geometric analysis on symmetric spaces, second edition · 2019-02-12 · from my previous books...

Geometric Analysis on Symmetric Spaces

Second Edition

http://dx.doi.org/10.1090/surv/039

Mathematical Surveys

and Monographs

Volume 39

^VDED

Geometric Analysis on Symmetric Spaces

Second Edition

Sigurdur Helgason

American Mathematical Society Providence, Rhode Island

Editorial Committee Jer ry L. Bona Michael G. Eas twood Ra lph L. Cohen Benjamin Sudakov

J. T . Stafford, Chair

2000 Mathematics Subject Classification. P r i m a r y 43A85, 53C35, 22E46, 22E30, 43A90, 44A12, 32M15; Secondary 53C65, 31A20, 43A35, 35L05, 14M17, 17B25, 22F30.

For addi t ional information and upda t e s on this book, visit w w w . a m s . o r g / b o o k p a g e s / s u r v - 3 9

Library of Congress Cataloging-in-Publicat ion Data Helgason, Sigurdur, 1927-

Geometric analysis on symmetric spaces / Sigurdur Helgason. — 2nd ed. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 39)

Includes bibliographical references and index. ISBN 978-0-8218-4530-1 (alk. paper) 1. Symmetric spaces. I. Title.

QA649.H43 2008 516.3/62—dc22 2008025621

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

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First Edition © 1994 by the American Mathematical Society. Reprinted with corrections 1997.

Second Edition © 2008 by the American Mathematical Society. All rights reserved.

Printed in the United States of America.

The American Mathematical Society retains all rights except those granted to the United States Government.

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10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08

To my Danish mathematical friends

past and present

Contents

Preface to Second Edition xiii Preface xv

CHAPTER I A Duality in Integral Geometry. 1

§1. Generalities 3 1. Notation and Preliminaries 3 2. Principal Problems 5

§2. The Radon Transform for Points and Hyperplanes. 8 1. The Principal Results 8 2. The Kernel of the Dual Transform 13 3. The Radon Transform and its Dual on the K-types 17 4. Inversion of the Dual Transform 20 5. The Range Characterization for Distributions and

Consequences 25 6. Some Facts about Topological Vector Spaces 29

§3. Homogeneous Spaces in Duality. 30 1. The Radon Transform for a Double Fibration 30 2. The Radon Transform for Grassmannians 39 3. Examples. 44

A. The d-Plane Transform 45 B . The Poisson Integral as a Radon Transform 49 C. Hyperbolic Spaces and Spheres 50

Exercises and Further Results. 52

Notes. 57

CHAPTER II A Duality for Symmetric Spaces. 59

§1. The Space of Horocycles. 60 1. Definition and Coset Representation 60 2. The Isotropy Actions for X and for E 62 3. Geodesies in the Horocycle Space 65

vn

CONTENTS

§2. Invariant Differential Operators. 70 1. The Isomorphisms 70 2. Radial Part Interpretation 74 3. Joint Eigenspaces and Eigenspace Representations 75 4. The Mean Value Operators 77

§3. The Radon Transform and its Dual. 82 1. Measure-theoretic Preliminaries 82 2. Integral Transforms and Differential Operators 84 3. The Inversion Formula and the Plancherel Formula

for the Radon Transform 89 4. The Poisson Transform 99 5. The Dual Transform and the Poisson Kernel 102

§4. Finite-dimensional Spherical and Conical Representations. 105

1. Conical Distributions. Elementary Properties 105 2. Conical Functions and Finite-Dimensional

Representations 113 3. The Finite-dimensional Spherical Representations 119 4. Conical Models and Spherical Models 120 5. Simultaneous Euclidean Imbeddings of X and of E.

Horocycles as Plane Sections. 122 6. Restricted Weights 127 7. The Component H(n) 131

§5. Conical Distributions. 134 1. The Construction of #'A s 134 2. The Reduction to RankOne 137 3. The Analytic Continuation of \£A,s 141 4. The Determination of the Conical Distributions 151

§6. Some Rank-One Results. 157 1. Component Computations 157 2. The Inversion of W 159 3. The Simplicity Criterion 165 4. The Algebra T)(K/M) 167 5. An Additional Conical Distribution for A = 0 169 6. Conical Distributions for the Exceptional A 171


Notes. 192

CONTENTS

CHAPTER III The Fourier Transform on a Symmetric Space. 197

§1. The Inversion and the Plancherel Formula 198 1. The Symmetry of the Spherical Function 198 2. The Plancherel Formula 202

§2. Generalized Spherical Functions (Eisenstein Integrals.) 227 1. Reduction to Zonal Spherical Functions 227 2. The Expansion of $>x,s 233 3. Simplicity (preliminary results) 241

§3. The Q^-matrices. 243 1. The if-finite functions in £X(E) 243 2. Connections with Harmonic Polynomials 244 3. A Product Formula for det QS(X) (preliminary version) 248

§4. The Simplicity Criterion. 255

§5. The Paley-Wiener Theorem for the Fourier Transform on X = G/K. 260

1. Estimates of the T-coemcients 261 2. Some Identities for Cs 264 3. The Fourier Transform and the Radon Transform.

if-types 266 4. Completion of the Proof of the Paley-Wiener Theorem.

The Range £'(X)~ 268 5. A Topological Paley-Wiener Theorem for the if-types 273 6. The Inversion Formula, the Plancherel Formula

and the Range Theorem for the 5-spherical Transform 279 §6. Eigenfunctions and Eigenspace Representations. 282

1. The if-finite Joint Eigenfunctions of D(X) 282 2. The Irreducibility Criterion for the Eigenspace

Representations on G/K 284

§7. Tangent Space Analysis. 285 1. Discussion 285 2. The J-polynomials 286 3. Generalized Bessel Functions and Zonal Spherical

Functions 292 4. The Fourier Transform of if-finite Functions 293 5. The Range V(p)~ inside W(a* x K/M) 298

§8. Eigenfunctions and Eigenspace Representations on XQ. 300 1. Simplicity 300 2. The if-finite Joint Eigenfunctions of D(G 0 / i f ) 303 3. The Irreducibility Criterion for the Eigenspace

Representations of G0/K 309

CONTENTS

The Compact Case. 310 1. Motivation 310 2. Compact Symmetric Spaces 313 3. Analogies 316 4. The Product Decomposition 316

Elements of T>(G/K) as Fractions. 322

The Rank-One Case. 327 1. An Explicit Formula for the Eisenstein Integral 327 2. Harmonic Analysis of if-finite Functions 332

The Spherical Transform Revisited. 335 1. Positive Definite Functions 335 2. The Spherical Transform for Gelfand Pairs 339 3. The Case of a Symmetric Space G/K 346


Notes. 358

CHAPTER IV The Radon Transform on X and on X 0 .

Range Questions. 363

The Support Theorem. 363

The Ranges £>(Xf, £'(Xf and £ (~ ) v . 365

The Range and Kernel Determined in terms of if-types. 369 1. The General Case 369 2. Examples: H2 and R2 379

The Radon Transform and its Dual for K-invariants. 381 The Radon Transform on X0. 387

1. Preliminaries 387 2. The Support Theorem 392 3. The Range and Kernel for the if-types 394 4. The Ranges £f{X0) and £(E0)v 395


Notes. 398

CHAPTER V Differential Equations on Symmetric Spaces. 401

Solvability. 401 1. Fundamental Solution of D 402 2. Solvability in £{X) 403

CONTENTS

3. Solvability in S'(X) 406 4. Explicit solution by Radon transforms 407

§2. Mean Value Theorems. 413 1. The Mean Value Operators 413 2. Approximations by Analytic Functions 416 3. Asgeirsson's Mean Value Theorem Extended to

Homogeneous Spaces 418 §3. Harmonic Functions on Symmetric Spaces. 421

1. Generalities 421 2. Bounded Harmonic Functions 421 3. The Poisson Integral Formula for X 425 4. The Fatou Theorem 430 5. The Furstenberg Compactification 439

§4. Harmonic Functions on Bounded Symmetric Domains. 442 1. The Bounded Realization of a Hermitian Symmetric Space 442 2. The Geodesies in a Bounded Symmetric Domain 444 3. The Restricted Root Systems for Bounded Symmetric

Domains 445 4. The Action of G0 on D and the Polydisk in D 451 5. The Shilov Boundary of a Bounded Symmetric Domain 453 6. The Dirichlet Problem for the Shilov Boundary 460 7. The Hua Equations 461 8. Integral Geometry Interpretation 466

§5. The Wave Equation on Symmetric Spaces. 468 1. Introduction. Huygens' Principle 468 2. Huygens' Principle for Compact Groups and Symmetric

Spaces X = G/K (G complex) 471 3. Huygens' Principle and Cartan Subgroups 477 4. Orbital Integrals and Huygens' Principle 482 5. Energy Equipartition 486 6. The Flat Case Revisited 490 7. The Multitemporal Wave Equation on X = G/K 493 8. The Multitemporal Cauchy Problem 497 9. Incoming Waves and Supports 506 10. Energy and Spectral Representation 511 11. The Analog of the Priedlander Limit Theorem 524

§6. Eigenfunctions and Hyper functions. 527 1. Arbitrary Eigenfunctions 527 2. Exponentially Bounded Eigenfunctions 531


Notes. 537

CONTENTS

CHAPTER VI Eigenspace Representations. 539

§1. Generalities. 539 1. A Motivating Example 539 2. Eigenspace Representations on Function- and

Distribution-Spaces 540 3. Eigenspace Representations for Vector Bundles 541

§2. Irreducibility Criteria for a Symmetric Space. 543 1. The Compact Case 543 2. The Euclidean Type 545 3. The Noncompact Type 546

§3. Eigenspace Representations for the Horocycle Space G/MN. 547

1. The Principal Series 547 2. The Spherical Principal Series. Irreducibility 548 3. Conical Distributions and the Construction of the

Intertwining Operators 554 4. Convolution on G/MN 557

§4. Eigenspace Representations for the Complex Space G/N. 562

1. The Algebra T>(G/N) 562 2. The Principal Series 564 3. The Finite-Dimensional Holomorphic

Representations. 565 §5. Two Models of the Spherical Representations 567


Notes. 571

SOLUTIONS TO EXERCISES 573

BIBLIOGRAPHY 599

SYMBOLS FREQUENTLY USED 627

INDEX 633

PREFACE TO SECOND EDITION

This book has been unavailable for some time and I am happy to follow the publisher's suggestion for a new edition.

