the guillemin–sternberg conjecture for noncompact groups ...sarathy/itseemspage3.pdf ·...

J. K-Theory 1 (2008), 473–533doi: 10.1017/is008001002jkt022

©2008 ISOPP

The Guillemin–Sternberg conjecturefor noncompact groups and spaces

by

P. HOCHS AND N.P. LANDSMAN

Abstract

The Guillemin–Sternberg conjecture states that “quantisation commutes withreduction" in a specific technical setting. So far, this conjecture has almostexclusively been stated and proved for compact Lie groups G acting on com-pact symplectic manifolds, and, largely due to the use of Spinc Dirac operatortechniques, has reached a high degree of perfection under these compactnessassumptions. In this paper we formulate an appropriate Guillemin–Sternbergconjecture in the general case, under the main assumptions that the Liegroup action is proper and cocompact. This formulation is motivated by ourinterpretation of the “quantisation commuates with reduction" phenomenonas a special case of the functoriality of quantisation, and uses equivariantK-homology and the K-theory of the group C �-algebra C �.G/ in a crucialway. For example, the equivariant index - which in the compact case takesvalues in the representation ring R.G/ - is replaced by the analytic assemblymap - which takes values in K0.C �.G// - familiar from the Baum–Connesconjecture in noncommutative geometry. Under the usual freeness assumptionon the action, we prove our conjecture for all Lie groups G having a discretenormal subgroup � with compact quotient G=� , but we believe it is valid forall unimodular Lie groups.

Key Words: Guillemin–Sternberg conjecture, quantisation, reduction

Mathematics Subject Classification 2000: 53D50, 19K35, 46L65

Contents

1 Introduction 474

2 Assumptions and result 4862.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4872.2 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3 Reduction by � . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4922.4 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

474 P.HOCHS & N. P. LANDSMAN

3 Differential operators on vector bundles 4953.1 The homomorphism VN . . . . . . . . . . . . . . . . . . . . . . . 4963.2 Spaces of L2-sections . . . . . . . . . . . . . . . . . . . . . . . . 4973.3 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . 5003.4 Multiplication of sections by functions . . . . . . . . . . . . . . . 5053.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

4 Dirac operators 5074.1 The isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 5084.2 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . 508

5 Abelian discrete groups 5125.1 The assembly map for abelian discrete groups . . . . . . . . . . . 5135.2 The Hilbert C �-module part of the assembly map . . . . . . . . . . 5155.3 The operator part of the assembly map . . . . . . . . . . . . . . . 5195.4 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

6 Example: action of Z2n on R2n 5216.1 Prequantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5216.2 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . 5226.3 The case nD 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

A Naturality of the assembly map 526A.1 The statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526A.2 Integrals of families of operators . . . . . . . . . . . . . . . . . . . 527

References 529

1. Introduction

In 1982 two fascinating conjectures appeared about group actions. Guilleminand Sternberg [26] gave a precise mathematical formulation of Dirac’s idea that“quantisation commutes with reduction" [13], in which they defined the formeras geometric quantisation. As it stood, their conjecture - which they provedunder special assumptions - only made sense for actions of compact Lie groupson compact symplectic manifolds. These compactness assumptions were left inplace throughout all later refinements in the formulation of the conjecture and theensuing proofs thereof under more general assumptions [33, 50, 51, 52, 54, 70, 72].Baum and Connes, on the other hand, formulated a conjectural description of theK-theory of the reduced C �-algebra C �

r .G/ of a locally compact group G in terms of

Guillemin–Sternberg conjecture 475

its proper actions [4]. Here the emphasis was entirely on the noncompact case,as the Baum–Connes conjecture is trivially satisfied for compact groups. Themodern formulation of the conjecture in terms of the equivariant K-homology ofthe classifying space for proper G-actions was given in [5]. In this version it hasnow been proved for all (almost) connected groups G [10] as well as for a largeclass of discrete groups (see [71]).

Although the two conjectures in question have quite a lot in common - suchas the central role played by index theory and Dirac operators,1 or their acuterelevance to modern physics, especially the quantisation of singular phase spaces(cf. [42, 44]) - the compact/noncompact divide (and perhaps also the sociologicaldivision between the communities of symplectic geometry and noncommutativegeometry) seems to have precluded much “interaction" between them. As weshall see, however, some of the ideas surrounding the Baum–Connes conjectureare precisely what one needs to generalize the Guillemin–Sternberg conjecture tothe locally compact case.

Being merely desirable in mathematics, such a generalization is actually crucialfor physics. As a case in point, we mention the problem of constructing Yang–Millstheory in dimension 4 - this is one of the Clay Mathematics Institute MillenniumPrize Problems - where the groups and spaces in question are not just noncompact,but even infinite-dimensional! Also, the work of Dirac that initiated the moderntheory of constraints and reduction in mechanics and field theory was originallymotivated by the problem - still open - of quantising general relativity, where againthe groups and spaces are infinite-dimensional. From this perspective, the presentwork, in which we attempt to push the Guillemin–Sternberg conjecture beyond thecompactness barrier to the locally compact situation, is merely an exiguous step inthe right direction.

