use the so-called natural Hilbert spaceL2(M) associated with the manifold M instead of L(M). By definition,the space L2(M) consists of square-integrable sections of the line bundleL of half-densities on M. In a local coordinate system (x', ... , xn), n =dim M, a section f of L has the form f (x) Idnxl and under the change ofcoordinates x = x(y) it transforms into f (y) Fjdnyj where

f(y) = f(x(y)) I detOXI z

8y? II III

For two sections f1 and f2 the inner product is given by the integral

(33) (fl, f2) =JM

Here the expression fl f2 is understood as a density on M, which in a coor-dinate chart U with coordinates (x1, ... , xn) looks like f1(x) f2(x)Idnx1.

Since L is a natural bundle (see Appendix 11.2.2), the group G CDiff (M) acts on L. If a point m E M has coordinates (x1, ... , x') in achart U and the point m g has coordinates (y', ... , yn) in a chart V, theny3 are functions of xi and vice versa (for a fixed g E G). Also a sectionf = f (x) ldnxl goes to the section ir(g) f, which has the form Y (y) Jdny[

in V where Y (y) = f (x(y)) I det II8xu/8y' II 112. From this definition it fol-lows that (ir(g) fl, ir(g) f2) = (fl, f2), so the representation (7r, L2(M)) isunitary.


In practical computations one usually uses just one chart U with coordi-nates (x1, ... , x"), which covers almost all of the manifold M except for a fi-nite union of submanifolds of lower dimensions. We can also define a smoothsection s : U -+ G of the natural projection p : G -+ M : g '-+ Hg. Then thebig part of G (also except for a finite union of submanifolds of lower dimen-sions) will be identified with the product H x U via g = hs(x), h E H, X E U.

Note that in general the fiber bundle H - G p+ M is non-trivial, sothat there is no smooth (or even continuous) section s : M - G of theprojection p on the whole M.

Example 4. Let M be the two-dimensional sphere S2 C 1R3. It is ahomogeneous manifold SO(2)\SO(3). Geometrically, the group SO(3) canbe viewed as the tangent sphere bundle3 Tl M over M. The absence of acontinuous section s : M -' G means that there is no continuous tangentvector field on S2 that has unit length everywhere. This is the well-known"Hedgehog Theorem" from algebraic topology.

Nevertheless, we can cover all of the sphere except for one point by acoordinate chart U (see Appendix 11.1.1) and identify p-1(U) C G withU x SO(2) = R2 X S1. 0

Let d G(g) and dH (h) be left-invariant volume forms on G and H, respec-tively. Choose a local smooth section s over U C M and identify p 'U C Gwith H x U. In the parameters (h, x) a left-invariant form on G looks like

dc(g) = r(h, x) dH(h) d"x

for some smooth function r on U x H. Taking into account the left invarianceof dc(g) and dH(h), we get r(x, h'h) = r(x, h). Therefore, r(x, h) = r(x).Let us define a special volume form µ8 on U by

(34) µ3 = r(x)Oc(s(x))d"x.

Then the following equality holds:

(35) OG(s(x))-1 dH(h) du8(x)

Using the relations dc(g) = /c(g)dc(g), dH(h) = AH(h)dH(h), we canrewrite (35) in the form

(35') dG(9) = dc(hs(x)) = AG(h)dy(h)dii8(x)

The advantage of the measure µ8 is its simple transformation rule.

3That is, the collection of tangent vectors of unit length to M.


Lemma 3. Under the action of G the measure µ9 is transformed accordingto the rule

(36)dµ4 (x . 9) _ AH(h., (x, g))

dp8(x) AG(It,(x, 9))

where h,(x, g) E H is defined by the Master equation

(37) s(x)g = ia9(x, g)s(x g).

Proof. We shall, as above, use the pair (h, x) E H x U to parametrize theelement hs(x) E G. Then the element hs(x)g is parametrized by the pair(h h4(x, g), y) where y = x g and hg(x. g) is defined by (37). Thereforefor the right-invariant form d G we have

dG(h, x) = d? (h . hq(x, g). x - g).

From this, using (35'). we obtain

AG(h)dpR(x)drt (h) _AGO h,,(x, 9)) dp,(x . g)d,'(h h(x. y)),

JH(h) IH(h hq(x, g))

which is equivalent to the desired formula (36). 0

Example 5. Let G = SL(2, R) act from the right on the projective linePI (R) with homogeneous coordinates (x° : xa) by the formula

g fl:(x0:r1).t(xO:x1)(0b

Let U be the chart with coordinate x = x°/xa. It covers all of theprojective line except for the point (1 : 0). In terms of this coordinate theaction of C is fractional linear: x :- j

The group G is unimodular (as any semisiinple Lie group) and the bi-invariant volume form is

daAd;3Ad7 d;3Ad-yAd6 doAdj3Ad6 daAd-yAd6dg=

a=-

b=

13

/

The subgroup H, the stabilizer of the point x = 0. is the upper triangularsubgroup with elements

h = Cap 0A.


This group is not unimodular and we have

dH h = dA A dµ,

We choose the section s : P1 G

s(x)= 1 ).(x 1

Then the Master equation (37) takes the form

.AH(h) _ .12

(11)

(-yCl 1" ) - (0 A - ') k1

The solution to this equation is

y=x J_ ax +y A= (dx+6)-1 µ=iix+b'

The parametrization y H (h, x) looks like

(y 6) -((6

01 6) ,

The equation (35') takes the form

b)'

d/3A dyAdb I _ I db a 2d` Ad1i,()I,6

which implies Idle,(x)I = IdxI.

The transformation law (36) in this case is

d /dx= (13x+6)--.(('-T+-' 2

lax+b

The natural Hilbert space L2 (P' (R)) consists of expressions 1(x) l dxl,

f E L2(R. dx), and the action of G in terms of functions f (x) has theform

I(x) = ji3:r+61-1 f(Ox+b

Cam

( 6) f \

This formula defines a unitary representation of SL(2, ]R) in L2(ilD1(R)).

The considerations above suggest the following general definition of theunitary induced representation Ind

H(p, W) for Lie groups.


The representation space L2(AI, W) - L2(AI)®W consists of IV-valuedsquare-integrable half-densities on A1= H\G.

In a local chart U a half-density f can be written as f (.r) dp,(x),f E L2(M, W, p,). So, for a given section s : U -+ G the representationspace is identified with L2(Af, W, p,).

The representation operators in this space are given by

(38) (ir(g)f)(x) _(AH (s(x, g))1 p(hs(x,

g))f (x ' g).OG(h.(x,

g))

This is the exact analog of formulas (16), (17) for finite groups. Note that

the additional factor(Au h, (:, g)

equals 1 for finite groups.Ac

We can also give an alternative definition of the space L2(M, W) interms of vector-functions on G, but it takes more preparation.

For any f E L2(M, W, p,) we define the W-valued function of on G by

(39)Of(hs(x))

=(AH(h)(h)) 51 p(h)f (x).

We denote the space of all of , f E L2 (Al. W, p,), by L2(G, H, p, W). It isclear that all 0 E L2(G, H, p, W) possess the property

(40) 0(hg) =OH(h)

Zp(h).(g)CSC; /

It turns out that this property is essentially characteristic for L2(G, H, p, W).To explain this we introduce a non-negative function p(g) such that4

1) fHp(hg)d,.h=1 for allg6G,2) supp p f1 Hg is compact for any coset Hg.

Lemma 4. Assume that a W -valued strongly measurable function 4)(g) sat-isfies (40) and also the finiteness condition

I4)(g)llp(g)dr(g) < oo.(41) IG I

Then 0 E L2(G, H, p, W).

4The existence of such a function is rather evident and the detailed construction is given inN. Bourbaki's book, Integration. Ch. VII, §2. no. 4.


Proof. Let f (x) = t(s(x)). If f E L2(M, W, µs), then 0 = Of and weare done. The function ip(g) = II0(g)IIt2v has the property z/i(hs(x)) _

(3(x)) = oc(h IIf(x)IIW. Therefore, after the change of variablesg = hs(x), the integral (41) becomes

ff p(hs(x))p(hs(x))dG(hs(x)) =XN

IIf(x)IIwp(hs(x))dH(h)dµs(x)

IIf(x)Ilwdis(x) = IllII2L2(ar.W,µ,)

We see that the finiteness condition (41) indeed is equivalent to finitenessof the norm IIfIIL2(1,f,VV,,j,) D

Note that in the course of the proof of Lemma 4 we get the explicitformula for the inner product in L2(G, H, p, W):

(42) (01, 02) = f(0i(), 0i(g))Wp(g)dG(g)

where p(g) is any function on G satisfying conditions 1) and 2) above.