While a related forthcoming book, "Integral Geometry and Radon Transforms" (here denoted [IGR]) deals with several examples of homogeneous spaces in duality with corresponding Radon transforms, the present work follows the direction of the first edition and concentrates on analysis on Riemannian symmetric spaces X = G/K. We develop further the theory of the Fourier transform and horocycle transform on X, also taking into account tools developed by Eguchi for the Schwartz space S(X). These transforms provide the principal methods for analysis on X, existence and uniqueness theorems for invariant differential equations on X, explicit solution formulas, as well as geometric properties of the solutions, for example the harmonic functions and the wave equation on X. On the space X there is a canonical hyperbolic system on X, introduced by Semenov-Tian-Shansky, which is multitemporal in the sense that the time variable has dimension equal to the rank of X. The solution has remarkable analogies to the classical wave equation on R n , summarized in a table in Chapter V, §5.

My intention has been to make the exposition easily accessible to readers with some modest background in Lie group theory which by now is rather widely known. To facilitate self-study and to indicate further developments each chapter concludes with a section "Exercises and Further Results". Solutions and references are collected at the end of the book. The harder problems are starred. Occasionally results and proofs rely on material from my previous books "Differential Geometry, Lie Groups and Symmetric Spaces" abbreviated [DS] and "Groups and Geometric Analysis", abbreviated [GGA].

Once again I wish to express my gratitude to my friends and collaborators, Adam Koranyi, Gestur Olafsson, Frangois Rouviere and Henrik Schlichtkrull and especially to my long-term colleague David Vogan for significant help at specified spots in the text. Finally, I thank Brett Coon-ley and Jan Wetzel for their invaluable help in the production and the editor Dr. Edward Dunne for his interest in the work and his patient and accommodating cooperation.

I would also like to express my thanks for the following permissions of partial quotations:

(i) To Academic Press concerning my papers [1970a], [1976], [1980a], [1992b], and [1992d] quoted at the end of the Preface to the first edition. (ii) To Elsevier concerning my paper [2005]. (iii) To John Wiley and Sons concerning my paper [1998a] and my paper with Schlichtkrull [1999].

xiii

P R E F A C E

Among Riemannian manifolds the symmetric spaces in the sense of E. Cart an form an abundant supply of elegant examples whose structure is particularly enhanced by the rich theory of semisimple Lie groups. The simplest examples, the classical 2-sphere S2 and the hyperbolic plane if2, play familiar roles in many fields in mathematics.

On these spaces, global analysis, particularly integration theory and partial differential operators, arises in a canonical fashion by the requirement of geometric invariance. On Rn these two subjects are related by the Fourier transform. Also harmonic analysis on compact symmetric spaces is well developed through the Peter-Weyl theory for compact groups and Car-tan's refinement thereof. For the noncompact symmetric spaces, however, we are presented with a multitude of new and natural problems.

The present monograph is devoted to geometric analysis on noncompact Riemannian symmetric spaces X. (The Euclidean case and the compact case are also briefly investigated in Chapter III, §§7-9, and Chapter IV, §5, but from an unconventional point of view). A central object of study is the algebra D(X) of invariant differential operators on the space. A simultaneous diagonalization of these operators is provided by a certain Fourier transform / —• / ~ on X which is the subject of Chapter III. Just as is the case with Rn the symmetric space X turns out to be self-dual under the mentioned Fourier transform; thus range questions like the intrinsic characterization of (C£°(X))~ in analogy with the classical Paley-Wiener theorem in Rn become natural and their answers useful.

Chapters II and IV are devoted to the theory of the Radon transform on X, particularly inversion formulas and range questions. The space E of horocycles in X offers many analogies to the space X itself and this gives rise to the study of conical functions and conical distributions on El which are the analogs of the spherical functions on X. They have interesting connections with the representation theory of the isometry group G of X, discussed in Chapter II, §4, and in Chapter VI, §3, where the conical distributions furnish intertwining operators for the spherical principal series. In Corollary 3.9, Ch. VI, these intertwining operators are explicitly related to the above-mentioned Fourier transform on X.

While the Fourier transform theory in Chapter III gives rise to an explicit simultaneous diagonalization of the algebra D(X), the Radon transform theory in Chapter II is considered within the framework of a general integral transform theory for double fibrations in the sense of Chapter I, §3. This viewpoint is extremely general: two dual integral transforms arise whenever we are given two subgroups of a given group G. In the introduction to Chapter I we stress this point by indicating five such examples

xv

X V I PREFACE

arising in this fashion from the single group G = SU(1,1) of the conformal maps of the unit disk, namely the X-ray transform, the horocycle transform, the Poisson integral, the Pompeiu problem, theta series, and cusp forms. When range results are considered, this viewpoint of the Poisson integral as a Radon transform offers a very interesting analogy with the X-ray transform in JR3 (Chapter I, §3, No. 5).

With the tools developed in Chapters I-IV we study in Chapter V some natural problems for the invariant differential operators on X, solvability questions, the structure of the joint eigenfunctions, with emphasis on the harmonic functions, as well as the solutions to the invariant wave equation on X. In Chapter VI we consider in some detail the representations of G which naturally arise from the joint eigenspaces of the operators in the algebra D(X) and the algebra D(E).

The length of this book is a result of my wish to make the exposition easily accessible to readers with some modest background in semisimple Lie group theory. In particular, familiarity with representation theory is not needed. To facilitate self-study and to indicate further developments each chapter is concluded with a section "Exercises and Further Results". Solutions and references are given towards the end of the book. The harder problems are starred. Occasionally, results and proofs rely on material from my earlier books, "Differential Geometry, Lie groups, and Symmetric Spaces" and "Groups and Geometric Analysis". In the text these books are denoted by [DS] and [GGA].

Some of the material in this book has been the subject of courses at MIT over a number of years and feedback from participants has been most beneficial. I am particularly indebted to Men-chang Hu, who in his MIT thesis from 1973 determined the conical distributions for X of rank one. His work is outlined in Chapter II, §6, No. 5-6, following his thesis and in greater detail than in his article Hu [1975]. I am also deeply grateful to Adam Koranyi for his advice and generous help with the material in Chapter V, §§3-4, as explained in the notes to that chapter. Similarly, I am grateful to Henrik Schlichtkrull for beneficial discussions and for his suggestions of Proposition 8.6 in Chapter III and Corollary 5.11 in Chapter V, indicated in the text. I have also profited in various ways from expert suggestions from my colleague David Vogan. I am grateful to the National Science Foundation for support during the writing of this book.

Many people have read at least parts of the manuscript and have furnished me with helpful comments and corrections; of these I mention Fulton Gonzalez, Jeremy Orloff, An Yang, Werner Hoffman, Andreas Juhl, Frangois Rouviere, Sonke Seifert, and particularly Frank Richter. I thank them all. Finally, I thank Judy Romvos for her expert and conscientious T^jX-setting of the manuscript.

A good deal of the material in this monograph has been treated in earlier papers of mine. While subsequent consolidation has usually led to a rewriting of the proofs, texts of theorems as well as occasional proofs

PREFACE xvii

have been preserved with minimal change. I thank Academic Press for permission to quote from the following journal publications of mine, listed in the bibliography: [1970a], [1976], [1980a], [1992b], [1992d], as well as the book [1962a].

SOLUTIONS TO EXERCISES

CHAPTER I

A. Radon Transform on Rn .

A. 1. By §2 (27) each Ek <g>p£ belongs to Kf. Conversely let ip eSf. Let G = M(n) with Haar measure dg, let £0 be the hyperplane xn — 0 in R n , and let $(g) = i/>(g • &>) for g € G. For F G D(G), £ = ft • &> put

<M0 = J FigWig^Qdg = j F(g)^{g-lh • &)dff G G

which lies in £ (P n ) nJ\f = Af. Let F run through a sequence (F») with ^i > 0, / ^ = 1, supp(Fi) —• e. Then ipFi —• \j> in C(P n ) so statement follows from Theorem 2.5.

A. 2. For the Fourier transform <p(s) we have

R

By the definition

(Yk®Vy(x) = ± J<p((x,V))Yk(r,)dr,

R Vs^-1 /

On the other hand, we have the classical formula (see e.g. [GGA], p. 25)

(A) j eiX^«)Yk(uj)du = Cn9kYk(ri)-A ( n / 2 ) - l '

g n - 1

where cn^k = (2ir)n/2ik and Jr is the Bessel function. Here we replace k by 0 and n by n + 2k. Then we obtain

k

J eiX^Yk(V)dr, = ( ^ j n H y eiA^dC S n - 1 S n+2fc- l

Finally, we put x = ru and get

( y f c ® V ' ) V M = ^ 1 ( r / 2 7 r ) f c n H | ±d<; J $(s)(is)keisr<1ds gn + 2/e-l R

573

574 SOLUTIONS TO EXERCISES

as desired.

A. 3. We know from §2, No. 3 that Pk(2,cos0) = Hk(sm6,cos6) where Hk(xi,x2) is the unique harmonic polynomial on R2 which is homogeneous of degree fc, is invariant under (#i, x2 —> (—xi,x2) and satisfies Hk(0,1) = 1. Since {xi+ix2)k and (x\ —ix2)k span the space of homogeneous kth degree harmonic polynomials we have

Hk(xux2) = Re((x2 + ixi)k),

which gives the desired result.

A. 4. If dfc is the normalized Haar measure on K we have

(/)*(*) = j ^Jtf(x + k>y)dm(y)>jdk K Co

= j dm(y) J f(x + k- y)dk = J (M^f)(x)dm(y), Co K Co

where (Mrf)(z) is the average of / over Sr(z). Hence

0 0

(fy(x) = ndJ(Mrf)(x)ra-'dr, 0

so, using polar coordiantes around x

{f)y{x) = ^j\x-y\d-nf{y)dy

and now the inversion formula follows from the standard inversion of the Riesz potential, ([GGA], Ch. I, Prop. 2.38).