To set the stage, we display the usual “quantisation commutes with reduction"diagram:

.G ˚M;!/� Q ��

�

RC

��

G ˚Q.M;!/�

RQ

��.MG ;!G/

� Q �� Q.MG ;!G/:

(1)

Here .M;!/ is a symplectic manifold carrying a strongly Hamiltonian action of aLie groupG (cf. Section 2), with associated Marsden–Weinstein quotient .MG ;!G/,

1Here the motivating role played by two closely related papers on the discrete series representationsof semisimple Lie groups should be mentioned. It seems that Parthasarathy [56] influenced Bott’sformulation of the Guillemin–Sternberg conjecture (see below), whereas Atiyah and Schmid [3] inpart inspired the Baum–Connes conjecture (see [12, 37]).


i.e.MG Dˆ�1.0/=G; (2)

where ˆ W M ! g� is the momentum map associated to the given G-action (cf.Subsection 2.1). This explains the term ‘RC ’ in the diagram: Classical Reductionas outlined by Dirac [13] is defined as Marsden–Weinstein reduction [1, 47, 48, 49].However - and this explains both the fascination and the mystery of the fieldof constrained quantisation - the mathematical meaning of the remaining threearrows in the diagram is open to discussion: see [13, 18, 21, 29, 39, 67] forvarious perspectives. In any case, Q stands for Quantization, RQ denotes QuantumReduction, and all authors seem to agree that, if at all possible, the arrows shouldbe defined so as to make the diagram commute (up to isomorphism as appropriate).We will return to the significance of this commutativity requirement below, but forthe moment we just remark that it by no means fixes the interpretation of the arrows.

For example, following the practice of physicists, Dirac [13] suggested thatQ.M;!/ - the quantisation of the phase space .M;!/ as such - should be aHilbert space, which subsequently is to carry a unitary representation U of G that“quantises" the given canonical G-action on M . We assume some procedure hasbeen selected to construct these data; see below. Provided that G is compact,the quantum reduction operation RQ then consists in taking the G-invariant partQ.M;!/G ofQ.M;!/. Similarly,Q.MG ;!G/ is a Hilbert space without any furtherdressing. Now assume that M and G are both compact: in that case, the reducedspace MG is compact as well, so that Q.M;!/ and Q.MG ;!G/ are typically finite-dimensional (this depends on the details of the quantisation procedure). In that case,commutativity of diagram (1) up to isomorphism just boils down to the numericalequality

dim.Q.M;!/G/D dim.Q.MG ;!G//: (3)

This equality becomes a meaningful conjecture once an explicit construction ofthe objects G ˚ Q.M;!/ and Q.MG ;!G/ as specified above has been prescribed.Guillemin and Sternberg [26] considered the case in which the symplectic manifoldM is compact, prequantisable, and equipped with a positive-definite complexpolarization J � TCM left invariant by the given strongly Hamiltonian G-action. Recall that a symplectic manifold .M;!/ is called prequantisable whenthe cohomology class Œ!�=2� in H 2.M;R/ is integral, i.e., lies in the image ofH 2.M;Z/ under the natural homomorphism H 2.M;Z/!H 2.M;R/. In that case,there exists a line bundle L! over M whose first Chern class c1.L!/ maps toŒ!�=2� under this homomorphism; L! is called the prequantisation line bundleover M . Under these circumstances, the quantisation operation Q may be definedthrough geometric quantisation (cf. [25] for a recent and pertinent treatment): one


picks a connection r on L! whose curvature is !, and defines the Hilbert spaceQ.M;!/ as the space Q.M;!/ D H 0.M;L!/ of polarized sections of L! (i.e. ofsections annihilated by all rX , X 2 J ). This Hilbert space carries a natural unitaryrepresentation of G determined by the classical data, as polarized sections of L!are mapped into each other by the lift of the G-action. Moreover, it turns out thatthe reduced space MG - assumed to be a manifold - inherits all relevant structureson M (except, of course, the G-action), so that it is quantisable as well, in the samefashion. Thus (3) becomes, in self-explanatory notation,

dim.H 0.M;L!/G/D dim.H 0.MG ;L!G

//; (4)

which Guillemin and Sternberg indeed managed to prove (see also [28]).Impressive as this is, it is hard to think of a more favourable situation for

quantisation theory then the one assumed in [26]. In the mid-1990s, various earlierattempts to generalize geometric quantisation - notably in a cohomological direction- and the associated Guillemin–Sternberg conjecture culminated in an unpublishedproposal by Raoul Bott to define quantisation in terms of the (equivariant) index of asuitable Dirac operator. See, e.g., [65]. As this definition forms the starting point ofour generalization of the Guillemin–Sternberg conjecture to the noncompact case,we consider it in some detail. (See [16, 20, 22, 25] for the theory of Spinc structuresand the associated Dirac operators).