Example 6. Keep the notation of Example 5. Let p,,E be a 1-dimen-sional unitary representation of H defined by

pa,c 0 ,1µi = I,\I`O(signa)F

where a E R, e = 0, 1. Then the induced representation 7r,,E acts in L2(R, dx)by the formula

f) (x) = Ifix+aI-i+:o (sign (fix+8))`f(ax+d)(7ro.E (aY [3X

This is the so-called principal series of the unitary representation of G.

2.3. *-representations of smooth G-manifolds. In this section wetransfer the results of Section 1 to the case of G-manifolds. The most im-portant results have to do with homogeneous manifolds.

Let M be a smooth manifold with an action of a Lie group G on it. Wecall M a G-manifold.

Definition 1. A *-representation H of a smooth manifold M in a Hilbertspace ?{ is an algebra homomorphism H : A(M) - End(f) satisfying thecondition:

H(¢) = II(0)`.A *-representation II is called non-degenerate if the set of vectors of theform II(O)x, 0 E A(M), x E 9{, is dense in N.


Definition 2. We say that a *-representation II of a smooth G-manifoldM in a Hilbert space 7d is compatible with a unitary representation (n, f)of the group G in the same space if the following condition is satisfied:

x(9) o I1(i) o x(9)-1 = II(R(g)0) where (R(g)0)(m) = O(m - g).

Let us discuss these definitions and draw some consequences from them.

First of all, observe that the non-degeneracy condition in Definition 1for compact manifolds is equivalent to the equality II(1M) = IN. It isintroduced mainly to avoid the trivial example 11(0) = 0.

Second, instead of operators 11(0) we can define the projector-valuedmeasure it on M so that the operators 11(0) are given by integrals:

(43) 17(0) = jt O(rn)dp(m).

For the reader's convenience we recall the main facts here about projec-tor-valued measures and the corresponding integrals (see, e.g., [KG] or anytextbook on spectral theory of operators in Hilbert spaces).

Let X be a topological space. Denote by B(X) the minimal collectionof subsets of X that contains all open subsets and is closed under countableunions and complements. Elements of B(X) are called Borel subsets.

A real-valued function f is called a Borel function if for all e E R thesets

Lc(f) :_ {x E X I f (x) < c}

are Borel sets. It is known that the set of all Borel functions is the minimalalgebra that is closed under pointwise limits and contains all continuousfunctions. In other words, any Borel function can be obtained from contin-uous functions by a sequence of pointwise limits.

Warning. Note that one limit is not enough! Actually, the situationhere is rather delicate. The functions that can be written as a pointwise limitof a sequence of continuous functions form the so-called 1st Baire class. Ina complete metric space the function of this class always has a dense subsetof points of continuity.

The pointwise limits of sequences of functions from the 1st Baire classform the 2nd Baire class. The functions of this class can be nowhere con-tinuous but still do not exaust the set of Borel functions.

We can inductively define the k-th Baire class for all k E N and call theunion of these classes the w-th Baire class. It still does not exaust the set ofBorel functions.


So. we can define new Baire classes: w + I-st, w + 2-nd.... , 2w-th, ... ,3w-th, ... , w2-th..... w"'-th. etc.

The collection of all Baire classes is a minimal uncountable set: anyBaire class contains only a countable family of smaller Baire classes.

A complex-valued function is called Borel if its real and f.maginary partsare Borel functions.

We denote by P(H) the set of all orthoprojectors in a Hilbert spaceN, i.e. operators P satisfying P2 = P = P.

By definition a projector-valued measure p on X is a map from B(X )to P(N) that satisfies the conditions:

1) il(X) = IN,2) p (U; ° l S1) = E' l p(S;) where U denotes the union of disjoint sets.The last property is usually called the countable additivity.Remark 5. The additivity property is a very strong restriction on a

projector-valued function. The point is that a suns PI + P2 of two orthopro-jectors can be an orthoprojector only if PI P2 = P2PI = 0.5 Geometricallythis means that subspaces Ni = PIR and N2 = P27{ are orthogonal.

Therefore, the additivity of p implies that for any decomposition of aset S into a union of disjoint sets Si, S2, ... we have a decomposition ofthe Hilbert space V = p(S)N into the orthogonal sum of subspaces Vk =p(Xk)H, i = 1, 2, .... C)

For a scalar function f on X we can define the integral of f with respectto a P(N)-valued measure p, which is denoted by A = f. Jr(x)dp(x).

By definition it is an operator A in 11 such that for any two vectorsv1, v2 E % we have

(Avi, v2) = fV f (.)dp,,,.,2(x)

where the complex measure p,,,,,,z is defined by p,,,,,,z(S) = (p(S)vi, v2).One can show that the integral exists for any bounded Borel function

f. If f is continuous and has a compact support, then the integral can bedefined as a uniform limit of the Riemanu integral sums

S({V}, {xi}; f) _ f(xi)µ(V)

where (Vi } is a covering of X by small Borel subsets, xi E V are arbitrarychosen representatives, and the limit is taken over any sequence of coveringswith shrinking elements.

5Check this using the algebraic definition of orthoprojector: P2 = P' = P.


For any *-representation (11, N) of a smooth manifold Al we can asso-ciate a projector-valued measure it on M as follows. For an open subsetO E Al we can define p(O) as the orthoprojector on the closed subspace inN spanned by vectors of the form IT(O)v, ¢ E A(O), V E N.

For more general Borel sets the measure it can be uniquely defined usingthe property of countable additivity.

Conversely, for any projector-valued measure p on AI we define a *-representation R of Al by (38).

In terms of the projector-valued measure it the compatibility conditiontakes the form

7r(g) o p(S) o 7r(g)-' = p(S g- 1) for any Borel subset S C Al.

It is possible to describe all representations of a smooth manifold Al. Com-paring this situation with the one described in Section 1, we can suggest thefollowing standard model for R.

Let W be a Hilbert space, and let p be any Borel measure on Al. Definethe new Hilbert space V = L2(Al. W, dp) as the space of square-integrableW-valued strongly measurable functions on Al with the inner product

(44) (ft , MV = fI

(fl (x). f2(x))tt'dp.

Then we define the operator R(¢) in V by the formula

(45) (R(O)f)(x) = O(x)f(x)

The projector-valued measure p in this case is defined by

(46) (p(S) f)(x) f (.r) when x E S,

0 otherwise.

In other words, the operator p(S) is the multiplication by the characteristicfunction of the set S.

It is rather clear that the equivalence class of the *-representation (R, V)thus obtained depends only on the equivalence class of the measure p andthe Hilbert dimension k of W. So, we denote it by (111,.k, Vt,k) or simplyRµ.k

It turns out that this is not a universal model. To obtain the mostgeneral type of representation, we have to make the space W dependent onthe point m E Al. This can be done and the final construction looks asfollows.


Proposition 11. Any non-degenerate *-representation H of Al is (equiva-lent to) a direct sum of representations II,,k, ., k = 1.2..... c.

The measures {ttk} are pairwise disjoint and defined by the representa-tion II uniquely up to equivalence.

Note that for Al = R' (resp. for Al = Sl) this is essentially the SpectralTheorem for a self-adjoint (resp. unitary) operator in a Hilbert space.

Fortunately, the initial model (with a constant W) is still universal in thecase of representations of homogeneous manifolds compatible with a givengroup representation. (Compare again with Section 1.)

Theorem 7. Let. AI = H\G be a homogeneous right G-manifold, and letti be a Borel measure on AI given by some non-vanishing smooth density p.Assume that (R, fl) is a *-representation of Al compatible with a unitaryrepresentation (7r. f) of the group G.

Then the Hilbert space rl has the form N = L2(AI, W dp) for someHilbert space W. the representation 11 is given by (40), and it has the form

(47) (n(g)f)(x) = - A(x, 9)f (x -.9)dIZ(x . g)

dp(x)

where A is a unitary operator-valued function on Al x G satisfying the co-cycle equation:(48)A(x, gl)A(x gl, ,q2) = A(x, gl 2) for almost all x, gl, 92E AI x G x G.