The statement (i) amounts to that if V is a fc-dimensional vector sub-space of C n then V — k -£0 for some k G U(n). This is obvious by choosing a basis of V orthonormal with respect to the standard Hermitian inner product ( , ) on C n .

Statement (ii) amounts to proving that if W is a Lagrangian vector subspace of R 2 n then W = k • £0 for some k G U(n). It is well known that dim W = n. Writing z = x + iy, w = u + iv with x, y, u, v G R n we have

(z,w) — x - u-\- y - v — i(x • v — y • u) = (x, y) • 0 , v) - i{(x, y), (u, v)}

so the action of U(n) on C n ~ R 2 n preserves both the standard inner product on R 2 n and the skew symmetric form { , }. If e i , . . . , e n is an orthonormal basis of W over R then the formula above shows, W being

Chapter I 575

isotropic, that (e^e^) = 5{j so the e* form a complex orthonormal basis of C n . Viewing the standard orthonormal basis of R n x 0 as a complex orthonormal basis of C n we see that W = k • £0 for a suitable k G U(n).

A. 5. Prom [GGA], Ch. I, Theorem 2.20 we have supp(T) c BA(0). For e > 0 let / G V{X) have supp(/) C BA_e(0). Then supp(/) C /?A_e where

[3R = iZ: d(o,t)<R}-

Also by the inversion formula cf = (A/) v , since A is now a differential operator,

T(cf) = T ( ( A / ) v ) = f ( A / ) - 0

so supp(T) H £ A - e (0) = 0-

A. 6. We have with a constant c

{$*f)(x)=c ( / <p(w, (w,a ; ) - (w,y))dw )f(y)dy

/ ( / ^ ( M ™ , x ) - p ) / ( w , p ) d p j d w i R

/ ((p*f)(w, (w,x))dw = (ip*f)v(x)

S " - 1 R

= c

g n - 1

(Natterer, [1986], p. 14).

B. Homogeneous Spaces. Grassmann Manifolds.

B. 1. For (ii) we may take X2 = x0 and write x\ = g\K, £ = ^H. Then

x0, £ incident <=$• ^h — k ( some h € H, k £ K) xi,£ incident «<=> g\k\ — yhi ( some hi G H, k\ £ K).

Thus if XQI X\ a r e incident to £ we have #i = kh 1h\k11. Conversely, if

g\ — k'h'k" we put 7 — k!h! and then x0, #i are incident to £ = 7 # . For (hi) suppose first X i J D UK — K U H. Let x\ 7 #2 in X. Suppose

£1 7 £2 hi H both incident to x\ and #2- Let a = (^if, £j = 7?#. Since x is incident to £j there exist kij G X, /iij G # such that

#;fc;j = 7?^j * = 1» 2; ,7 = 1,2.

By eliminating gi and 7^ we obtain

&22 ^21^21 ^11 = ^22 ^12^12 &11 •


This being in KH n HK it lies in K U H. If the left hand side is in K, then h^hn G K so we get

g2K = jih2iK = ~iihiiK = gxK

which is a contradiction. Similarly, if the mentioned left hand side is in H we have k2~2

1k2i G H which gives the contradiction 72 # = 71 i?. Conversely, suppose KHnHK ^ KUH. Then there exist ^1,^2,^1? &2

such that /ii/ci = A/2 2 and /iifci ^ KUH. Put xi = /ii-K", £2 — &2#- Then %o 7 #i5 £0 7 £2? yet both £0 and £2 are incident to both x0 and #i.

B . 2 - 3 . For the first statement see [GGA], Cor. 4.10, Ch. II. For the other suppose the generators Di = Dpi were not algebraically independent. Let

P = Ea n i . . . ntxlY ...xntl

be a nonzero polynomial such that P(D±,..., Di) = 0. Let di = degree (Pi) and AT = max(E^n^), the maximum taken over the set of ^-tuples ( n i , . . . , ni) for which ani . . . n^ ^ 0. We write the polynomial

S = E a n i . . . n , P 1n i . . . P ; i <

as the sum S = Q + R, where

Q= Yl ani...niP?1...P?< T,diTii=N

and degree (R) < N. Also Q ^ 0 by assumption. Consider the operator

Vani...niD?...D?-Ds

whose order is < N ([GGA], p. 287). This operator equals 0 - DQ — DR which by the definition in Exercise B2 has order N. This gives the desired contradiction.

B . 10. Method of Helgason [1957] or [GGA], Ch. V, Lemma 2.6. First show that it suffices to compute

j \vl3\2\vM\2dV U(n)

and that this integral is given by

(i) (n(n + 1)) _ 1 if (i,j) and (k,£) are either in the same row or the same column (not both).

(ii) 2 ( n ( n + l ) ) - 1 i f ( i , j ) = ( M )

Chapter II 577

(hi) (n2 — 1) x if (i,j) and (fc, £) are neither in the same row nor the same column. See also Faraut-Koranyi [1993], p. 237.

B . 11. The proof is obtained by expanding in a Fourier series on T 2 (also observed by Gindikin).

B . 12. If U/K has rank one see [GGA], Ch. I, Cor. 4.19. If U/K has higher rank the result is immediate from Exercise 11 as pointed out by Grinberg.

B . 13. d is a if-orbit containing (1,0) so equals B. Also H • o is two-dimensional so equals D.

CHAPTER II

A. The Spaces X = G/K and S = G/MN.

A. 1. If kN C NK then k • £0 C £0 so k G M by text. If nK c KN then n • o belongs to each horocycle through o. If n ^ e, n • o = ka • o (a ^ e). But fc • £0 does not contain ka - o = n - o.

Let g = ! + a + n be the usual Iwasawa decomposition of g = sC(2, R) (as before Lemma 4.9). Let g = m + n + q where q is MA^-invariant. Let H e a have the component Hi in q. Then [#i,n] C n is a contradiction.

A. 2. Use (4) §3.

A. 3. Recall proof of Lemma 4.9 (ii).

A. 4. Consider V = C n + 1 with the Hermitian form

(y, w) = y0w0 - yiwi ynwn

and put V+ = {y G C n + 1 : (y, y) > 0}. The Hermitian hyperbolic space can be taken as F + / C * . With non-homogeneous coordinates zi = yi/y0, 1/+/C* is identified with the ball

B+ = {zeCn: |Zl |2 + . . . + | ^ n | 2 < 1 }

and the unitary action U( l ,n ) = U(V) on V induces the action of the projective group PU(V) on B+ (SU(l ,n) mod its center, cf. [DS], X, Exercise Dl). Let n : V —> V/C* be the natural map. Choose t G dB+ and choose y* ^ O o n f . The Iwasawa subgroup N (the unipotent radical of the isotropy group PXJ(V)i*) viewed as a subgroup of SU(1, n) fixes y* and hence also the function

\{y,y)\2


Thus the equation dy* — c, that is,

\(y*,y)\2 = \(y,y)\c2

is a horocycle. In non-homogeneous coordinates this is

r2

2\ c |1 - z\zx z*nzn\ = (1 - \zx\ \zn\ ) — Wo

12

which is an ellipsoid in the Euclidean metric. A PU(V)-invariant metric on £ + is given by (cf. Mostow [1973], p. 136)

\{w,w)2{y,y)2 J >(w,w)2(y,y)l

so the sphere Sr(ir(w)) is

r — |(w, w/|2cn r. I(J / ,J />I

Let it? —> y*,r —-> oo with (w,w)*ch. r = c (where (y*,y*) = 0). Then the sphere converges to the horocycle above.

Another verification in terms of the notation of [DS], IX, (§3 and Exercise B4). The horocycle N • o is given by

2it-\z\2 -2z K2(l - it)+ \z\2' 2(1-it)

and therefore equals the ellipsoid

2K + i|2 + |W2|2 = i.

Similarly the horocycle TV • o equals

(*) 2 K - | | 2

Let ^chr 0 shr^

o 1 0 ^shr 0 chry

Then the sphere Sr(o) equals Kar • o which is given by

\Zl\* + \Zi\2=\tfr. The image ar • 5 r(0) is by [DS], IX, Exercise B4 given by

Chapter II 579

so the equation for ar • Sr(o) is

(1 + th2r)|it;i|2 - thr(>i + w{) + \w2\2 = 0.

Thus as r —~> oo the sphere a r5 r(0) converges to the horocycle (*).

A. 5. First reduce the problem to the case X ~ H 2 as follows. Let Xa be a root vector in the Lie algebra of N and let Ga denote the analytic subgroup of G with Lie algebra KXa + H0Xa -f R[X a , 0-Xa], Then Ga - o is a totally geodesic submanifold of X isometric to H 2 and the horocycle exptXa • o in Ga • o equals (Ga - o)n(N • o). This reduces the problem to H 2 with metric

y2

where the geodesies are the semicircles

7u,r • x — u-\-rcos6, y ~ r s i n # , 0 < 0 < TT,

We have 7T

f(lu,r)= / f (u + r cos Q,r sin 0) (sin 0)~1d6 o

so taking £ as the line y = 1 our assumption amounts to

/ H*£dw = 0, r < l ,

2/

where c/u> is the Euclidean arc element. The rapid decrease of / implies that / (# , 2/)/y extends to a smooth function F on R2 by F(x, y) = f(x, \y\)/\y\. Then

(*) / F(s)dw(s) = 0 x e R5 r < 1. Sr(a;)

This implies for the corresponding disk Br(x)

I F(u, v)dudv = 0,

whence / (d1F)(x + u,v)dudv = Q

Br(o)

with #i ~ d/du. Using the divergence theorem on the vector field F(x + u, v)d/du we get

F(x + u, i>)u dw(u, v) = 0. /

Sr(o)


Combining this with (*) we deduce

(**) / F(s)sidw(s) s = (s1,s2). Sr(x)

Iterating the implication (*) = > (**) we obtain

f F(s)P{Sl)dw(s) = 0 Sr(x)

where P is any polynomial so we get the desired conclusion / = 0 on the strip 0 < y < 1.