The first step in Bott’s definition of quantisation is to canonically associate aSpinc structure .P;Š/ to a given symplectic and prequantisable manifold .M;!/[25, 50]. First, one picks an almost complex structure J on M that is compatiblewith ! (in that !.�;J �/ is positive definite and symmetric, i.e. a metric). This Jcanonically induces a Spinc structure PJ on TM [16, 25], but this is not the rightone to use here. The Spinc structure P needed to quantise M is the one obtained bytwisting PJ with the prequantisation line bundle L! . This means (cf. [25], App.D.2.7) that P D PJ �ker.�/ U.L!/, where � W Spinc.n/ ! SO.n/ is the usualcovering projection. We denote the associated Spinc Dirac operator byD/ LM . See forexample [16, 20]. Since M is even-dimensional, any Dirac operator on M worth itsname decomposes in the usual way as

D/ D�

0 D/ �D/ C 0

�; (5)

and we abuse notation in writing

index.D/ /D dimker.D/ C/� dimker.D/ �/: (6)

WhenM is compact, the Dirac operators .D/ LM /˙ determined by the Spinc structure

.P;Š/ have finite-dimensional kernels, whose dimensions define the quantisation


of .M;!/ asQ.M;!/D index.D/ LM / 2 Z: (7)

This number turns out to be independent of the choice of the Spinc structure onM , as long as it satisfies the above requirement, and is entirely determined by thecohomology class Œ!� (this is not true for the Spinc structure and the associatedDirac operator itself) [25]. If the symplectic manifold .M;!/ is Kähler and theline bundle L is ‘positive enough’, then the index of D/ LM equals the dimension ofthe space H 0.M;L/ of holomorphic sections of the prequantum line bundle. Thisprovides some justification for Bott’s definition of quantisation.

So far, quantisation just associates an integer to .M;!/. Bott’s definition ofquantisation gains in substance when a compact Lie group G acts on M in stronglyHamiltonian fashion. In that case, the pertinent Spinc structure may be chosen to beG invariant, and consequently the spaces ker.D/ ˙/ are finite-dimensional complexGmodules; we denote their isomorphism classes by square brackets. In this situationwe write

G-index.D/ /D Œker.D/ C/�� Œker.D/ �/�; (8)

which defines an element of the representation ring2 R.G/ of G. Thus thequantisation of .M;!/ with associated G-action may be defined as

Q.G ˚M;!/DG-index.D/ LM / 2R.G/: (9)

As before, this element only depends on Œ!� (and on theG-action, of course). WhenG is trivial, one may identify R.e/ with Z through the isomorphism

ŒV �� ŒW � 7! dim.V /� dim.W /;

so that (7) emerges as a special case of (9).In this setting, the Guillemin–Sternberg conjecture makes sense as long as M

and G are compact. Namely, in diagram (1) the upper right corner is now construedas an element of R.G/, whereas the lower right corner lies in R.e/ Š Z; as in theoriginal case, the geometric quantisation of the reduced space .MG ;!G/ is definedwhenever that of .M;!/ is. The quantum reduction map RQ W R.G/! Z is simplydefined by

RQ W ŒV �� ŒW � 7! .ŒV �� ŒW �/G WD dim.V G/� dim.W G/; (10)

2R.G/ is defined as the abelian group with one generator ŒL� for each finite-dimensional complexrepresentation L of G, and relations ŒL� D ŒM � when L and M are equivalent and ŒL�C ŒM � DŒL˚M�. The tensor product of representations defines a ring structure on R.G/.


where V G is the G-invariant part of V , etc. Thus the Guillemin–Sternbergconjecture in the setting of Bott’s definiton of quantisation simply reads�

G-index�D/ LM

��G D index�D/ LG

MG

�: (11)

In this form, the conjecture was proved by Meinrenken [50], who merely assumedthat M and MG are orbifolds.3 Also see [25, 33, 54, 70, 72] for other proofsand further references. A step towards a noncompact version of the Guillemin-Sternberg conjecture has been taken by Paradan in [54], where he considers actionsby compact groups on possibly noncompact manifolds. He proves that in this settingquantization commutes with reduction under certain conditions, that are met if themanifold in question is a coadjoint orbit of a semisimple Lie group, and the groupacting on it is a maximal compact subgroup. Our generalisation is more or less inan orthogonal direction: we assume that the quotient of the group action is compact,rather that the group itself.