Proof. For any g E G the operator R(g) defines an automorphism of the al-gebra A(M). The compatibility condition implies that the *-representations11 and 11 := 1T o R(g) are equivalent. It follows (see Proposition 11) that allmeasures 12k are quasi-invariant under the action of G on Al. We now usethe following fact from measure theory.6

Proposition 12. All non-zero Borel measures on the homogeneous mani-fold AI = H\G that are quasi-invariant under the action of G are pairwiseequivalent. In particular, they are equivalent to the measure it from Theorem7 and to all measures p. introduced by formula (33) above.

In the case under consideration we conclude that from the measurespk, k = 1, 2, ... , oo, only one measure, say p,,, is non-zero and equivalent tothe measure p.

6This result is a generalization of Lemma 2 in Chapter 2 and can he proven in a similar way.


Fix a Hilbert space W of dimension n. The representation space 7lcan be identified with L2(M, W, p) so that operators I1(Q,), 0 E A(M), actaccording to formula (47).

Now compare the representation operator 7r(g) with the unitary operator

(T(9)f) (x) = dd i(x)) f (x g)'

It is clear that these two operators have the same commutation relationswith operators II(O), namely

(25') ir(g)HI(0)ir(g)-I = II(R(g)O) where (R(g)O)(x) = O(x g).

It follows that the operator ir(g)ir(g)-' commutes with all multiplication op-erators 11(0), ¢ E A(M). Therefore, this operator itself is multiplication byan operator-valued function A(x, g). (We leave it to the reader to prove thisstatement using the scheme followed in Chapter 2.) So, our representationacquires the desired form (47).

The cocycle equation for A(x, g) has the same form as in (23) for fi-nite groups. It is deduced in the same way from the equality ir(glg2) =lr(gl)7r(g2), but now we can only assert that for any 9i. g2 E G the equationis satisfied for almost all m E M. 0

2.4. Mackey Inducibility Criterion.The main result of this section is the criterion for when a representation

of a Lie group G is induced from a representation of a closed subgroupH C G. It has the same form as in the finite group case (see Section 1), butthe proof is much more involved.

Theorem 8 (Mackey Inducibility Criterion). Let G be a Lie group, letH C G be a closed subgroup, and let Al = H\G be the smooth manifold ofright H-cosets in G. Assume that the unitary representation (7r, V) of Gis compatible with a non-degenerate *-representation (II, V) of the algebraA(M). Then there exists a unitary representation (p, W) of H such that Tris equivalent to the induced representation Ind H p.

Actually, Mackey proved a more general theorem (instead of smooth ho-mogeneous manifolds he considered arbitrary homogeneous spaces for locallycompact groups).

Proof. According to Theorem 7, we can write it in the form (47). So. weonly need the following result.


Proposition 13. Any solution to the cocycle equation (48) has the form

(49) g) = C(m)-1 p(h9(rn g))C(m . g)

where C is a unitary operator-valued function on Al. p is a unitary rep-resentation of H in W. s : Al -, G is a section of the natural projectionp : G Al. and h., (m, g) is the solution to the master equation (37).

Proof. The proof of the statement for finite groups was given in Section 1.Here we shall follow the same scheme.

Technically. it is convenient to consider instead of the operator-valuedfunction A(x, g) on Al x G another operator-valued function B on Al x H x Algiven by

(50) B(.r. h, y) = A(.r. s(.r)-'hs(y))

where s : Al -- G is a section of the natural projection p : G -- AI : g t-+Hg.

The cocycle equation in terms of B acquires a more simple form(51)B(.r, hl. y)B(y, h2. ;,) = B(r. hmh2..) for almost all values of argumnents.

We can interpret B(.r, h. y) as a --transfer operator' acting from souse spaceWy attached to the point y E Al to the space W., attached to the pointr E Al and dependent on the auxiliary parameter h E H. Equation (51) isa compatibility condition: the transfer in two steps with parameters hl, h2is the same as the direct transfer with the parameter h1 hl.

Proposition 13 can be rewritten as follows:

Proposition 13'. The general solution to (51) has almost everywhere theform.

(52) B(x, h, y) = C(r)-1p(h)C(y)

where C is a unitary operator-valued function on Al and p is a unitaryrepresentation of H in W.

It is clear that Proposition 13' implies time Inducibility Criterion: afterthe substitution f (r) C(r)-' f (r) the factor A(r. g) in (47) takes thedesired forum A(r, g) = p(hx(.r. g)).


Proof of Proposition 13'. We can delete from Al finitely many subman-ifolds of smaller dimension, so that. the rest can be covered by one chart Uand choose s smooth everywhere on U.

The main difficulty is that the main relation (51) is valid only almosteverywhere. To get a true relation we have to integrate it against some testfunctions.

So, let a, $ be some W-valued functions on U, and let y, , 77 be somescalar functions on U, H, H, respectively. Also, we introduce the followingnotation:

NB(x, h, y)a(y)dp(y)(h)dH(h),fa,E(x) = J

x

001413 = j (B(x, h, y)a(y),x HxU

Applying (51) to a(y), taking the inner product with O(x), and using theproperty B(x, h, y)* = B(y, h-1, x), we get

fuor

(53)

(f«,&), fj,n(x))u.'Y(x)dx = J r..4*,j,s1'(x)dx

(fe,t(x), fe,a(x))(v = ca.F*n,r.

Note that the right-hand side in (53) does not depend on x E U. Ge-ometrically, it means that the systems of vectors {f g (x) } for different isare congruent. In other words, there exist a system of vectors {v0, } in Wand for any x E U a unitary operator C(x) in W such that

(54) ff,t(x) = C(x)vv,c.

Let us replace the cocycle B(x, h, y) by an equivalent cocycle

B(x, h, y) = C(x)B(x, h, y)C*(y)

and modify accordingly the definitions of fa,4 and Then equation(54) becomes

fa,f (x) = vn.

It follows that B(x, h, y) actually does not depend on x. Since B(x, h. y)*= B(y, h-1, x), it also does not depend on y. Finally, by (53) B(x. h, y) =p(h) for some unitary representation (p, W) of the subgroup H. 0


We shall use the Mackey Criterion to show that all unirreps of exponen-tial Lie groups and many unirreps of more general Lie groups are inducedfrom representations of smaller subgroups. To this end we need some addi-tional information about *-representations of homogeneous manifolds.

Let Al be a G-manifold. We say that a partition of M into G-orbits istame if there exists a countable family { Xi }iErq of G-invariant Borel subsetsin Al which separates the orbits. (This means that for any two differentorbits we can find an index i such that Xi contains one of these orbits but notthe other.) For brevity we shall simply say that Al is a tame G-manifold.

The following result plays a crucial role.

Theorem 9. Let G be a Lie group. let Al be a tame G-manifold, and let(II, 7I) be a *-representation of Al compatible with an irreducible unitaryrepresentation (rr, 11) of G. Then

a) The corresponding projector-valued measure p on M is concentratedon a single G-orbit f E M.

b) The unirrep it is induced from a unirrep p of the stabilizer H of apoint in f2.

Proof. a) For any G-invariant subset X E Al the orthoprojector P(X)commutes with er(g), g E G. Therefore the range of p(X) is a G-invariantclosed subspace in 71. But it is irreducible, hence the only such subspacesare {0} and W. We conclude that p(X) is either the identity or the zerooperator.

Now consider the family {Xi}iEN of G-invariant Borel subsets in M thatseparates the orbits. Replacing, if necessary. Xi by M\Xi we can assumethat p(Xi) = 0 for all i E N. Then u(Ui Xi) = 0. Let it = M\ vi X.We claim that fl is a single G-orbit. Indeed, suppose that 11 contains twodifferent orbits ill and 02. By the assumption, there is an Xt. that separatesthese orbits, say ill c Xk, 02 ft Xk. But this contradicts the definition ofQ. By the construction we have p(f2) = 1.

b) The set Q. being a G-orbit. has the structure of a smooth homo-geneous G-manifold i2 = G/H where H is a stabilizer of a point in Q.Moreover, since Al is a tame G-manifold. Q is a locally closed subset in M,hence a smooth submanifold.

According to the Inducibility Criterion the unirrep rr 14 induced fromsome representation p of H. This representation is necessarily irreducible,since if p = pl E) p2, then we would have rr = IndHpt ® Ind yP2, whichcontradicts the assumption that rr is irreducible. 0


We now use Theorem 9 and the Mackey Inducibility Criterion to provethe following statement.