A. 7. Because of Theorem 2.9 it suffices to prove that the convolution algebra C^(MN) of M-bi-invariant functions in CC(MN) is commutative. This result from Koranyi [1980] follows (for m2Q ^ 1) from Kostant's theorem (Exercise D3 below) which implies that for each n £ N there exists an me M such that mum,-1 = n" 1 . Thus f(n) = fin'1) for / G C\(MN) which implies the commutativity. For the case rri2a — 1 s e e [GGA], Ch. IV, Exercise BIO.

A. 8. With the customary notation we have (as m*k(n)M = fc(n(ra*n))M),

J F{k{n)M)e-2p{H^))dn= f F(kM)dkM

N K/M

J F(k(n{m*n))M)e-2piH{fl))dfi, N

and since by §6, H(n) = H{n{m*n)) + B(m*n), this integral equals

/ F ( f c ( J n ) M ) e - 2 ^ H ( J n ) ) e - 2 ^ B ( m * n » 3 ^ d ( J n ) , J d(Jn) N

proving the result. A. 9. The vector v is in the center of t0 so is fixed under Adc0(K0); also e is in the highest root space so, Adc0 being spherical, e is M0-fixed. By computation

( sht i 0 —cht £\ 0 0 0

cht i 0 —sht ij

(i 0 - ^ Ad(at)e = e2t 0 0 0

\i 0 -ij

Chapter II 581

Put

-\ o o v0 = I 0 %

0 0 0 vi v2

Then the curve

t —• Ad(a t)v = v0 + §ch2t v\ + | s h 2t v2

lies in the intersection of X0 with the plane (si,S2) —> vo + si^i + £2^2-

A. 10. Consider a0 as in [DS], Cor. 7.6, Ch. VIII. The geodesic Ad(exp £(Xy+ X-^))v is easily computed and lies in the plane

( s i ,5 2 ) — • v + 5 i ( X 7 - X _ 7 ) + s2H1.

B. Conical Functions.

Part (i) is immediate from Theorem 4.8. For (ii) recall that by Corollary 4.13, —s*/x is the highest weight of the contragredient TTL . For m* we choose

/o

V

0 e \ 1 0

0 0/

where e = ±1, the sign determined by det(m*) = 1. Also M consists of the diagonal matrices m with diagonal elements ±1 satisfying det(m) = 1. If g G NMAN, g = n(g)m(g) expB(g)nB[g) then by [DS], IX, Exercise A2, the diagonal matrix expB(g) has entries

expB(g)u = |Aifo)| iA*_i(s)r

where A;(#) = det((#m)i<*,m<i) with g = (gem). By Theorem 4.7

ip(m*g • £o) = ip(m*n(g) expB(g) • &,) =(^(exp(- JB(^))n(^)- 1 (m*)- 1 )e ,e / ) = (^ ( (m*) - 1 ) e , ^ (exp(B(^ ) )e / )

=^(£*)e(~sV)(£?0?)).

Now if h = ra*# so g = (m*)_1/i then |Aj(g)| = | A ( ^ ) | so the desired formula for \j;(h - ^0) follows.

The conical functions in this case are related to "conical polynomials" studied in the book by Faraut and Koranyi [1993].


C. Hyperbolic Space; Inversion and Support Theorems.

C. 1. (i) By orthogonality with the geodesies, the horocycles are the (n — l)-spheres tangential to the boundary \x\ = 1. The induced metric on the horocycle is flat. This is obvious for example in the upper half-space model where N - o is a horizontal plane.

(ii) We see that if £0 == N • o and dq the volume element on £0 then

(/)VG/ >o) = Jdkj f(gk • q)dq = j [M<°^ f] (p) dq,

where (Mrf)(p) is the average of / over Sr(p), Thus

oo

where r = d(o,q) (d = distance in H n ) and p = d'(o,q) {d' = distance on horocycle).

It suffices to prove p = sinh r when q is in the xixn-plane so we are in the two-dimensional case. From [GGA] p. 36 (R and N • o are isometric under x —• - ~ ) we see that

HlogGqlt9 P = N' The first formula means

, , — tanh r or p(l + p2)~ * = tanh r \x + %\

so p = sinh r. Hence

oo

(/)v(p) = Q n_! J(Mrf)(p)shn-2rchrdr. O

(v) (vi) Since the area of Sr(p) is proportional to shn _ 1(2r) the formula in (v) follows from [GGA], Ch. II, Prop. 5.26. For (vi) we can write

oo

( / ) » = i O n _ ! j(Mrf)(p)sh(2r)shn-3(r)dr. 0

(vi)-(vii) Let F(r) = (M r / ) (p) , let A r = A(£) and assume k even

Chapter II 583

> 0. Then

fshkrsh(2r)&rF(r)dr=(k + 2)(k-2n + 4) f F(r)shkrsh(2r)dr o

oo

+fc(fc - n + 2) / F(r)shk~2rsh(2r)dr. o

If k = 0 £/iis should be

-2 (n - 2) I / 2F(r)sh(2r)dr + F(0) J .

Proof. By the Darboux equation, L applied to (/)v(p) amounts to the application of A(L) = A r to F(r). Now

fshkrsh(2r) (^ + 2(n - 1) c o t h ( 2 r ) ^ ) dr = \shkr sh(2r)F /

oo

- f F'[shkr ch(2r)2 + fcsh^rch rsh(2r) - 2(n - l)shfcreh(2r)]dr o

oo oo

= 2(n - 2) fshkrch(2r)F'dr - | fshk-2r sh2(2r)F'dr o o

= 2(n - 2) { [sh/srch(2r)Fl °° IL Jo

oo

- f F[2sh(2r)shkr + A; shfc-Vchrch(2r)]dr} o

-^{[Shfc-2rsh2(2r)Fl°° 2 IL Jo

oo

- />F[sh/c~2r4sh(2r)ch(2r) + (k - 2)shk~3rch rsh2(2r)]dr} o

oo

= -2 (n - 2) / F[2sh(2r)sh*y + | sh*- 2 rsh(2r) + k shfc r sh(2r)]dr o

oo

+ | / F[4shfe_2rsh2r + 8shfersh(2r) + (k - 2)(2shfe_2r + 2shkr)sh(2r)}dr


oo

= [ F shkrsh(2r)dr {(k + 2)(fc - 2n + 4)} dr

oo

/ F(r) shfc_2rsh(2r) {ife(fc - n + 2)} dr

o oo

+ 0

This means oo

(L + (A; + 2)(2n - k - A)) j F(r) shkr sh(2r)dr

o oo

•2)Jfc / F ( r -(n - fe - 2)Jfc / F{r) shk-2rsh(2r)dr.

By iteration, A; = n — 3, n — 5, • • • , we obtain (L + (n - l)(n - 1)) . . . (L + 2(2n - 4 ) ) ( / ) v = ( - l ^ Q ^ n - 2)!/.

For a different inversion method see Gelfand, Graev and Vilenkin [1966], Ch. V, §2. C. 2. Use [GGA], Ch. IV, Exercise C3 (for the case of a hyperbolic space) and combine with [GGA], Ch. I, Lemma 4.4. (For full details see Helgason [1980b]).

D. Conical Distributions.

D. 1. (Sketch) To see first that the theorem is local let {Va}aeA be a locally finite covering of V by coordinate neighborhoods and 1 = ^2 ipa a

a partition of 1 subordinate to this covering. Then T = Y,(pa(T\Va) where each restriction T\Va is assumed to have the indicated representation with distributions Tn i v . .n p ) a on Va. In order to move the ipa past the Xi over on Tni...n a we repeatedly use the formula (p(Xf) = X(f<p) — fX<p. For the local version of the theorem let exp tXi be a local 1-parameter group of local diffeomorphisms of a neighborhood oiwEWinV. Then

(X1X2^)(v) = | A(X2( /p)(exp(-tX1) • V ) | ^

= 1 -jr-rr(P(eM-t2X2)ex.p(-t1X1) • v) \ [dt1dt2 Jt!=t2=o

and if di = d/dti,

((X?...X?)(<p))(v) = {d^...d;^(exp(-tpXp)...exp(-t1X1)-v)}t=0.

Schwartz' theorem representing T in terms of the di (Schwartz [1966], Th. XXXV) therefore gives the result of the exercise.

Chapter II 585

D. 2. Let ga be the subalgebra generated by ga and 0_Q,. Then ga is semisimple of real rank one and

ga = 0-2a + 0 - a +Qa+ 02c* + (0°%

([DS], IX, §2). Let ej G 0 ^ . Then Cj,6ej and w = [ej,6ej] span sl(2, C) which operates on ( 0 a ) c . By [GGA], Appendix, Cor. 1.5, gja C [ ( s 0 ) 0 , ^ ] so

[(fla)o,ej] =9ja-

A fortiori [m + a, e ] = Qja so the orbit M • ej has codimension 1 so if the sphere is connected it must be M • ej (cf. Kostant [1975], Ch. II).

D. 3. (Sketch following Wallach [1973] and Lepowsky [1975].) (a) 0 = 02c* + 0-a + 0o + Qa + 02a 0o = m + a. Select X G g a , y =

—#X Gg_Q , such that the vector H = [X, F J G a satisfies

[H,X] = 2X, [ff,y] = - 2 y .

The algebra s = HX + R y + R # is isomorphic to $£(2, R) and 7r = adg \s is a representation of 5 on 0. Deduce from [GGA], Appendix, Lemma 1.2 (ii) that since the eigenvalues of ad H on 0 are 0, ±2, ±4 the dimensions of the irreducible components of TT can only be 1,3 or 5.

(b) Let g1 denote the sum of all the (2i + 1)-dimensional irreducible components of 0 and put

fl^jfriflja ( 0 < i < 2 , - 2 < j < 2 ) .

Then

0* = ®jtfj, Q±2a = 0±2>S±a = 9±1 © 5±2>9o = 0o ® So ® 9o>

and the decomposition 9 = 0° 0 01 0 02

is both 5 - and f?0-orthogonal. (c) Using

[XaiX-a] — B(XOCiX-a)Aa G m,

show that

0o Cm, 0O = R A a 0 (QI n m), 0° c m.