As alluded to above, using standard ideas from the context of the Baum–Connes conjecture one can formulate the Guillemin–Sternberg conjecture also fornoncompact groups and manifolds. We specify our precise assumptions in Section2 below; for the moment we just mention that it seems impossible to even formulatethe conjecture unless we assume that the strongly Hamiltonian actionG ˚ .M;!/ isproper and cocompact (or G-compact, which means that M=G is compact). Whenthis is the case, we may pass from the compact to the noncompact case by makingthe following replacements (or lack of these) in the formalism:

1. Symplectic reduction is unchanged.

2. The definition of the Spinc Dirac operator D/ LM associated to .M;!/ isunchanged.

3. The quantisation of the reduced space (which is compact by our regularityassumptions) is unchanged.

4. The representation ring R.G/ is replaced by K0.C �.G//,4 i.e. the usual K0group of the group C �-algebra of G.5

5. The equivariant index G-index.D/ LM / 2 R.G/ is replaced by �GM .ŒD/LM �/ 2

K0.C�.G//, where

�GM WKG0 .M/!K0.C�.G// (12)

3Even that assumption turned out to be unnecessary [51].4 The use of K0.C�.G// instead of K0.C�

r .G// is actually quite unusual in the context of theBaum–Connes conjecture; we will clarify this point in footnote 7 below.

5See [7, 22, 63, 73] for the general K-theory of C�-algebras, see [14, 57] for group C�-algebras,and see [5, 64, 71] for the K-theory of group C�-algebras.


is the analytic assembly map [5, 53, 71], KG0 .M/ is the equivariant analyticalK-homology group defined by G ˚ M [5, 31, 35, 71], and ŒD/ LM � is the classin KG0 .M/ defined by the Spinc Dirac operator D/ LM .

6. Accordingly, the quantisation of the unreduced space .G ˚M;!/ is now givenby

Q.G ˚M;!/DG-index.D/ LM / 2K0.C �.G//; (13)

whereG-index.D/ LM / WD �GM .ŒD/ LM �/ (14)

purely as a matter of notation.6

7. The map RQ WR.G/! Z given by (10) is replaced by the map

RQ D�RG

�� WK0.C �.G//! Z (15)

functorially induced by mapRG W C �.G/ ! C given by f 7! R

Gf .g/dg

(defined on f 2 L1.G/ or f 2 Cc.G/ and extended to f 2 C �.G/ bycontinuity).7 Here we make the usual identification of K0.C/ with Z. Again,purely as a matter of notation we write this map as x 7! xG .

With these replacements and the notation (14), our generalized Guillemin–Sternberg conjecture is formally given by its original version (11). More precisely:

Conjecture 1.1 (Quantisation commutes with reduction) Let G be a unimodularLie group, let .M;!/ be a symplectic manifold, and let G ˚M be a proper stronglyHamiltonian action. Suppose 0 is a regular value of the associated momentum map.Suppose that the action is cocompact and admits an equivariant prequantum linebundle L. Assume there is an almost complex structure J on M compatible with!. Let D/ LM be the Dirac operator on M associated to J and coupled to L, and letD/ LG

MGbe the Dirac operator on the reduced space MG , coupled to the reduced line

bundle LG . Then �RG

�� ı�GM

�D/ LM

D index�D/ LG

MG

�:

In this paper we will prove:

Theorem 1.2 Under the assumptions listed in Subsection 2.1, Conjecture 1.1 istrue.

6This notation is justified by the fact that for G and M compact one actually has an equality in(14), provided one identifies K0.C�.G// with R.G/.

7 This extension would not be defined on C�r .G/ (unless G is amenable). The continuous

extension to C�.G/ is a trivial consequence of the fact thatRG is just the representation of

C�.G/ corresponding to the trivial representation of G on C by the usual correspondence betweennondegenerate representations of C�.G/ and continuous unitary representations of G [14, 57].


A special case of the situation described in Subsection 2.1 is the case whereG isa torsion-free discrete group acting freely and cocompactly onM . Then Conjecture1.1 follows from a result of Pierrot ([58], Theorème 3.3.2).

Example 1.3 SupposeG is a semisimple Lie group with maximal compact subgroupK, and suppose T � K is a maximal torus which is also a Cartan subgroup of G.Then by a theorem of Harish-Chandra, G has discrete series representations. LetO� � g� be the coadjoint orbit of G through the element � 2 t�. Then, if i� isa dominant integral weight, we would expect the quantisation of O�, coupled to asuitable line bundle, to be the class in K0.C �.G// that corresponds to the discreteseries representation H� whose lowest K-type has highest weight i�. In this caseConjecture 1.1 reduces to the uninteresting equality

Q�.O�/G

�DQ.;/D 0DRQ.Q.O�//:However, we can try to generalise Conjecture 1.1 so that the reduction map RQ

is replaced by a reduction map R�Q, which amounts to taking the multiplicity ofthe discrete series representation whose lowest K-type is the dominant weight i�instead of the multiplicity of the trivial representation. Furthermore, we note thatthe symplectic reduction .O�/� of O� at the value � is a point if �D �, and emptyotherwise. Therefore, we would expect that

Q�.O�/�

�D ı��DR�Q.H�/DR�Q.Q.O�//;

where ı�� is the Kronecker delta. A version of Conjecture 1.1 for discrete seriesrepresentations of semisimple Lie groups is proved in [32].