Theorem 10. Any unitary irreducible representation of an exponential Liegroup G is induced by a 1-dimensional unitary representation of a closedsubgroup H C G.

Proof. We start by listing some properties of exponential Lie groups.It is known that the class of exponential Lie groups is strictly between

nilpotent and solvable simply connected Lie groups. A useful criterion isgiven by

Proposition 14. Let G be a simply connected Lie group with Lie (G) = g.Then G is exponential if for any X E g the operator ad X has no non-zeropure imaginary eigenvalues.

It is also known that any non-abelian exponential Lie group C containsa non-central connected abelian normal subgroup A. We use this subgroupto construct a G-manifold M and a *-representation II of M compatiblewith a given irreducible representation (7r, W) of G.

Since G is exponential, the subgroup A has the form A = exp a, a =Lie (A). Hence, A . R with coordinates x1, ..., xp. Let A be the Pontrya-gin dual to A, i.e. the group of all characters (1-dimensional unirreps) of A.These characters are labelled by points of the dual vector space (RP)' withcoordinates \I, ..., 1p and have the form

x.%(x) =e21ri(a.x) where (A, x) = JAkxk.

k

Now let (rr, N) be a unitary representation of C. Put Al = A and definethe *-representation (1I, N) of M by the formula

(55) H((6) = f (x)ir(x)dpxA

where

0(a)xa(x)dpAfi(x) := fA

is the Fourier transform of 0. That this is indeed a representation of A(AI)follows from the known property of the Fourier transform:

(01-02)42.Moreover, this representation is non-degenerate, since for an appropriaterb E A(A) the function 0 is almost concentrated in a given neighborhood of0 E A, so that H(O) is close to 1 in the strong operator topology.

394 Appendix V. Represenlatioiu Theory

Finally, the representation (55) is compatible with 7r if we define theG-action on A in a natural way:

(56) Xa(9-1x9). SEA, \ E A.

From Proposition 14 we obtain that the action of G on A is a linear actiongiven by matrices without pure imaginary eigenvalues. It follows that thisaction is tame.

Therefore, if 7r is irreducible, then, according to Theorem 9, 7r is inducedfrom some representation of a stabilizer of a point in .1 E A. Since A is anon-central normal subgroup, the action of G is non-trivial and most pointshave stabilizers that are proper subgroups of G. In this case we can useinduction on dim G to prove our statement.

As for exceptional points A E A that are fixed by the whole groupG, they correspond to degenerate representations of G, which are actuallyrepresentations of some quotient group. So, we can again use induction ondim G. (We omit the details since they are discussed in the main part ofthe book.) 0

References

[AW] S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices andquantum Schubert calculus, Math. Res. Lett. 5 (1998). 817 836.

f AS] A. Alexeev and S. Shatashvili, Path integrrtl quantization of the coadjoint orbit ofthe Virusoro group and 2d gravity, Nuclear Phys. B323 (1989). 719-733: Fromgeometric quantization to conformal field theory. Commun. Math. Plays. 128(1990), 197 212.

(Ail] V. I. Arnold. Snake calculus and the combinatorres of the Bernoulli. Eider andSpringer numbers of Coxeter groups, Uspekhi Alat. Nauk 47 (1993), no. 1. 3 15:English transl., Russian Math. Surveys 47 (1992), 1 51.

[Ar2] , On the teaching of mathematics, Uspekhi Mat. Nauk 53 (1998). no. 1.229 234: English transi., Russian Math. Surveys 53 (1998), 229-236.

[AG] V. 1. Arnold and A. B. Giveutal. Symplectic geometry. Encyclopaedia of Mathe-matical Sciences. vol. 4. Dynamical systems. IV. Springer-Verlag. Berlin Heidel-berg New York, 1990, pp. 1 136.

[A] M. Atiyah. Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14(1982). 1-15.

(AK] L. Auslander and B. Hostaut, Polarization and unitary representations of solvableLie groups, Invent. Math. 14 (1971), 255 354.

[Bar] I). Barbasch, The unitary dual for complex classical Lie groups, Invent. Math.96 (1989), 103 176.

[BV] D. Barbash and D. Vogan, Urn potent representations of complex senetstneplegroups, Ann. of Math. (2) 121 (1985). 41 110.

[Ba] V. Bargmann. Irreducible unitary representations of the Lorentz group. Ann. ofMath. (2) 48 (1947), 568 640.

(BPZ] A. Belavin, A. Polyakov, and A. Zamolodchikov. Infinite conformal symmetry intwo-dimensional quantum field theory. Nuclear Phys. B241 (198.1), 333-380.

[Bel] F. A. Berezin, Some remarks on the associative envelope of a Lie algebra. Funk-cional. Anal. i Prilozhen. 1 (1967), no. 2. 1 14: English transl. in FunctionalAnal. Appl. 1 (1967).

395

396 References

[Be2] , Introduction to algebra and analysis with anticommuting variables, Izd.Moskow. Univ.. Moscow, 1983; English transl., Introduction to supersymmetricanalysis, Reidel, Dordrecht.

[BK] F. A. Berezin and G. I. Kac, Lie groups with commuting and anticommutingparameters, Mat. Sb. 82 (1970), 343-359; English transl. in Math. USSR-Sb.

(BGV] N. Berlin, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Springer-Verlag, Berlin-Hiedelberg-New York, 1992.

[BCD] P. Bernat, N. Conze, M. Duflo, M. Levy-Nahas, M. Rais, P. Renouard, and M.Vergne, Representations des groupes de Lie resolubles, Dunod, Paris, 1972.

[BS] R. Berndt and R. Schmidt, Elements of the representation theory of the Jacobigroup. Birkhauser, Basel. 1998.

(BL] 1. N. Bernstein and D. A. Leites, Irreducible representations of finite dimensionalLie superalgebras of series W and S, C. R. Acad. Bulgare Sci. 32 (1979). 277-278.

[B] R. J. Blattner, On a theorem of G. W. Mackey, Bull. Amer. Math. Soc. 68 (1962),585: Positive definite measures, Proc. Amer. Math. Soc. 14 (1963), 423-428.

[Br] 1. D. Brown. Dual topology of a nilpotent Lie group, Ann. Sci. Ecole Norm. Sup.6 (1973), 407-411.

[Bor] A. Borel, Linear algebraic groups, Springer-Verlag, New York. 1991.

[BW] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups and repre-sentations of reductive groups, Princeton Univ. Press, Princeton. NJ, 1980.

(Bou] N. Bourbaki, Lie groups and Lie algebras, Chapters 1--3, Springer-Verlag, Berlin-Hiedelberg-New York, 1989, 1998: Chapters 4-6. Springer, Berlin-Hiedelberg-New York, 2002.

(Bu] I. K. Busyatskaya. Representations of exponential Lie groups, Funkcional. Anal.i Prilozhen. 7 (1973), no. 2. 151-152: English transl. in Functional Anal. Appl.7 (1972).

[Di 1] J. Dixmier, Sur les representation unitaires des groupes de Lie nilpotents, I, Amer.J. Math. 81 (1959), 160-170; II, Bull. Soc. Math. France 85 (1957), 325-388; III,Canal. J. Math. 10 (1958), 321-348: IV, Canad. J. Math. 11 (1959), 321-344:V, Bull. Soc. Math. France 87 (1959), 65-79; VI, Canad. J. Math. 12 (1960),324 352.

[Di2] , Enveloping algebras, North Holland, Amsterdam, 1977.

[Di3] , C. -algebras. North-Holland. Amsterdam, 1977.

(D«'] A. H. Dooley and N. J. Wildberger, Harmonic analysis and the global exponentialmap for compact Lie groups, Canal. Math. Bull. 27 (1993), 25-32; N. J. Wild-berger. On a relationship between adjoint orbits and conjugacy classes of a Liegroup, Canal. J. Math. 33 (1990), no. 3. 297- 304.

[Dr] V. G. Drinfeld, Quantum groups, Proc. Internat. Congress Math. (Berkeley, CA,1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798-820.

[DKN] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, Integrable systems. I, En-cyclopaedia of Mathematical Sciences. Dynamical systems. IV, vol. 4, Springer-Verlag, Berlin-Hiedelberg-New York. 1990, pp. 173-280.

[Dul] M. Duflo, Fundamental-serves representations of a semisimple Lie group, Funk-cional. Anal. i Prilozhen. 4 (1970). no. 1 38-42: English transl. in FunctionalAnal. Appl. 4 (1970).