Let m* = 0* n m (i = 0,1, 2). The m0 is the Lie algebra of M0. (d) For Z G 0 put Z* = [X, Z], Z** - (Z*)*, Z* = [y, Z], Z** =

(Z*)*. Prove that if Z G m2,

(Z**)* = 4Z*, (Z**)* = 4Z*, (Z*)* = 6Z, (Z*)* = 6Z


and deduce for Y, Z G tri2

[Y,Z**] = [Y*\Z] = l[Z*,Y*\, [Y,Zr = j[Y*,Z*}.

(e) Given Z G 0, let Z^ be the component in gl in the decomposition 0 = 0° 0 01 0 02. Then if y, Z G m2,

[y, z]x = o, [[r, z]0 + 2[y, z]2 , z**] - o. (f) Suppose y, Z G m2 and ^ ( y * * , Z**) - 0. Then

[y**,z**] - -[z**,y**] Gm, [y**,z**]x = o and

[y*,z„] = -6[y,z]*, [[y?z]0z**] = i^5(y,^y)y**.

(g) Let [/ G 02a: and select Z G tri2 such that f/ = Z**. Let V be in the orthocomplement (for Be) of U in 02a and select Y G rri2 such that y** = V. Deduce from (f) that [W, U] = V for some W G m0 and consequently M0 • J7 fills up a sphere in 02a-

D. 4. For the existence of Sy one can just repeat the proof of Prop. 4.4. Part (a) is obvious. For Part (b) we have by the definition of \I>0, Lemma 3.1 and Cor. 6.2,

*o(y>) = y , ( ^ - ^ o ) ( 0 e p ( l o g f l ( 0 ) d e

= / (<p - <p0)(ria • o)e-P(^a+B(m*n))e2p(\oga)da ^

NA

Now take tp of the form (p(na • £0) = f(n)g{a) where f g(a)ep^oga^da —. 1. Then (b) follows.

D. 5. (i) Use Theorem 4.1 and Corollary 6.2. (ii) Use Cor. 6.2. (iii) Use the M-invariance of S and S^. (iv) Prove

(u2+v2)£W®T0eCon(Z>£)

as an intermediary result, (vi) With the particular g chosen one finds (with r~\n) = f(n(nn)))

*((f®g)n~1) = (S + cA5)r~1

and for the particular choice of / , (AS)(fn~1) = 0. Thus h(s) = 5 ( / n _ 1 ) -S(f) and the contradiction h'(0) ^ 0 is obtained by an elementary computation.

Chapter III 587

D. 6. Solution is similar to that of Exercise D5. For (iv) it is useful to remark the following. Let

( 9n 912 9i3\ 921 922 923 € SU(2 , 1) 931 932 933/

and fp -Q 0\

a=\q p 0 \ e K \0 0 l)

such that k(g)M = aM> Then

V = {911 + 913)1'{931 + £33), Q = (921 + 923)/{921 + #33)

and k(n(nn))M = k{nn)M.

E. The Heisenberg Group.

E. l . - E . 2. See Faraut and Harzallah [1987].

E. 3. The homogeneity and the left invariance are obvious. Since d(g,e) = \\g\\ only the inequality ||<7i#2|| < ||#i|| + ||#2|| remains to be proved and this just involves the Schwarz inequality (Cygan [1981], Koranyi [1983] or Faraut and Harzallah [1987]).

For E. 4, E. 5. and E. 6. see Cowling [1982], Folland [1973] and Koranyi [1982b]. For an exposition of these results see Faraut and Harzallah [1987]. Much of the theory is generalized to N for G/K of rank one in Cowling, Dooley, Koranyi and Ricci [1992].

CHAPTER III

A. Differential Operators.

A. 1. We have for k G K, g e G, n e AT, a G A

Vx(kgn) = r,x(g), rjx(ga) = e^iX-^^^rjx(g).

In the decomposition

D(G) = (*D(G) + D(G)n) 0 D(A)

let D —• DA denote the projection of D(G) onto D(A). If T G t, X G n and Di,D2 G D(G) we have

DiXrix = 0, {TD2r]x){e) = 0, (Dr]X)(e) = (DArix)(e),


and if / G V{G) is right invariant under K,

j(TD2Vx)(g)f(g)dg = J (D2rlx)(g)((-T)f)(g)dg = 0.

Hence

j(DVx)(g)f(g)dg = j (DAVx)(g)f(g)dg G G

= (DArix)(e) J Vx(g)f(9)dg = (Drjx)(e) J Vx(g)f(g)dg. G G

A. 2. See Helgason [1992a].

B. Rank One Results.

B. 1. By the Fourier expansion for a F G £(K/M) (see e.g. [GGA], Ch. V, §3, (13)) we have

r ~ d(6)

F(e)= J ^ d(S) F(k)Y,(S(k)vi,vl)dk seKM K 2 = 1

where F(k) = F(kM), (v^ is an orthonormal basis of V$ such that v — v\ span VS

M. Replacing /c by /cm and integrating over M the sum over i can be restricted to i = 1.

D. The Compact Case.

D. 1. (i) By calculation ( x ^ x _ 1 ) i = cos#. Alternatively, note that u —> xux"1 is a rotation fixing to and tn. (ii) The area of a sphere in S3

of radius 6 is a constant multiple of sin2 6. (iii) Calculate lim Ff(6)/0. o—•()

(iv) The basis zpwq(p + q — £) diagonalizes irt{te) giving the formula for Xi(to)- Then note that by (ii) and the fact that Ff is odd,

Xe(f) = J f{u)xM)du = ^ J(e-ie - eie)Ff{6)Xl{te)de

u 2TT

= UF^ e-W+WM.

Part (v) follows from the fact that \e has L2 norm on U equal to 1 as a result of (ii) and (iii). Part (vi) follows from (iv). For (vii) suppose IT € U is not of the form ne; using (vi) on / = Trace(-7r) we get a contradiction.

Chapter III 589

D. 2. likeK we have

(lP*f)(u)= ¥>(uv 1)f(v)dv = (p(uv 1)f(vk)dv u u

= / (f(uk~1v~1)f(v)dv u

which by averaging over K becomes

(p(u) / {p{v~l)}(y)dv. u

The generalization follows from [GGA], Proposition 2.4 in Ch. IV.

D. 3. The dual of the symmetric space G/K is now (U x U)/U* where the diagonal U* is isomorphic to K. Formula (24) in §9 gives

*"-{n^}'-Here d(fi) is the degree of the irreducible representation r^ of U x U which has a fixed vector under the diagonal group U* and highest weight /i. The irreducible representations r of U x U are of the form

T{UI,U2) = TTI(UI) (8)7r2(u2)

where 7TI,7T2 G U (cf. Weil [1940], §17). Here r has a fixed vector under U* if and only if there is a nonzero vector A G V\ ® V2 such that

7Ti(u) <g> 7r2(ix)A = A, u G 17,

V being the representation space of 71 . This means for the tensor product 7Ti ® 7 T 2

(TTI 0 7r2)(u)A = A

Because of the identification V\ 0 F2 = Hom(Vr2/5 Vi) A is a linear transfor

mation of V2 into Vi so this equation amounts to 7TI(U)AK2(U~1) — A which means TTI and 7r2 contragredient, i.e., iri ~ 7r, 7r2 ~ 7r. Thus /i = (1/, — sz/) where 1/ is the highest weight of n (relative to a maximal abelian subal-gebra t C u) and s is the "maximal" Weyl group element. Considering the relationship between the root system A(u c , t c) and the restricted root system of u x u with respect to t* - {(H, -H) : H G t} ([DS], Ch. VII, §4), where each restricted root has multiplicity 2. Note also for the Killing forms

BUXU((H, -H), (H\ H')) = 2BU(H, H').


Thus TT (M + A <*) = TT (V + Po,0)

M+ <**> & bo,® where on the left (,) refers to -BUxu? on the right to Bu, (5 runs over the positive roots in A(uc , tc) and p0 half their sum. Since d{n) = d(y)2 the formula above gives the formula for d(y) the degree of it.

E. The Flat Case.

E. 2. See Helgason [1980a], §6.

E. 3. We have

(MyMxf)(z)= J f f(z + £-x + k-y)dkd£ K K

I f f(z + £-x + £k-y)dkd£ = f(Mx+kyf)(z)dk. K K K

Here we take x = r e n , y = sen where e n = (0, ••• ,1). Then the last integral is constant for k in the subgroup fixing e n so the integral equals

L J (Mx+swf)(z)dw. s^-^o)

Letting 0 denote the angle between e n and w we integrate this last integral with w first varying in the section of S n _ 1 (0) with the plane (en , y) = cos 6. Since

\x -f sw\2 = r2 + s2 — 2rs cos 6

this gives the second expression for (MyMxf)(z). The last is obtained by the substitution t — (r2 + s2 — 2rs cos 6) 2. (For a different proof see John [1955], p. 80; see also Asgeirsson [1937]).

F . The Noncompact Case.

F . 1. If A G a* then |c(A)|2 = c(A)c(-A) = c(sA)c(-sA).

F . 2. The formula

J f{g)<P-x(9)dg = J Ff(a)e-iX^^da G A

converts the statement into an analogous one for the exponentials elX for which it is obvious.

F . 4. Clearly <p x / e L2(Xf. If F G L2(X)^ is orthonal to all if x / then

yF(A)^(A)/(A)|c(A)|-2dA = 0. a*

Chapter IV 591

Since the functions </? form a uniformly dense subalgebra of C0(a*/W) and since / is analytic on a*, F = 0 a.e.

For the general case let F G L2(X) be orthogonal to fT^ for all g G G. Then

f F<g\x)fT^h\x)dx = 0 g,heG. x

Here we can replace FT^ and fTW by their if-averages (Fr^)^ and (fr(h))b. Integrating against <p(h) (p € T^{G)) then gives (FT^)f = 0 by the first part. Hence F — 0 a.e.

F .5 . Let As(#) G a be given by

geNsexpAs(g)K

and as in (3) §1 put As(gK, kM) = As{k-Xg). Then

As(gK, kM) = sA(gK, kmsM).

The p which corresponds to Ns is sp so the formula

fs(\,kM) = f{s-1\,kmsM)

follows easily.