The truth of our generalized Guillemin–Sternberg conjecture for a special classof noncompact groups may be some justification for our specific formulation ofthe generalization, but in fact there is a much deeper reason why the “quantisationcommutes with reduction" issue should be stated in precisely the way we have given.Namely, in the above formulation the Guillemin–Sternberg conjecture is a specialcase of the (conjectural) functoriality of quantisation. This single claim summarizesa research program, of which the first steps may be found in the papers [40, 41, 43].In summary, one may define a “classical" category C and a “quantum" category Q,and construe the act of quantisation as a functor Q W C!Q. While the categories inquestion haven been rigorously constructed in [40] and [30, 34], respectively,8 theexistence of the functor Q is so far hypothetical. However, the picture that emerges

8The objects of C are integrable Poisson manifolds and its arrows are regular Weinstein dual


from the cases where Q has been constructed should hold in complete generality:deformation quantisation (in the C �-algebraic sense first proposed by Rieffel [61,62]) is the object side of Q, whereas geometric quantisation (in the sense of Bottas explained above) is the arrow side of Q. Moreover, in the setting of stronglyHamiltonian group actions as considered above, the “quantisation commutes withreduction" principle is nothing but the functoriality of quantisation (in cases wherethe functor has indeed been defined).9

Outline of the proof

We believe our generalized Guillemin–Sternberg conjecture to be true for allunimodular Lie groupsG, but for reasons of human frailty we are only able to proveit in this paper when G has a discrete normal subgroup � , such that the quotientgroupK WDG=� is compact.10 This incorporates a number of interesting examples.Our proof is based on:

1. The validity of the Guillemin–Sternberg conjecture in the compact case [33,50, 52, 51, 54, 70, 72];

2. Naturality of the assembly map for discrete groups [53];

3. Symplectic reduction in stages [39, 46, 49];

4. Quantum reduction in stages.

In this paper we show, among other things, that the Guillemin–Sternberg conjecturefor discrete (and possibly noncompact) groups G is a consequence of the secondpoint. For G as specified in the previous paragraph, naturality of the assembly mapimplies a K-equivariant version thereof. The third and fourth items are used in analmost trivial way, namely in setting up the following diagram, which provides an

pairs; see [40] for the meaning of the qualifiers. The arrows are composed by a generalization ofthe symplectic reduction procedure, and isomorphism of objects in C turns out to be the same asMorita equivalence of Poisson manifolds in the sense of Xu [76]. The category Q is nothing but theKasparov–Higson category KK, whose objects are separable C�-algebras and whose sets of arrowsare Kasparov’s KK-groups, composed with Kasparov’s intersection product. See [7, 30, 34].

9Let G ˚ .M;!/ define the Weinstein dual pair pt M ! g� in the usual way [74], the arrow! being given by the momentum map. Functoriality of quantisation means that Q.pt M !g�/�KK Q.g� - 0! pt/ D Q..pt M ! g�/�C .g� - 0! pt//. This equality is exactlythe same as (11). See [43].

10Such groups are automatically unimodular. The Guillemin–Sternberg conjecture may not hold inthe non-unimodular case; see [19].


outline of our proof.

Preq.G ˚M;!/

hD=�

M

i��

R.�/C

��

KG0 .M/�G

M �� K0.C�.G//

R.�/Q

��Preq.K ˚M� ;!�/

�D=�

M�

��

R.K/C

��

KK0 .M�/�K

M� �� K0.C�.K//

R.K/Q

��Preq..M�/K/;.!�/K/

�D=�

MG

�� K0

�.M�/K

� index �� Z

(16)

Here the following notation is used. We write

KDG=�I (17)

M� DM=�; (18)

as � is discrete (so that its associated momentum map ˆ� is identically zero,whence ˆ�1

� .0/ D M ). Furthermore, Preq.G ˚ M;!/ is defined relative to agiven Hamiltonian action of G on a symplectic manifold .M;!/, and consists ofall possible prequantisations .L;r;H/ of this action. A necessary condition forPreq.G ˚ M;!/ to be nonempty is that the cohomology class Œ!�=2� 2H 2.M;R/

be integral. (If the group G is compact, this condition is also sufficient.) We makethis assumption. Similarly, Preq.K ˚ M� ;!�/ is defined given the K-action onM� induced by the G-action on M , and Preq..M�/K ;.!�/K/ is just the set ofprequantisations of the symplectic manifold�

.M�/K ;.!�/K�Š .MG ;!G/I (19)

this isomorphism is a special and almost trivial case of the theorem on symplecticreduction in stages [39, 46]. The maps R.�/C and R.K/C denote Marsden–Weinsteinreduction (at zero) with respect to the groups � and K, respectively. We define thequantum counterparts of these maps by