References 397

[Du2] , Representations irreductibles des groupes semi-simples complexes. Anal-yse harinonique sur les groupes de Lie (Sera. Nancy-Strasbourg. 1973 75). Lec-ture Notes in Math., vol. 497, Springer-Verlag, Berlin-Hiedelberg- New York.1975. pp. 26-88.

[Du3] , Operateurs dtfjerentieLs bi-in variants sur an groupe de Lie, Ann. Sci. EcoleNorm. Sup. 4 (1977), 265-283.

[EFK] P. Etingof, 1. Frenkel, and A. Kirillov. Lectures on representation theory andKnizhnik-Zaniolodchikov equations. Amer. Math. Soc.. Providence. RI. 1998.

[EK] P. Etingof and A. Kirillov. Jr.. The unified representation-theoretic approach tospecial functions. Funkcional. Anal. i Prilozhen. 28 (1994). no. 1. 91 94; Englishtransi. in Functional Anal. Appl. 28 (1994).

[FF1] B. L. Feigin and D. B. Fuchs. On invariant differential operators on the tine,Funkcional. Anal. i Prilozhen. 13 (1979), no. 1, 91 92: English transi. in Func-tional Anal. Appl. 13 (1979); Homology of the Lie algebra of vector fields on theline, Fumkcional. Anal. i Prilozhen. 14 (1980). no. 3, 45-60; English transi. inFunctional Anal. Appl. 14 (1980).

[FF2] , Representations of the Vimsono algebra, Advanced studies in contempo-rary mathematics (A. M. Vershik and D. P. Zhelobenko. eds.), Representation ofLie groups and related topics, vol. 7. Gordon and Breach. London. 1990.

[Fe] J. M. G. Fell, Weak containment and induced representations of groups, Canad..1. Math. 14 (1962), 237-268.

[Fr] I. Frenkel, Orbital theory for affine Lie algebras, Invent. Math. (198-1), :301 352.

[FH] W. Fulton and J. Harris. Representation theory. A first course, Spri nger- Verlag,Berlin- Hiedelberg New York. 1991.

[Ge] I. M. Gelfand. Collected papers. vol. 11. Springer-Verlag. Berlin-Hiedelberg-NewYork, 1988.

[GK] I. M. Gelfand and A. A. Kirillov, Sun les corps lies aux algebms enveloppantesdes algebms de Lie, Publ. Math. IHES (1966). no. 31. 509-523.

(Gi] V. A. Ginzburg, The method of orbits in the representation theory of complex Liegroups, Funkcional. Anal. i Prilozhen. 15 (1981). no. 1. 23-37: English transl.in Functional Anal. Appl. 15 (1981): 9-modules, Spongers representations andbivanant Chern classes. Adv. Math. 61 (1986), 1 -18.

[GO] P. Goddard and D. Olive (eds.). Kac-Moody and Virasoro algebras. A reprintvolume for physicists, World Scientific. Singapore, 1988.

[GWW] C. Gordon, D. Webb, and S. Wolpert. One can't hear the shape of a drum. Bull.Amer. Math. Soc. 27 (1992), no. 1. 1:34-138.

[GW) C. Gordon and S. Wilson. The spectrum of the Laplacian on Riemannian Heisen-bery manifolds, Michigan Math. .1. 33 (1986), 253 271; Isospectral deformationsof compact solvnianifolds. J. Differential Geom. 19 (1984), 241 256.

(GG] M. Goto and F. Grosshans, Semisimple Lie algebras. Marcel Dekker. New YorkBasel. 1978.

[Gr] A. Grothendieck, Sur quelques points d'algebrr homologique, Tohoku Math. J. 9(1957), 119-221.

[Grz] P. Ya. Grozinan. Local invariant bilinear operators on tensor fields on the plane.Vcstnik Mask. Univ. Ser. I, Mat. Mekh. 6 (1980), 3-6: Classification of bilinearinvariant operators on tensor fields. Funkcional. Anal. i Prilozhen. 14. no. 2,58-59; English transl. in Functional Anal. Appl. 14 (1980).

398 References

[Gu) A. Guichardet, Theone de Mackey et methode des orbites selon M. Dufto. Expo-sition. Math. 3 (1985), 303-346.

[GLS] V. Guillemin, E. Lerman, and S. Sternberg, Symplectic fibrations and multiplicitydiagrams, Cambridge Univ. Press, Cambridge, 1996.

[GSI] V. Guillemin and S. Sternberg, Geometric asymptotics, Amer. Math. Soc.. Prov-idence, RI, 1977.

[GS2] , Convexity properties of the moment mapping, Invent. Math. 67 (1982),491-513.

(Hl Harish-Chandra, Differential operators on a sem:simple Lie algebra, Amer. J.Math. 79 (1957), 87-120; Collected Papers, vols. I-IV, Springer-Verlag, Heidel-berg, 1989.

(H2SVl M. Hazewinkel, W. Hesselink, D. Siersma, and F. D. Veldkamp, The ubiquity ofCoxeter-Dynk:n diagrams (an introduction to the A - D - E problem). NieuwArch. Wisk. (3) 25 (1977), no. 3, 257-307.

(Hec] G. J. Heckmann, Projections of orbits and asymptotic behavior of multiplicitiesfor compact connected Lie groups, Invent. Math. 67 (1983), 333-356.

[Hell S. Helgason, Differential geometry and symmetric spaces, Academic Press, 1978,1984; Corrected reprint of the 1984 original, Amer. Math. Soc., Providence, RI,2000.

(Ho] R. Howe, On Frobenius reciprocity for unipotent algebraic groups over Q, Amer.J. Math. 93 (1971), 163-172.

[Hu] J. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972, 1980.

(1] I. M. Isaacs, Characters of groups associated with finite algebras, J. Algebra 177(1995), 708-730; I. M. Isaacs and D. Karagueuzian, Conjugacy in groups of uppertriangular matrices, J. Algebra 202 (1998), 704-711.

(J] N. Jacobson, Lie algebras, Interscience, New York, 1962; Dover, 1979.

[Kal] V. G. Kac. Infinite dimensional Lie algebras, Cambridge Univ. Press, Cambridge,1990.

[Ka2] , Lie superalgebras, Adv. Math. 26 (1977), 8-96.[KPI V. Kac and D. H. Peterson, Defining relations of certain infinite-dimensional

groups, Asterisque, Numero Hors Serie (1985), 165-208; Regular functions on cer-tain infinite-dimensional groups, Arithmetic and Geometry, vol. II, Birkhiiuser,Boston, 1985. pp. 141-166.

[KV] M. Kashiwara and M. Vergne, The Campbell-Hausdorff formula and invarianthyperfunctions, Invent. Math. 47 (1978), 249-272.

[KO) B. Khesin and V. Yu. Ovsienko, Symplectic leaves of the Gelfand-Dikii bracketsand homotopy classes of nonftattening curves, Funkcional. Anal. i Prilozhen. 24(1990), no. 1, 38-47; English transl. in Functional Anal. Appl. 24 (1990).

[Kill A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspekhi Mat.Nauk 17 (1962), 57-110; English transl. in Russian Math. Surveys 17 (1962).

[Ki2] , Elements of the theory of representations. "Nauka", Moscow, 1972, 1978;English transl., Springer-Verlag, Berlin-Hiedelberg-New York, 1976.

References 399

[Ki3] , Constructions of unitary irreducible representations of Lie groups. VestnikMosk. Univ. Ser. I 25 (1970), 41-51. (Russian)

[Ki4] , Local Lie algebras, Uspekhi Mat. Nauk 31 (1976), 55-75; English transl.in Russian Math. Surveys 31 (1976).

[Ki5] , Introduction to the theory of representations and noncommutative har-monic analysis, Encyclopaedia of Mathematical Sciences, Noncoinmutative har-monic analysis. I, vol. 22, Springer-Verlag, Berlin-Hiedelberg-New York, 1994.

[Ki6] , Introduction to the orbit method. I, II, Contemp. Math, vol. 145. Amer.Math. Soc., Providence, RI, 1993.

[Ki7] , Geometric quantization, Encyclopaedia of Mathematical Sciences, Dy-namical systems. IV, vol. 4, Springer-Verlag, Berlin-Hiedelberg-New York, 1990,pp. 137-172.