CHAPTER IV

1. Writing h in G as h = kan according to the Iwasawa decomposition and using the K-invariance of jfe we have

(A x f2)(g • o) = jhigh-1. o)/2(ft • 6)dh G

= I fiignâ-1 • o)f2(an • o)e 2 / ) l o g a âdn. AN

Hence

(A x A)A(î«i • &>) = / (A x AHîm • o)*i,

f fiikKunxa,-1 • o)dni W a n • o)e2p{loga)dadn AN N

Interchanging ni and a - 1 in the inner integral cancels out the factor e2p(loga")

so the expression reduces to

/ A(*i t t ia _ 1 • £o)A>(a ' £o)cfo A

/


as desired. Since * is commutative whereas x is not the if-invariance condition cannot be dropped.

2. Because of the K-invariance of <p we write <p(H) instead oi(p(k exp H-^0). Then by Ch. II, §3, (56),

( / x $)(x) = f f{g-o) j ^(Aig-1 • x, bVeWte-^Mdbdg. G B

Using loc. cit. (47) and (51) this becomes

J f(9 • o) j <p(A(x, g(b)) - A(g • o, g(b)))eMA(x>9(b)))dg(b)dg G B

= J e2'W*'kM»dkM J f(g • o)<p(A(x, kM) - A(g • o, kM))dgK. K/M G/K

Now use the formula

/ F(kan • o)dadn = / F(kg • o)dgx = / F(g • o)dgx AN G/K G/K

on the function F(y) = f(y)(p(A(x, kM) — A(y, kM)) whereby our integral over G/K becomes

/ f(kan • o)(p(A(x,kM) — loga)dadn — (f x cp)(kexpA(x1kM)) AN

Substituting and using (56) again this gives

(/ x $){x) = (/* vy

as stated.

3. By definition

(Anh)(F) = fih(A*F) = fF(expH(hk))e-p(Hihk))dk

K

and {H(hk) : k G K} = C(h) ([GGA], Ch. IX, Theorem 10.5).

5. (i) The Fourier series (20) §3 converges in the topology of £ (R x S1) SO

(ii) By Theorems 2.4 and 3.4, if a G £'(£) then the following conditions are equivalent:

Chapter V 593

(a) aG<?'(H2)A .

(b) CT(I/J) = 0 for each ij; G £(H) satisfying

(1) Dnie^n) is odd (n G Z)

where D n denotes (£> -f 1) • • • (D + 2|ra| - 1), D = d/dt. (c) cr(ip) = 0 for each ip G £(H) satisfying

(2) eVn e ( J D;^(R) ) ± (n G Z)

* denoting adjoint and subscript e indicating "even", and _L denoting an-nihilator.

If a G £'(H) is such that crn has the form in (ii) then etan = £^Tn

where rn G £'e(R>)- If ^ G £(H) satisfies (1) then

(e 2V n ) (^_ n ) = (D;r n ) (eVn) = 0

so a(^) = 0 by (i). Thus by (b) we have a G £ '(H2)A . On the other hand, suppose a G £'(H) satisfies (c), that is

<r($) = 0 whenever efyn G ( D ^ R ) ) ^ (n G Z).

Fix /cGZ and use this on the function ip(£,t,e) = ip-k(t)e~lk0 • Then (T(I/J) = 0 implies (e2t<Tk)(il>-k) = 0> that is (et(Jk)(eti/j-k) = 0. This means that etcik belongs to the double annihilator ( ^ ( ^ ( R ) ) ) 1 - 1 , which equals D%(£'e(R)), this latter space being closed in £'e(R) (cf. Theorem 2.16 in Ch. I). Since k G Z was arbitrary this shows property (ii) for <r.

CHAPTER V

1. By the symmetry of L

X

so the conditions are necessary. For the sufficiency, consider the Fourier transform

/(A, 6 ) = f f(x)e(-iX+pKA(x>h»dx. x

The conditions amount to / (± ip , b) = 0 so /(A, 6) is divisible by (A, A) + (p, p) and the quotient is holomorphic of uniform exponential type and satisfies (3) in Ch. Ill, §5. By the Paley-Wiener theorem, u exists.

3. (i) See Helgason [1976], (Theorem 8.1); another proof is in Dadok [1979].

(ii) See Helgason [1973b] and Eguchi [1979a].


5. See deRham [1955], Ch. V.

4. ,6 .-7. See Theorems 5.3, 6.1-6.3 in Helgason [1964a].

8. One has to verify

TV

— 7T

and using (d2/d02)(\0\) = 28 this is a simple matter. 10. (i) If T G u, [T, Ui] — 2 cijUj where (c^) is skew symmetric. Hence

3

[7>] = ]T[T, JC/ill/i + JUi[T, Ui] i

= J2cij(JUj)Ui + J2cij(JUi)Uj

= ^Cij{JUj)Ui -Y,cv(Jui)ui = 0-i,j i>3

Similarly,

[JT, u}} = E[JT, J ^ ] ^ + Yl JUilJT> Ui\ i

= - E ^ ^ + E ciiijuwuj) i,3 i,j

= l^caiUiUj ~ UjUi) + lY^CiMJUiHJUj) - (JUj)(JUi)) i,3 i,j

= \J2 *i v» ui\ +1E °n vu*> Jui\ = °-ij i,3

This proves (i). For (ii) observe that UJ annihilates all C°° functions / on G which are right invariant under K. Thus if um = f we find a contradiction by averaging over right translations by K.

11. (From a discussion with Schlichtkrull). Let v : D(G) —> E(X) be the homomorphism (from Ch. Ill, §10) given by the action of G on X. Then T commutes with each v{D) so by (1) loc. cit. TZ = ZT for each Z G Z(G/K). Let D e T>(G/K). By Theorem 10.1 in Ch III, DZX = Z2 for some Zx ± 0, Z2 e Z(G/K). Then TDZX = TZ2, DTZX = Z2T so (TD - DT)(Zif) = 0 for / e £(X). By the surjectivity of Zx (Theorem 1.4) we conclude TD = DT.

12. The first statement is immediate from the theorem quoted. For the necessity of the condition and for the compact case see Helgason [1992a].

Chapter V 595

13. The equation holds for all / of the form f(kan) = fi(k)f2(a)fs(n), hence for all / .

14. Suppose first / holomorphic on all of D. Since the rotations z —• e%e z belong to the center of K we have (replacing / by fr^)

2?r 2TT 27r

/(0) = - ^ I f(eiez)d6 = ± j f(eiek • z)d0 = / / ( * • z)dk. 0 0 0

Applying this to the composite function fog(gEGo)we see that / satisfies the mean value theorem (Cor. 2.2) so is harmonic. This argument can be localized since Cor. 2.2 can.

15. We have by (38) in §4,

AQ • &ri = \ ^ tanh?/7X_7 + bTl : y1 G R > S e r - r ! J

proving (i). Part (ii) follows from the fact that the Weyl group consists of all signed permutations.

17. See Proposition 5.2 in Helgason [1987]. The flat case is proved in Menzala and Schonbeck [1984] on the basis of the spherical support theorem [GGA], Ch. I, Lemma 2.7.

18. It suffices to prove this for b = eM and then the function v is TV-invariant. If D G *D(G/K) then AN(D), the AT-radial part of D, is given by &N(D) = ePT(D)oe-P ([GGA], Ch. II, Cor. 5.19); the statement about v is then easily verified.

19. For this we use the transmutation property

(1) A(D)D(p = T(D)A(p, (p K - invariant,

(Ch. IV, Theorem 4.1) and the Darboux equation

(2)

Putting

we have

DgK ( j f(gk • x) dk\ =DX J f(gk -x)dk. K K

f\x) = Jf(k.x)dk, K

F(gK,\oga) = ep(loga) f (/^'^(an - o) dn.


Applying T(D)a and using (1) this becomes

eP(log

N

which by (2) becomes

a) J D(fT(g 1))^an.0)dn

P(\oga) J fDgK J f(gkan . 0) dk\ dn = (DF)(gK, a). e N ' K

CHAPTER VI

1. Using a if-invariant Laplace-Beltrami operator on K/M we see that each joint eigenspace E is finite-dimensional. Let E = 0 Ei be the direct

decomposition into irreducible subspaces. Pick fi G Ei such that fi(eM) = 1 and fi is M-invariant. Then each fi is a spherical function and Dfi — x(D)fi, where the homomorphism \ : "D(K/M) —> C is the same for all i. Using [GGA], Ch. IV, Cor. 2.3 we find that all fi coincide so E is irreducible.

Taking K = SU(2), M = e, each joint eigenspace has to contain a character \ °f K- If T is a maximal torus with Lie algebra spanned by a vector H it is easily seen that H\ is not a constant multiple of \ (°f-e.g. [GGA], Ch. V, Ex. A7).

2. This is a basic step in Bruhat's analysis [1956] §6) of the principal series for G. By Schur's lemma (for unitary rpresentations) (i) is equivalent to the statement that all bounded linear operators A : JC\ —> JC\ commuting with all r\(g) (g G G) are scalars. Let A be one such operator, consider the sesquilinear form

B(<p,i/;)= J (p(kMN)con]{(Ai/;){kMN))dkM K/M

and the form B(f,g) = B{f\g*) f,geV(G),

where f\xMAN)= I f{xamn)e{-iXJrp)^aUmdadn.

MAN

Then

B(fL(x)R(pi) qL(x)R(p2)\ _ e - (2A+p)( logai) e ( iA-p)( loga 2 ) jg/Y „ \

Chapter VI 597

and by the Schwartz kernel theorem (Hormander [1983], Ch. V)

B(f,g)= J f(x)con](g{y))df(x,y), GxG

where f eV'(GxG). Then

jiL(x,x)R(plip2) _ e(iA+/9)(logai)e(-iA+p)(loga2)^

where L(x,x)R(pi,p2) denotes the diffeomorphism (u,v) —> {xup\,xvp2) of G x G. Consider the diffeomorphism cp : (x,y) —> (y~1x,y) of G x G. Then, if h G V{G x G), we have by the left invariance of T,

fv(h) = f (h^1) = T((h<p~1)L(z>z)).