R.�/Q WD

�P�

��I (20)

R.K/Q WD �RK��: (21)

Here�X

�

��W K0.C �.G//! K0.C

�.K// is the map functorially induced by the

mapP� W C �.G/! C �.G=�/ given by�P

�f�.�g/D

X�2�

f .�g/; (22)


initially defined on f 2 Cc.G/; see [23] for the continuity of this map.11 Finally, themaps involving the symbol ŒD/ �� are defined by taking the K-homology class of theDirac operator coupled to a given prequantum line bundle, (as outlined above andas explained in detail in the main body of the paper below). Thus the commutativityof the upper part of diagram (16) is the equality

�KM�

�D/ L�

M�

DR.�/Q

��GM ŒD/

LM ��; (23)

whereas commutativity of the lower part yields

indexD/ .L� /K.M� /K

DR.K/Q

��KM�

�D/ L�

M�

�: (24)

It is easily shown that RK ı

P� D

RG ; (25)

so that by functoriality of K0 one has

R.K/Q ıR.�/Q DR.G/Q ; (26)

with R.G/Q DRQ as in (15). Using (26) and

R.K/C ıR.�/C DR.G/C WDRC ; (27)

which is a mere rewriting of (19), we see that the outer diagram in (16) is equal to

Preq.G ˚M;!/

RC

��

Q �� K0.C�.G//

RQ

��Preq.MG ;!G/

Q �� Z:

(28)

Clearly, commutativity of (28) is precisely the commutativity of diagram (1) withthe post-modern meaning we have given to its ingredients. Since diagram (28)commutes when the two inner diagrams in diagram (16) commute, the latter wouldprove our generalized Guillemin–Sternberg conjecture. Now the lower diagramcommutes by the validity of the Guillemin–Sternberg conjecture for compact K,whereas the upper diagram decomposes as

Preq.G ˚M;!/

hD=�

M

i��

R.�/C

��

KG0 .M/�G

M ��

V�

��

K0.C�.G//

R.�/Q

��Preq.K ˚M� ;!�/

�D=�

M�

�� KK0 .M�/

�KM� �� K0.C

�.K//;

(29)

11This map can more generally be defined for any closed normal subgroup N of G, cf. AppendixA.


where V� is a map defined in Subsection 3.1 (the V stands for Valette, who was thefirst to write this map down in a more special context). Verifying the commutativityof the two inner diagrams of diagram (29), then, forms the main burden of our proof.

The commutativity of the right-hand inner diagram follows from a general-ization of the naturality of the assembly map for discrete groups as proved byValette [53] to possibly nondiscrete groups. This is dealt with in Appendix A. Thecommutativity of the left-hand inner diagram is Theorem 2.9:

V� ŒD/ LM �D�D/ L�

M�

: (30)

The proof of this result occupies Sections 3 and 4.In Section 3, we compute the image under the map V� of a K-homology

class associated to a general equivariant, elliptic, symmetric, first order differentialoperator D on a �-vector bundle E over a �-manifold M . If the action of � onM is free, as we assume, then the quotient space E=� defines a vector bundle overM=� . The operator D induces an operator D� on this quotient bundle. It turns outthat the homomorphism V� maps the class associated toD to the class associated toD� .

In Section 4, we show that ifD/ LM is the Dirac operator on a symplectic manifoldM , coupled to a prequantum line bundle L, then the operator

�D/ LM

�� from theprevious paragraph is precisely the Dirac operator on the quotient M=� coupled tothe line bundle L=� .

As an encore, in Section 5 we give an independent proof of our generalizedGuillemin–Sternberg conjecture for the case that G is discrete and abelian. Thisproof, based on a paper by Lusztig [45] (see also [5], pp. 242–243) givesconsiderable insight in the situation. It is based on an explicit computation of theimage under ��M of a K-homology class ŒD� associated to a �-equivariant ellipticdifferential operator D on a �-vector bundle E over a �-manifold M . Because inthis case C �.�/ Š C. O�/ (with O� the unitary dual of �), this image corresponds tothe formal difference of two equivalence classes of vector bundles over O� . Thesebundles are described as the kernel and cokernel of a ‘field of operators’

�D˛�˛2 O�

on a ‘field of vector bundles’�E˛!M=�

�˛2 O� . The operators D˛ and the bundles

E˛ are constructed explicitly from D and E, respectively. The quantum reductionof the class ��M ŒD� is the index of the operator D1 on E1 ! M=� , where 1 2 O�is the trivial representation. Because D1 is the operator D� mentioned above, theGuillemin–Sternberg conjecture follows from the computation in Section 4.

Finally, in Section 6 we check the discrete abelian case in an instructiveexplicit computation. We will see that the quantisation of the action of Z2 onR2 corresponds to a certain line bundle over the two-torus T2 D OZ2. Thequantum reduction of this K-theory class is its rank, the integer 1. This is also


the quantisation of the reduced space T2 D R2=Z2, as can be seen either directlyor by applying Atiyah-Singer for Dirac operators. Although this is the simplestexample of Guillemin-Sternberg for noncompact groups and spaces, the details arenontrivial and, in our opinion, well worth spelling out.