[Ki8] , Invariant operators on geometric quantities. Itogi Nauki i Techniki. Sovre-mennye Problemy Matematiki. vol. 16, VINITI, Moscow, 1980, pp. 3-29; Englishtransl. in J. Soviet Math. 18 (1982), no. 1; On invariant differential operationson geometric quantities, Funkcional. Anal. i Prilozhen. 11 (1977). no. 1. 39-44:English transl. in Functional Anal. Appl.

[Ki9J , Merits and demerits of the orbit method, Bull. Amer. Math. Soc. 36(1999), 433-488.

[Kil0] , Variation on the triangular theme, E. B. Dynkin Seminar on Lie Groups,Amer. Math. Soc. Transl. Ser. 2, vol. 169, Amer. Math. Soc., Providence, RI,1995, pp. 43-73.

[Kill] , Characters of unitary representations of Lie groups: Reduction theorems,Funkcional. Anal. i Prilozhen. 3 (1969), no. 1, 36-47; English transi. in FunctionalAnal. Appl. 3 (1969).

[Ki121 , Infinite dimensional Lie groups: their orbits, nnvariants, and represen-tations; the geometry of moments, Lecture Notes in Math., vol. 970, Springer-Verlag, Berlin-Hiedelberg-New York, 1982, pp. 101-123.

[K113] , Introduction to family algebra, Moscow Math. J. 1 (2001), 49-63.

[KG] A. A. Kirillov and A. D. Gvishiani, Theorems and problems in functional anal-ysis, "Nauka", Moscow, 1979, 1988; English transl., Springer-Verlag, Berlin-Hiedelberg-New York, 1982.

[KK] A. A. Kirillov and M. L. Kontsevich, The growth of the Lie algebra generated bytwo generic vector fields on the line, Vestnik Moskov. Univ., Ser. I Math. Mech.1983, no. 4, 15-20. (Russian)

[KKMJ A. A. Kirillov, M. L. Kontsevich, and A. 1. Molev, Algebra of intermediategrowth, Preprint, vol. 39, Institute Appl. Math., Acad. Sci. SSSR, 1983.

[KM] A. A. Kirillov and A. I. Melnikov, On a remarkable sequence of polynomials,Algebre non commutative, groupes quantiques et invariants (Reims, 1995). Soc.Math. France, Paris, 1997, pp. 35-42.

[KN] A. A. Kirillov and Yu. A. Neretin, The manifold An of structure. constants ofn-dimensional Lie algebra, Problems in Modern Analysis, Moscov. Gos. Univ.,Moscow, 1984, pp. 42-56. (Russian)

[KYJ A. A. Kirillov and D. V. Yuriev, Representations of the Virasoro algebra by theorbit method, J. Geom. Phys. 5 (1988), 351-363.

400 References

[KI] W. H. Klink. Nilpotent groups and anharynonic oscillators, Noncompact Liegroups and some of their applications (E. A. Tanner and R. Wilson, eds.), Kluwer,Dordrecht, 1994. pp. 301- 313; P. E. T. Jorgensen and W. H. Klink, Spectral trans-form for the sub-Laplacian on the Heisenberg group, J. Analyse Math. 50 (1988),101-121.

[Kly] A. Klyachko, Stable bundles, representation theory and Hermitian operators. Se-lecta Math. (N.S.) 4 (1998), no. 3, 419-445; Vector bundles, linear represen-tations. and spectral problems, Proc. Internat. Congress Math. (Beijing, 2002).vol. 11, Higher Ed. Press, Beijing, 2003, pp. 599-613.

[Kn] A. Knapp, Representation theory of semisimple groups. An overview based onexamples. Princeton Univ. Press. Princeton, NJ, 1986.

[KT] A. Knutson and T. Tao. The honeycomb model of tensor products. I.Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055-1090.

[KMS] I. Kolar , P. Michor. and J. Slovak, Natural operations in differential geometry,Springer-Verlag, Berlin-Hiedelberg-New York, 1993.

[Kon] M. Kontsevich, Formality conjecture, Deformation theory and symplectic ge-ometry (Ascona. 1996), Math. Phys. Stud., vol. 20, Kluwer, Dordrecht, 1996,pp. 139-156; Deformation quantization of algebraic varieties, Lett. Math. Phys.56 (2001), no. 3. 271-294.

[Kol] B. Kostant, Quantization and unitary representations, Lecture Notes in Math.,vol. 170. Springer-Verlag, Berlin- Hiedelberg- New York, 1970, pp. 87-208; Gradedmanifolds. graded Lie theory and prequantization, Lecture Notes in Math., vol.570. Springer-Verlag, Berlin-Hiedelberg-New York, 1977. pp. 177-306.

(Ko2] _. The solution to a generalized Toda lattice and representation theory, Adv.Math. 34 (1979), 195-338.

[La] S. Lang. SL2(R), Addison-Wesley, Reading, MA, 1975.

(Le] D. A. Leites, Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk35 (1980), 3-57; English transl. in Russian Math. Surveys 35 (1980).

[LL] H. Leptin and J. Ludwig, Unitary representation theory of exponential Lie groups,Walter de Gruyter, Berlin, 1994.

[LV] G. Lion and M. Vergne, The Weil representation, Maslov index and Theta senes,Birkhiiuscr, Boston. 1980.

[Mac] G. W. Mackey. Impnmitivtty for representations locally compact groups, I, Proc.Nat. Acad. Sci. USA 35 (1949), 537 -545: Unitary representations of group ex-tensions. I, Acta Math. 99 (1952), 265-311.

[Nil S. Mac Lane, Categories for the working mathematician, Second Edition, Springer-Verlag, New York, 1998.

(Mull Yu. I. Manin, Gauge fields and complex geometry, Springer-Verlag, Berlin Hiedel-berg New York. 1988.

(NIn2] . New dimensions in geometry, Uspekhi Mat. Nauk 39 (1984), no. 6, 47-73; Lecture Notes in Math.. vol. 1111. Springer-Verlag, Berlin- Heidelberg- NewYork. 1985, pp. 59 101.

[MPR] W. G. McKay, .1. Patera, and D. W. Rand, Tables of representations of simpleLie algebras. vol. 1. Exceptional simple Lie algebras, CRM, Montreal, 1990.

References 401

[Mo] C. C. Moore. Decomposition of unitary representations defined by a discrete sub-group of nilpotent groups, Ann. of Math. (2) 82 (1965), 146-182.

[Nel] E. Nelson. Analytic vectors, Ann. of Math. (2) 70 (1959), 572-615; E. Nelsonand W. Stinespring, Representation of elliptic operators in an enveloping algebra,Amer. J. Math. 81 (1959), 547-560.

[Nerd] Yu. A. Neretin, Categories of symmetries and infinite-dimensional groups. OxfordUniv. Press, Oxford. 1996.

(Ner2] , An estimate for the number of parameters defining an n-dimensionalalgebra, Math. URSS-Izv. 30 (1988). no. 2. 283-29.1.

[Ni] A. Nijenhuis, Geometric aspects of formal differential operations on tensor fields,Proc. Internat. Congress Math. (Edinburgh. 1958). Cambridge Univ. Paws. NewYork, 1960, pp. 463-469.

[0k] A. Okounkov, Why would multiplicities be log-concave?, The orbit method ingeometry and physics (Marseille, 2000). Birkhuser. Boston. 2003. pp. 329-3.17.

(OPR] hl. A. Olshanetskii, A. Ni. Perelomov. A. G. Reyman, and Ni. A. Semmiiov-Tian-Shanskii, Integrable systems. II, Encyclopaedia of Mathematical Sciences, Dy-namical Systems. VII. vol. 16. Springer-Verlag. Berlin Hiedelberg-New York.1994, pp. 83 259.

[OV] A. Onishchik and E. Vinberg. A seminar on Lie groups and algebraic groups,Nauka, Moscow, 1988. 1995; English trattsl.. Springer-Verlag. Berlin-Hiedelberg-New York. 1990.

[O] B. Orsted, Induced representations and new proof of the imprimzhmty theorem,J. Funct. Anal. 31 (1979), 355 359.

[PT] R. Palais and C. L. Terng, Natural bundles have finite order, Topology 16 (1977).271-277.

[PWZ] M. Petkovsek, H. Wilf, and D. Zeilberger, A = B, A. K. Peters, Wellesley, MA.1996.

[PS] A. Presley and G. Segal. Loop groups. Clarendon Press, Oxford. 1986.

[Pr] R. Proctor, Odd symplectzc groups. Invent. Math. 22 (1988). 307-332.