However

( / ^ - y ^ O c y ) = h^iz-'x^z-'y) = h(y-lx,z-ly)

SO, ^hcp-^L(z,z) = (hL(e,z)y-\

Thus Ttp(h)=T(p(hL^z))

so T*(/i) = y j h{x,y)dS{x)dy,

G G

where S G V(G). Since (^_1(x,?/) = (yx,y) this implies

T(f®g) = J j f(yx)g(y)dS(x)dy. G G

Using the homogeneity of T under R{p\,p2) we obtain the homogeneity condition for S in (ii). The converse follows by reversing the steps. All the commuting operators A are proportional if and only if the corresponding S are proportional and then they must be the example stated.

3. See Helgason [1970a], Ch. Ill, §6.

4. Using Lemma 3.10 and (39) we see quickly that ^\ e = \I/_A e- Thus if <p € 2>(3)

(if x *Aie)(fcaMAr) = V-\,e(<p o r{ka))

e(i\-p)(loS a) f ip{kcMN)e(-iX+pW°zc)dc.


Taking (f(kaMN) = (3(kM)j(a) the result follows.

5. In the solution below d and C[ denote compact sets and —» A denotes the interior of a set A. Let C\ C—> G2 C H, let 1 )^ (5) denote the set of <p € £>(£) with support in Gi, and let G2' C G satisfy 7r(G|) = G , C[ C (C2)0 C G, n : G —• G/MN being the natural mapping. Let C0

be a compact neighborhood of e in MTV and put Ci — C^C0 (i = 1,2). Let /1 € £>(G) be > 0 on G, > 0 on Gi, and supp(/i) C G2. Then the function

satisfies / = 99 (cf. (36) §3). Also y? —• / is a continuous mapping of ©Ci(3) into T>c2(G)' T h u s b ^ (37) i n §3, # x 77 is a distribution. For the last part one must show

Jl>(t)(<P x »?*)(Ode = /(V> x r / ) ( 0 ^ ( 0 ^ -

Let /1 G D(G) satisfy fi = i[>. Then this last equation amounts to

J fi(g) j f(gh-1)d(r]*r(h)dg = I f(g) J Mgh^drj^dg. G G G G

However, (77*)~ = 77/ so this last equation is obvious.

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S Y M B O L S F R E Q U E N T L Y U S E D

Ad: adjoint representation of a Lie group, 5 ad: adjoint representation of a Lie algebra, 5 A(r): spherical area, 420, 484 A(g): component in g = nexpA(G)k) 86, 99 A(B): space of analytic functions on JB, 530 A'{B)\ space of analytic functionals (hyperfunctions) on B, 530 a, ac, a*, a*: abelian subspaces and their duals, 61 a': 68 a*{6): subset of a*, 237 a+ , a+: Weyl chambers in a and a*, 61, 202 la\ transpose, 29 Ay. vector in ac, corresponding to A, 61 A(x,b): composite distance, 99 AQ\ projection from p to a, 289 A: Abel transform, 381 A*: dual Abel transform, 382 A-*'- space of analytic vectors, 416 Br(p), Br(p): open ball with radius r, center p, 3 B: Killing form, 61 *B: set of bounded spherical functions, 341 pR: ball in 3 , 364 BC(G): space of bounded continuous functions on G, 336 CI: closure, 3 conj: complex conjugate, 93 Cn: complex n-space, 298 Cn: special set, 6 C(X): space of continuous functions of X, 3 CC(X): space of continuous functions of compact support, 3 C\(G): space of iif-bi-invariant functions in CC(G), 80 CK(X): space of continuous functions with support in K, 3 Co(X): space of continuous functions vanishing at oo, 3 C°°(X)1C^°(X): set of differentiable functions, set of differentiable functions of compact support, 4 c(A): Harish-Chandra's c-function, 90 cs(X): partial c-function, 141 Cs: generalized c-functions, 234 C + ,~C , + C,C~: closures of Weyl chambers and their duals, 129 r , T: isomorphisms of differential operators, 74 rS)y- intertwining operator, 240 Tx(A): Gamma function for X, 284

627

628 SYMBOLS FREQUENTLY USED

di'. partial derivative, 3 6: density, 213 V(X): C?(X), 4 V(X): set of distributions on X, 4 Vr

x\ eigenspace, 76 VK(X): set of / <E V with support in K, 4 VH(Pn): subspace of £>(Pn), 11 VH(G(d,n)): subspace of V(G(d,n)), 45 Vs(X): space of X-commuting functions, 273 T>s(X): K-Snite functions of type 5, 273 V*(X),V[(X): space of X-invariant elements in V(X),V'(X), 207, 381 V\G),V[[G): space of iC-bi-invariant members of £>(G),£>'(G), 90 D(G): set of left-invariant differential operators on G, 70 DH(G): subalgebra of D(G), 70, 71 D(G/H): set of G-invariant differential operators on G/H, 71, 75 DW(A): ^-invariants in D(A), 70 D(X),D(E): invariant operators on X, H, 70, 71 d(S) or d$: dimension (= degree) of a representation, 14 A(£>): radial part of D, 70 AMN(D), AK(D), AN(D): radial parts of L>, 75, 70 A(£jc,J)c): set of roots, 128 ds(A),es(A): factors in cs(A), 142 E{M): set of all differential operators on M, 36 £(X): C°°(X), 4 £'{X): space of distributions of compact support, 4 £*{X),£[(X): space of K-invariant elements in £(X),£'(X), 207, 381 £\,£(\),£^),£*,£x',£x16' eigenspaces, 76, 229, 282, 531 £^(G): space of if-bi-invariant members of £(G), 381 Ekm- eigenspace of Laplacian, 11 e\}b'- plane wave eigenfunction, 99 F(a,b;c;z): hypergeometric function, 328 / -> / : map from CC(G) to CC(G/H), 26, 155 f6: if-commuting function, 266 T\ spherical transform, 220 T{X)\ function space, 376 g^T^D^. i m a g e s o f g e £{M)^T € £>'(M), operator D under <p, 4 y?A- spherical function, 76, 86 <&\j: generalized spherical function, 228 Go: a group of linear transformations of Xo, 285 G(d,n),Gd,n'' manifolds of d-planes, 39, 41 H A ( C n ) , % ( a * ) , « ( a * x B)w: exponential type, 261, 275, 567 Hom(V, W): space of linear transformations of V into W, 273 Hn: hyperbolic space, 50 Hi,H(p): space of harmonic polynomials, 16, 230

SYMBOLS FREQUENTLY USED

WA: Hilbert space inside £A(X),£A(p), 284, 309, 552 H6(a*): special holomorphic functions on a*, 275 H: Hilbert transform, 6, 390 rjx' if-fixed vector in £A(£), 243 H(g): component in g = k expif(g)n, 99 Im: imaginary part, 261 1(E): space of invariant polynomials on E1, 229 I7: Riesz potential, 6 I(X): group of isometries of X, 50 J2(G): if-bi-invariant Schwartz space, 220 I'x s: intertwining integral, 244 7A,s- normalized intertwining operator, 245 J: inversion, 162 JS(X): polynomial matrix, 287 Jn(z): Bessel function, 285 /CV(a*),/C(a* x B)w' exponential type, slow growth, 271 K,KM- unitary dual and subset, 227, 370 /CA: Hilbert space inside P^(S), 548 X8'. character of 5, 13 £: algebra in Cartan decomposition, 77 L1(X): space of integrable functions on X, 85 LP(X): space of / with |/ |* G LX(X), 433 L — Lx'. Laplace-Beltrami operator on X, 5 L(g) = Lg: left translation by g, 5 1(6): dimension, 228 A: operator on P , 7, on H, 93, weight lattice, 240 i: orthocomplement of m in £, 71 A0: operator on Ho, 390 Mp: the tangent space to a manifold M at p, 3 m*: element, 64 Mr: mean-value operator, 77, 484 M(n): group of isometries of Rn, 1 m: centralizer of a in I, 61 Wl: set of continuous homomorphisms, 339 ffll(B): space of measures on JB, 439 J\f: kernel of dual transform, 13, 367 n: part of Iwasawa decomposition, 61 0(n),0(p,q): orthogonal groups, 1, 352 ftn: area of S71'1, 9 Pn: set of hyperplanes in jRn, 8 Pi: space of homogeneous polynomials of degree /, 16 P\,P\: Poisson transform, 300, 100 P6(X): inverse of Qs(\), 236 93: set of positive definite spherical functions, 340 7r(A): product of roots, 91, 154

630 SYMBOLS FREQUENTLY USED

p: part of a Cart an decomposition, 61 Q5(\): polynomial matrix, 232 71: ring of functions on A+, 234 R: modified Radon transform, 220 Rn: real n-space, 1 JR+: set of reals > 0, 3 Re: real part, 90 Rg or R(g): right translation by #, 5 Res: residue, 6 p, po> P*: half sum of roots, 61, 323 S*: element, 64 Sn: n-sphere, 7 Sr (p): sphere of radius r and center p, 3 <S(jRn): space of rapidly decreasing functions on fin, 5 S(X),S'(X): 214, 532 S*(Rn),S0(Rn): subspacesof<SGRn), 10 Sf(Rn): space of tempered distributions, 5 5(S): 91 S(V): symmetric algebra over V, 230, 231 S\l8,S'Xa: distributions on B, 136, 142 S(G): 214 S(D): Shilov boundary of £>, 453 sgn(x): signum function, 7 SH(Pn): subspace of <S(Pn), 8, 11 sh x: sinh x, 2 cr(F, G): weak topology, 29 cr(a): diffeomorphism of 5 , 107 £(g, a): set of restricted roots, 129 <TR: sphere in S, 364 £,£+,£(5", £+,!!*: sets of restricted roots, 61, 90, 129, 138 tA: transpose of A, 29 Tr(A): trace of A, 14 T\,T\,T\: eigenspace representations, 77, 284 r: homomorphism of D(G), 232 T ( # ) : translation on G/H, 5 ^: Cart an involution, 61 U(n): unitary group, 53 Vs: representation space of 5, 227 V6

M: space of fixed vectors under 5(M), 228 W: Weyl group, 61, 323 H: dual space, 62 3 : special spherical function, 214 3*: open orbit in 3 , 64 So: space of horocycle planes, 387 £*: origin in 3*, 64