2. Assumptions and result

We now state the assumptions under which we will prove our generalisedGuillemin–Sternberg conjecture, i.e. Theorem 1.2. These assumptions are mainlyused in the proof of our key intermediate result, Theorem 2.9, which is proved inSections 3 and 4.

We first fix some notation and assumptions. If M is a manifold, then the spacesof vector fields and differential forms on M are denoted by X.M/ and ��.M/,respectively. The symbol y denotes contraction of differential forms by vector fields.If the manifold M is equipped with an almost complex structure, then we have thespace �0;�.M/ of differential forms on M of type .0;�/. Unless stated otherwise,all manifolds, maps and actions are supposed to be C1.

If a vector bundle E ! M is given (which is supposed to be complex unlessstated otherwise), then the space of smooth sections of E is denoted by C1.M;E/.If M is equipped with a measure, and the bundle E carries a metric, then L2.M;E/the space of square integrable sections of E. The space of differential forms on Mwith coefficients in E is denoted by ��.M IE/, and similarly we have the space�0;�.M IE/ for almost complex manifolds. If F ! M is another vector bundle,and ' W E ! F is a homomorphism of vector bundles, then composition with 'gives a homomorphism of C1.M/-modules

Q' W C1.M;E/! C1.M;F /:

If a group G acts onM , and if E is a G-vector bundle overM , then we have thecanonical representation of G on C1.M;E/ given by .g � s/.m/D g � s.g�1m/, forg 2 G and s 2 C1.M;E/. A superscript ‘G’ denotes the subspace of G-invariantelements. Thus we obtain for example the vector spaces C1.M;E/G , �0;�.M/G ,etc. The Lie algebra of a Lie group is denoted by a lower case Gothic letter, so thatfor example the group G has the Lie algebra g.

In the context of Hilbert spaces and Hilbert C �-modules, we denote the spacesof bounded, compact and finite-rank operators by B, K and F , respectively.

The spaces of continuous functions, bounded continuous functions, continuousfunctions vanishing at infinity and compactly supported continuous functions on atopological space X are denoted by C.X/, Cb.X/, C0.X/ and Cc.X/, respectively.


2.1. Assumptions

Let .M;!/ be a symplectic manifold, and let G be a Lie group. Suppose that G hasa discrete, normal subgroup � , such that the quotient group K WDG=� is compact.For example,

� G DK, � D feg with K a compact Lie group,

� G D � discrete,

� G DRn, � D Zn so that G=� is the torus Tn,

or direct products of these three examples. In fact, if G is connected, then thesubgroup � must be central, and G is the product of a compact group and a vectorspace.

The assumption that G=� is compact is not needed in the proof of Theorem 2.9;it is only made so that we can apply the Guillemin–Sternberg theorem for compactgroups in diagram (16).

Suppose that G acts on M , and that the following assumptions hold.

1. The action is proper.

2. The action preserves the symplectic form !.

3. The quotient space M=G is compact.

4. The action is Hamiltonian,12 in the sense that there exists a map

ˆ WM ! g�;

that is equivariant with respect to the co-adjoint representation ofG in g�, suchthat for all X 2 g,

dˆX D�XMy!:

Here ˆX is the function on M obtained by pairing ˆ with X and XM is thevector field on M induced by X .

5. The discrete subgroup � acts freely on M , and the whole group G acts freelyon the level set ˆ�1.0/.

6. The symplectic manifold .M;!/ admits a G-equivariant prequantisation. Thatis, there is a G-equivariant complex line bundle

L!M;

12Sometimes an action is called ‘Hamiltonian’ as opposed to ‘strongly Hamiltonian’ if it admits amomentum map that is not necessarily equivariant or Poisson. We will not use this terminology; forus the word ‘Hamiltonian’ always means ‘strongly Hamiltonian’.


equipped with a G-invariant Hermitian metric H , and a Hermitian connectionr with curvature two-form r2 D 2�i!, which is equivariant as an operatorfrom C1.M;L/ to �1.M IL/.

7. There is a G-invariant almost complex structure J on TM , such that

B.�;�/ WD !.�;J �/

defines a Riemannian metric on M .

8. The manifold M is complete with respect to the Riemannian metric B .

Ad 3. Because the quotientG=� is compact, compactness ofM=G is equivalentto compactness of M=� . The latter assumption is used in the proof of Lemma 3.3(where � is replaced by a general closed normal subgroup N ), but it is not essential.Furthermore, in the definitions of K-homology in [5, 53] it is assumed that theorbit spaces of the group actions involved are compact.13 Finally, compactness ofM=� allows us to apply the Guillemin-Sternberg conjecture for compact groupsand spaces to this quotient.