[P1] L. Pukanszky. On the theory of exponential groups, Trans. Amer. Math. Soc. 126(1967). 487 507.

[P2] , Unitary representations of solvable Lie groups. Amt. Sci. Ecole Norm.Sup. IV (1971), 457-608.

[QFS] P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan.D. R. Morrison, and E. Witten (eds.), Quantum fields and strings: A course formathematicians, vols. L II, Amer. Math. Soc., Providence. RI, 1999.

[RS] A. G. Reiman and M. A. Semenov-Tian-Shanskii, Group-theoretic methods in thetheory of integrable systems, Encyclopaedia of Mathematical Sciences, vol. 1.

Dynamical systems. VII, Springer-Verlag. Berlin-Hiedelberg- New York. 1994.pp. 1 161- 969

[R] R. W. Richardson. Conjugacy classes in Lie algebra and algebraic groups. Anti.of Math. (2) 86 (1967), 1-15

[Ro] W. Rossmann, Kir zllov's character formula for reductive Lie groups. Invent.Math. 48 (1971). 207-220.

[Sch] Y. A. Schouten, Tensor analysis for physicists, Clarendon Press, Oxford, 1959.

402 References

[Sh] L M. Shchepochkina, On representations of solvable Lie groups, Funkcional. Anal.i Prilozhen. 11 (1977), no. 2 93_94: English transl. in Functional Anal. Appl. 11(1977); Orbit method for the restriction-induction problems for normal subgroupof a solvable Lie group, C. R. Acad. Bulgare Sci. 33 (1980), 1039-1042.

[So] Ya. Soibelman, Orbit method for the algebra of functions on quantum groups andcoherent states. L Int. Math. Res. Notices 6 (1993), 151-163.

[S] J.-M. Souriau, Systemes dynamiques, Dunod, Paris, 1970.

[T] C. L. Terng, Natural vector bundles and natural differential operators, Amer.Math. J. 100 (1978), 775-828.

[To] P. Torasso, Quantification geometnque, operateurs d'entrelacement et reprdssenta-tion,s unitaires de SL3(IR), Acta Math. 1550 (1983), 153-242.

[Veb] 0. Veblen, Differential invariants and geometry. Atti Congr. Int. Mat. Bologna,1928.

[Verl] M. Vergne. Construction de sous-algr bres subordonee a un clement du dual dunealgebre de Lie resoluble. C. R. Acad. Sci. Paris 270 (1970), 173-175; 704-707.

[Ver2] , On Rossman's character formula for discrete series, Invent. Math. 54(1979),11-14.

[Vol] D. Vogan, Representations of real reductive groups, Birkhauser, Boston, 1981.

[Vo2] , The orbit method and unitary representations of reductive Lie group, Al-gebraic and Analytic Methods in Representation Theory (Sonderborg, 1994),Perspect. in Math., vol. 17, Academic Press, San Diego, CA, 1997, pp. 243-339.

[Wa] N. Wallach, Real inductive groups. L Academic Press, 1988.

[We] A. Weinstein, Lectures on symplectic manifolds, CBMS Lecture Notes, vol. 29Amer. Math. Soc., Providence, RI, 1977.

[Wh] EL Whitney, The self-intersection of a smooth it-manifold in 2n-space, Ann. Math.43 (1944), 220-246.

[Will E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. (1982), 661-692.

[Wi2] , Coadjoint orbits of the Virasoro group, Commun. Math. Phys. 114 (1988),1 53

[Y] K. Yosida, Functional analysis, Springer-Verlag, Berlin-Hiedelberg-New York,1994.

[Za] D. Zagier, Values of zeta-functions and their applications, First European Con-gress Math. (Paris, 1992), vol. II, Birkhauser, Basel, 1994, pp. 497-512.

[Ze] A. Zelevinsky, Representation of finite classical groups, Lecture Notes in Math.,vol. 861, Springer-Verlag, Berlin-Hiedelberg-New York, 1981.

[Zhl] D. P. Zhelobenko, Compact Lie groups and their representations, "Nauka", Mos-cow, 1970; English transl., Amer. Math. Soc., Providence, RI, 1973.

[Zh2] , Representations of reductive Lie algebra, "Nauka", Moscow, 1994. (Rus-sian)

Index

*-representation. 374. 384o-string. 312e-neighborhood. 209

abelianization. 324addition rule, 168adjoint, 340, 34.1Adler-Kostant scheme. 200admissible, 125Ado Theorem, 272almost complex structure, 252almost product structure, 253amendment to Rule 6 1.34amendments to Rules 3 4. 5, 132angle, 338annihilation operators, 411anti-Hermitian, 340anticanonical class, 172antihomomorphism, 319atlas, 231

hallopen, 209

Banach space, 333-335basic group. 1. 3.Bessel inequality, 338Betti number

i-th. 324Betti numbers. 221Borel

function. 385subgroup. 1153subset. 254. 385

boundary operator. 217boundary point, 208bounded. 335

bundlecotangent. 2.11trivial, 239

Campbell-Hausdorff formula, M.canonical

anticotnmutation relations. 211commutation relations. 39coordinates, !13 142. 256decomposition. 311embedding, 215form, 163map, 223projection, 215relations, 258transformation, 2.56

CAR. 211Cartan matrix. 302Cartan subalgebra. 311Cartan subgroup. 153category, 212

dual. 213Cauchy sequence. 209CCR, 38

Fock realization, 48Weyl form, 42

CechCcch cohotnology. 223

chain complex, 217character, 369

distributional. 369generalized, 369infinitesimal, 368multiplicative. 73

chart. 231Chevalley basis, 1.32

403

404

classical compact groups. 13f1closecl.:1.14closed set, 208coauljoint orbit, 1cochain, cocycle, coboundary, 217cocycle equation, 22. 218. 372.388coefficients. :138cohomology group. 217conunutant, :352e o1111nut alive

diagram, 213Fourier analysis. 135

comm tator, 270compact real form. 314compatible. :371, 385complete. 209. 3.38completion. 210complcxificat ion.:315composition, 212connected space. 208continuous basis, iiicontinuous map. 208continuous sutn. 351contractible chart. 232contraction. 292countable additivity, :386covariance condition. 330covector field, 241Coxeter element, 307('oxeter number, Uicreation operators, 40cross-noun. 341curvature form, 13cyclic vector, lit. 346

de Rham cohontology. 221decomposable. 351deformation (of a Lie algebra). 292degenerate series, 1134derivation, 211

of degree k, 251diagonalizable. 352diamond Lie algebra, 43. 12fidiffeornorphic. 232diffeomorphisnl, 232differential

k-forth. 2411-form, 241

directintegral, 351

of algebras. 352limit, 222product, 215suns. 214

direct product. 320direct suns, 361

directed set. 222disjoint measures, 350distance. 208divergence, 250division algebra. 272domain of definition, 343

natural. 43dominant weight, 2.16dual cone. 300dual hypergroups, l.2dual root, 312dual root system, 3121)ynkin diagram, 302D,ynkin graph. 302

extended. LU

effective, 319Einstein's rule. 242elementary cell. 204elementary system, 197equivalent

atlases. 231deformations. 292

essential homomorphism, 2:36essentially self-adjoin, 344Euler-Bernoulli triangle, 191exceptional groups, 1311exponential. 142

coordinate system, 282exponential coordinates, 33exponential map, 283exponents, 307extension. 297, 344

central, 297trivial, 297

exterior point. 208

factor manifold, 229factorization problem. 200faithful, 319fiber. 238. 239fiber bundle, 238

base of, 238section of. 239

fibered product, 120over C. 320

field of geometric objects, 243filtration, 143first amendment to Rule 2. 124first integral. 267fixed point, 320flag, 192flag manifold, 192. 323

degenerate, 149Fourier analysis, 132Fourier coefficients, 364

Index

Index

Fourier transform, 113modified. 114non-commutative. 2fi

free boson operators, 44

Frobenius Integrability Criterion. 24Frobenius Theorem (on division algebras),

273Fubini-Study form, 241functional dimension, Z5. 82. 201functionals. 335functor, 213

left adjoint. 321reprsentable. 215

fundamental sequence. 209fundamental weight. 307

C-bundle, 326G-equivariant maps. 319C-manifold. 319. 384

tame, 392C-orbit. 320G-space. 319GK-dimension. Zfig-module, 291Gelfand-Girding space, 367generalized exponents. 307generating function, 257generator. 257generic, 1.55geometric quantization, 196graph. 211, 344Grassmannian, 143group

unitary dual. 73groups of Heisenberg type, 94

Hamiltonian field, 257Hausdorff. 231Heisenberg group, 33

generalized, 33Heisenberg Lie algebra, 32

generalized, 32Hernite functions. 51Hermite polynomials. 51Hermitian, 340, 368Hermitian form, 260highest weight. 1ST., 246Hilbert basis, 338Hilbert cross-norm, 341Hilbert dimension, 339Hilbert space, 337holomorphic induction, 123, 11.1homogeneous