SYMBOLS FREQUENTLY USED 631

£(#, b): horocycle determined by x and 6, 99 ^ A , S ^ A , S

: conical distributions, 135, 142 ^A,<5- generalized Bessel function, 289 Z, Z+: the integers, the nonnegative integers, 3 Z{G): center of D{G), 322 Z(G/K): image of Z(G) in D(G/K), 322 ~: Fourier transform, spherical transform, lift of functions, distributions, 4, 77, 155, 198 A: Radon transform, incidence, 1, 31 V: Dual Radon transform, incidence, 1, 31 *, x: convolutions, adjoint operation, pullback, star operator, Fourier transform, 6, 9, 26, 80, 82, 96, 137, 200, 557 ®: direct sum, 527 0 : tensor product, 12, 108, 112 {,): inner product, 29 fcj,£^: space of if-invariants in E, 86, 90 • : operator, 8, 97 • p : operator on G(p, n), 41 —: closure, 3, restriction, 116 o: interior, 3 _L: annihilator, 16

I N D E X

A

Abel transform, 381 Adjoint representation, 5 Analytical functionals, 529, 530 Analytic vector, 416 Annihilator, 17 Antipodal mapping, 162

B

Banach algebra topology, 339 Base, 541 Bessel function, 285, 289, 292 , 478 Bessel transform

generalized, 293 Borel imbedding, 444 Boundary component, 68 Bounded growth, 418 Bounded, 29 Bruhat decomposition, 63

C

Cartan involution, 61 Cartan subalgebras

conjugacy of, 59, 97, 480 Casimir operator, 534 Cauchy problem, 410, 469, 497 Centralizer, 60 Character of a representation, 13 Compatibility with projection, 542 Composite distance, 64, 99 Conformal diffeomorphism, 486 Conical

distribution, 105, 186 distribution, exceptional, 171

distributions, parametrization of, 106

function, 105, for SL(n ,R) , 183 representation, 106, model for, 121 vector, 106

Contraction, 422 Contragredient representation, 118 Convolutions, 6, 9, 26, 96, 98, 137 Cusp form, 3

D

d-plane transform, 45 ^-spherical transform, 274, 279, 293 Darboux equation, 185 Differential operator

image of, 4 invariant, 35 radial part, 70, 74

Dirichlet problem, 422, 460 Distributions, 4

spaces of, 12 of compact support, 4

Double fibration, 30, 32, 388 Dual transform, 1, 9, 85

inversion of, 20 Duality

topology compatible with, 29 for a symmetric space, 62

E

Eigenfunctions of slow growth, 531 exponentially bounded, 531

Eigenspace, 11 representation, 75

633

634 INDEX

for distribution spaces, 540 for vector bundles, 541 of function spaces, 540

Eigenvalue, 11 Eisenstein integral, 228, 327 Energy

conservation of, 487 equipartition of, 488 kinetic, 487 potential, 487

Euclidean imbeddings, 122 Evaluation mapping, 29 Exponential type

of a holomorphic function, 261 uniform, 261

Extreme point, 338 weight vector, 558, 569 weight, 569

F

G

Gamma function of a symmetric space, 284

Gelfand pair, 340 Gelfand transform, 340 Generalized Bessel function, 289, 292

reduction to

zonal spherical function, 292 Geodesic

in a horocycle space, 65 symmetry, 62 in bounded symmetric domains, 444

Grassmannian, 39 Green's kernel, 533 Green's function, 533

H

Haar measure, 26 Harish-Chandra c-function, 90 Harish-Chandra imbedding, 443 Harmonic

function, 100, 421 polynomials, 16

Heisenberg group, 190 Hilbert transform, 6, 390 Holomorphic,

representation, 565 function of exponential type, 261 function of uniform exponential type, 261

Homogeneous spaces in duality, 31 Horocycle, 60

as plane section, 122, 127 interior of, 182 normal to, 64, 99 parallel, 65 plane, 387 transform, 2, 85

Hua equations, 461 Huygens principle, 468, 471, 473, 474,

477, 482, 485, 504, 538 converse of, 536

Hyperbolic space, 184, 378, 398 Hyperfunction, 530 Hypergeometric functions, 328 Hyperplane, 5

I

Incident, 31, 53

Fatou theorem, 430, 432, 438 Fiber, 541 Fixed point property, 426 Flat in a symmetric space, 65 Fourier transform, 4, 9, 82

Euclidean, 92 on a symmetric space, 197, 199, 202,

315 self duality under, 197, 208

Frechet space, 4, 30 Functional on the boundary, 528 Fundamental solution, 402 Funk-Hecke theorem, 18 Furstenberg compatification, 439

INDEX 635

Indivisible roots, 90 Inductive limit, 4 Inner product, 3 Intertwining operator, 554, 556 Intertwining, 10 Invariant differential operator, 35, 55 Invariant

operator, 25, 55 distribution, 25

Inversion problem, 7 Inversion

constant curvature spaces for, 51 ^-spherical transform for, 279 for N, 159 Fourier transform for, 201 Grassmann manifolds for, 42, 45 horocycle planes for, 391 horocycle transform for, 89 hyperplane transform for, 5 spherical transform for, 221, 342

Isotropic vector, 298 Iwasawa decomposition, 60

J

J-polynomials, 286 Jacobi

functions, 352 transform, 352

Joint eigenspace, 75, 76

K

K-finite joint eigenfunctions, 527 K-type, 14 Kelvin transformation, 192 Kernel, 13, 394 Killing form, 61 Klein-Gordon equation, 472

L

Lagrangian subspace, 53 Laplacian, 5 Lebesgue differentiation theorem, 432 Legendre polynomial, 18, 52

Lift of a function, 155 distribution, 155

Light cone, 482 Line bundle, 541 Local trivialization, 541 Lorentzian manifold, 482

M

Maximal flat, 65 Maximal theorem, 473 Maximum principle, 422 Mean value, 77, 413 Mean value operator, 77

expansion for, 78 commutativity, 80, 415

Measure, 3 Moment condition, 11 Multiplicity, 61

N

Normal to a horocycle, 64 Normalizer, 61

O

Open mapping theorem, 29 Orbital integral, 482

P

Paley-Wiener theorem for the Fourier transform on X, 260 for the K-types, 275 for the Radon transform on X, 365

Parallel horocycles, 65 Peter-Weyl expansion, 14 Pizetti formula, 193 Poisson

integral, 2, 49 kernel and the dual transform, 100 kernel, 49, 456 transform and the dual transform, 102, 300

636 INDEX

Polar coordinate representations, 62 Polydisk in a

bounded symmetric domain, 451 Pompeiu problem, 2 Positive definite, 335 Principal series, 564 Projection map, 541 Pseudo-Riemannian manifold, 482

Q

Q-matrices, 232

R

Radial, part of a differential operator, 70, 74

Radon inversion formula, 5 Radon transform, 1, 7, 9, 85

double fibration for, 30 horocycle planes, for, 388 horocycles, for, 2, 85 injectivity, for, 85 range problem for, 7 inversion formula for 89 Plancherel formula for, 89

Range theorems, 11 d-plane transform, 46 ^-spherical transform, 275 Fourier transform, 261, 271, 275,

281 horocycle plane transform, 394 hyperplane transform, 11 Poisson transform, 529, 530, 531

Rank-one reduction, 137 Rapidly decreasing function, 4 Reduced expression, 138 Reductive homogeneous space, 70 Reflection of a symmetric space, 67 Regular

geodesic, 65 vector, 153, 301

Representation adjoint, 5 character of, 13

conical, 105 contragredient, 119 eigenspace, 75, 540, 541 holomorphic, 565 irreducible, 550 spherical, 106 scalar irreducible, 571 weights of, 127

Restricted roots, 61 for bounded symmetric domain, 445

Restricted weight vector, 127 Restricted weight, 127 Retrograde cone, 482 Riesz potential, 6 Roots,

indivisible, 90 restricted, 61 strongly orthogonal, 442 unmultipliable, 129

S

Scalar irreducible, 571 Scattering theory, 90 Schwartz kernel theorem, 597 Schwartz spaces, 4, 8, 214, 384 Schwarz' theorem, 2 Semi-norms, 4 Semireflexive, 30 Shilov boundary 453

Go-homogeneous, 454 Ko-homogeneous, 454 Poisson kernel for, 456

Simple, 151, 165, 242, 300 Singular, 301 Singular support, 536 Slow growth, 91 Solvability, 401, 403, 534, 535 Solvable groups, 426 Space of p-planes 39

through the origin, 41 Sphere, area of, 9 Spherical

function, 86, 340

INDEX 637

function, generalized, 228 reduction to zonal

spherical function, 228 functional equation for, 240, 278 principle series, 549

compact models for, 550 irreducibility, 550

representation, 106 model for, 121

transform, 90, 335, 340 transform of type J, 275 vector, 106 zonal spherical function, 292

Strong topology, 29 SU(2,l)-reduction, 257 Sub-Laplacian, 191 Support problem, 7 Support theorem, 12, 182, 185, 392

Tempered distribution, 5 Theta series, 3 Totally geodesic submanifold, 68 Transform

Abel, 381 Bessel, 293 d-plane transform, 45 Double fibration, 32, 57 Dual transform, 1 Fourier, 4, 82, 199, 315 Gelfand transform, 340 Hilbert, 6, 390 Horocycle transform, 2, 85 Horocycle plane transform, 388 Kelvin, 192 Poisson transform, 102

Radon transform, 1, 5, 85 Spherical transform, 90, 274 Twisted Radon transform, 44 X-ray transform, 2

Transmutation operator, 402 Transpose map, 29 Transversal manifold, 70 Transversality, 33 TschebyschefT polynomial, 52 Tube type, 462, 465 Twisted transform, 44

U

Ultrahyperbolic operator, 50 Ultraspherical polynomial, 18

Vector bundle associated to a representation, 542 complex, 541

W

Wave equation, 468 propagator, 479

Weak topology, 29 Weak* topology, 29 Weight, 127 Weight vector, 123, 127 Weyl chamber, 61, 430, 439 Weyl group, 61 Whittaker vector, 194

X-ray transform, 2

geometric analysis on symmetric spaces, second edition · 2019-02-12 · from my previous books...

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