G-bundle. 326C-set. 321k-cochain. 217coordinates. 230spectrum, 349

homology class, 217homotopy class, 216homotopy theory, 216homotopy type, 216

induction functor, 320inner automorphism. 276integrability condition, 253integrable subbundle, 23integration. 253interior point, 207intermediate. 202intertwiner, 291, 360intertwining number. IlL 3110intertwining operator. IM . 291intertwining operators. 360invariant

relative. Uinverse limit. 223irreducible, 362isomorphic. 211isotropic. 258

Jacobi identity, 270

k-boundaries, 217k-chains. 217k-cycles. 217k-irreducible. 363k-th homology group, 217Khhler form, 261Kidder polarization, 125Ki hler potential, 151. 261Killing form, 310. 316Kirillov-Kostant form, 1113Kostant partition function. 133Kronecker Theorem. 285

Lagrangian, 258Lagrangian fibration, 258lattice, 307

dual, 134leaves. 24left C,-set. 318left H-coset, 322left action. 319Leibnitz rule. 241length. 156, 305. 338Leray atlas, 232lie

over, 1.32under. 132

Lie algebra. 270homomorphism, 291abelian, 297classical simple, 315exceptional. 315

405

406

Lie algebra (continued)graded. 2i4nilpotent, 297semisimplc, 297solvable. 297

Lie bracket. 248Lie derivative. 248Lie group, 269

locally isomorphic, 269, 284matrix. 269

Lie superalgebra, 249. 289limit of the sequence, 208linear continuous functionals. 335linear geometric objects of first order, 245linear order. 300linear relations. 344local coordinates. 228, 231locally projective structure, 22logarithm map. 1.21loop group. 181Lorentz transformation, 181

manifoldk-smooth n-dimensional, 231abstract. 230algebraic, 236algebraic, degree of, 262analytic, 231complex. 235Kahler, 150 261non-orientable. 234oriented, 233Poisson, 263quotient, 229reduced. 266reduced symplectic, thseparable, 231smooth. 231smooth, structure of. 232symplectic. 256

mapm-smooth, 232

Master equation, 372matrix element, 358matrix notation, 269maximal torus. 323mean value property, 4fimeasure

equivalent, 349disjoint, 350projector-valued, 385quasi-invariant, .52spectral, 350

metaplectic group, 184metric space, 209metrizable, 209mirror. 299

Index

modifiedRule L 124Rule 14. 134Rule 2. 111. L31Rule 6. 114Rule L 11$Rule $, 134

moduleprojective, 243

moduli space, 235moment map, 17, 267morphism, 212multiplicity, 165. 346, 349multiplicity function, 350multiplicity of the weight,multiset, 10

n-th homotopy group, 216natural

bundle, 326domain of definition, 344Hilbert space, 379operation. 247

negatively related charts, 233

neighborhood. 208Nelson condition, 42Nijenhuis bracket. 251nilpotency class. 22nilpotent

Lie algebra. 71Lie group, 22

non-degenerate, 374-384non-homogeneous form for cochains, 218norm, 335normal, 340normal real form, 314normalized, 113normalizer. 322

object, 212couniversal, 214final, 214geometric, 243initial, 214representing, 215universal, 214

octonion, 196odd symplectic group, 28. 182open set, 208operator

adjoint, 335compact, 341dual, 335Hilbert-Schmidt, 341irreducible, 363smoothing, 366unbounded. 343

Index

operator calculus, 347opposite Borel subalgebras, 153orbit

integral, 1.51orientable, 233orientat ion. 233orthogonal. 338orthonormal, 338orthoprojector. 340. 386

P-admissible. 24p-adic integers. 210p-adic north. 210p-adic numbers, 210parabolic, 153parallelogram identity, 337partition of unity, 255Plancherel measure. 76, 370Planck constant. 39

normalized. 39Poincare polynomial, 151Poisson bracket. 258, 263polarization

algebraic admissible, 27complex, 25complex algebraic, 21real. 24real algebraic. 21

polyvector field, 242Pontryagin dual, 1i5positively related charts, 233principal series. 384proper, 297proper map, 236pseudo-Kihler form. 261Pukanszky condition, LUPythagorean Theorem. 3.38

quantization rules. 196quaternions. 273

real form. 316reduced decomposition, 306reflexive, 336regular, 155. 300relative cohomology, 220relative trace, 122representation

absolutely irreducible, 246

adjoint, 276

algebraically irreducible, 363induced, 371linear. 291metaplectic, 184topologically irreducible, 363unitary. 357unitary induced. 382weights of, 246

representing function, 330restriction functor, 320restriction-induction principle, fz4rigged coadjoint orbits, 123rigged momentum, 123right G -set, 319right action. 319rigid, 292root. 298

decomposable, 30(1negative, 314positive. 314simple, 300

root lattice, 307root system. 298root vector, 314

scalar product, 337

of quaternions. 273Schouten bracket, 249

Schur functor. 1.85second amendment to Rule 2. 125second index, 138section, 372

G-invariant, 327self-adjoint, 344semidirect product, 297separable, 339separate, 392sequentially continuous map. 208Shatten ideal. 185sign rule, 289signature. 294simple spectrum, 346. 349singular, 300singular k-cycle

real, integral, Uskew field, 272skew product, 239skew-gradient. 257small categories. 213smooth vector, 363spectral data, 350spectral function, 348spectrum

homogeneous, 346spinor, 211split real form, 314stabilizer, 322Stokes formula, 254strong integral, 336subcategories, 212subexponential. 202submanifold

Lagrangian, 24smooth, 228

subordinate, 26

407

408

subrepresentation, 361sum of rigged orbits, 132super Jacobi identity, 252super Leibnitz rule, 251supercommutative, 251superconunutator, 248. 252supergroup. 194support of a function, 236symmetric, 344symmetric algebra. 352symmetric coordinate system, 279symmetrization map. 35.287symplectic leaf. 264symplcctic reduction, 198, 266symplectic vector field, 256sym plect omorphism. 256

tame, 35.1, 392tangent space, 240tangent vector. 240tensor density of the first kind, 245tensor density of the second kind, 245tensor field of type (k, 1). 244topological space. 207topology

uniform, 358weak, strong, 358

trace, 342trace class, 342trace norm. 341transition functions, 231transitive action, 321trivial deformation, 292twisted product, 239type L 354

Index

unimodular coordinate system. 250unipotent, 112unipotent radical, l.5 iunirrep. I3unit morphism, 212unitary. 340unitary dual. Zgunitary representation

of a Lie group, 360universal enveloping algebra, 286universal property of coadjoint orbits, 1Z

vacuum vector, 45vector bundle, 239

natural, 243vector differential form. 251vector field. 240vector product of quaternions, 273velocity, 240Virasoro-Bott group. 22virtual coadjoint orbit. 122von Neumann algebra, 352

decomposable, 352

weak integral, 336weakly containment, 1116weight, 246weight lattice, 307weight vector, 245Weyl algebra. 4AWeyl chamber. 300

open, 300positive, 300

Weyl group, 154, 299. 324wild, 354Witt theorem, 28

Isaac Newton encrypted his discoveries in analysis in the form of an anagram,which deciphers to the sentence "it is worthwhile to solve differential equa-tions". Accordingly, one can express the main idea behind the Orbit Method bysaying "It is worthwhile to study coadjoint orbits".

The orbit method was introduced by the author, A.A. Kirillov, in the 1960s andremains a useful and powerful tool in areas such as Lie theory, group represen-tations, integrable systems, complex and symplectic geometry, and mathematicalphysics.This book describes the essence of the orbit method for non-experts andgives the first systematic, detailed, and self-contained exposition of the method.It starts with a convenient "User's Guide" and contains numerous examples. Itcan be used as a text for a graduate course, as well as a handbook for non-experts and a reference book for research mathematicians and mathematicalphysicists.

9

ISBN 0-8218-3530-0

78082111835302

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