Ad 4. The map ˆ is called a momentum map of the action. For connectedgroups, the assumption that ˆ is equivariant is equivalent to the assumpion that itis a Poisson map with respect to the negative Lie-Poisson structure on g� (see forexample [39], Corollary III.1.2.5 or [47], §11.6). Note that if G D � is discrete,then the action is automatically Hamiltonian. Indeed, Lie.�/D f0g, so that the zeromap is a momentum map.

Ad 5. Freeness of the action of � onM implies that the quotientM=� is smooth,and that a �-vector bundle E ! M induces a vector bundle E=� ! M=� . Andif G acts freely on ˆ�1.0/, then it follows from de definition of momentum mapsthat 0 is a regular value of ˆ (Smale’s lemma [66]), so that the reduced spaceMG isa smooth manifold. If freeness is replaced by local freeness in Assumption 5, thenM=� and MG are orbifolds. In that case, a quotient vector bundle E=� ! M=�

can be replaced by ‘the vector bundle over M=� whose space of smooth sections isC1.M;E/� (see Proposition 2.8).14

Ad 6. Example 2.3 below shows that it is not always obvious if this assumptionis satisfied. Equivariance of the connection r implies that the Dolbeault–Diracoperator on M , coupled to L via r (Definition 2.5) is equivariant.

The Kostant formulaX 7! �rXM

C 2�iˆX13K-homology can be defined more generally, but the compactness assumption makes things a little

easier.14It is not a good idea to make the stronger assumption that G acts freely onM . For in that case, G

must be discrete (see Remark 2.2).


defines a representation of g in the space of smooth sections of L. In the literatureon the Guillemin-Sternberg conjecture, it is usually assumed that the action of G onL is such that the corresponding representation of g in C1.M;L/ is given by theKostant formula. Then, if the group G is connected, the connection r satisfiesgrvg�1 D rg �v for all g 2 G and v 2 X.M/. This property is equivalent toequivariance of the connection r in the sense of assumption 6.

If the manifold M is simply connected and the group G is discrete, thenHawkins [27] gives a procedure to lift the action of G on M to a projective actionon the trivial line bundle overM , such that a given connection is equivariant. Undera certain condition (integrality of a group cocycle), this projective action is an actualaction.

Ad 7. An equivalent assumption is that there exists a G-invariant Riemannianmetric on M . (See e.g. [25], pp. 111-112.)

Ad 8. This assumption implies that the Dirac operator on M is essentially self-adjoint on its natural domain (see Subsection 2.2).

The assumptions and notation above will be used in this section and in Section 4. InSection 3, we will work under more general assumptions.

Remark 2.1 If the group G is compact, then some of these assumptions are alwayssatisfied. First of all, the action is automatically proper. Furthermore, if thecohomology class Œ!� 2 H 2

dR.M/ is integral, then a prequantum line bundle exists,and the connection can be made equivariant by averaging over G. Also, averagingover G makes any Riemannian metric G-invariant, so that assumption 7 is alsosatisfied. And finally, sinceM=G is compact, so isM . In particular,M is complete,so assumption 8 is satisfied.

Remark 2.2 If the action of G on M is (locally) free and Hamiltonian, and M=Gis compact, then G must be discrete. Indeed, if the action is locally free then bySmale’s lemma the momentum map ˆ is a submersion, and in particular an openmapping. And since it is G-equivariant, it induces

ˆG WM=G! g�=Ad�.G/;

which is also open. So, since M=G is compact, the image

ˆG.M=G/� g�=Ad�.G/

is a compact open subset. Because g�=Ad�.G/ is connected15, it must therefore becompact. This, however, can only be the case (under our assumptions) when G is

15If G DK is a compact connected Lie group, then k�=Ad�.K/ is a Weyl chamber.


discrete. Indeed, we have

Ad�.G/Š Ad�.K/�GL.k�/ŠGL.g�/:

So Ad�.G/ is compact, and g�=Ad�.G/ cannot be compact, unless g� D 0, i.e. G isdiscrete.

Example 2.3 Let M D C, with coordinate z D q C ip, and the standard complexstructure. We equip M with the symplectic form ! D dp^ dq.

Consider the group G D � D ZC iZ�C. We let it act on M by addition:

.kC i l/ � z D zC kC i l;

for k;l 2 Z, z 2C.Consider the trivial line bundle L D M � C ! M . We define an action of

ZC iZ on L by letting the elements 1;i 2 ZC iZ act as follows:

1 � .z;w/D .zC 1;w/Ii � .z;w/D .zC i;e�2�izw/;

for z;w 2C. Define a Hermitian metric H on L by

H..qC ip;w/;.qC ip;w0//D e2�.p�p2/w Nw0:

This metric is ZC iZ-invariant, and the connection

r D d C 2�ipdzC�dp

is Hermitian, ZC iZ-invariant, and has curvature form 2�i!.The details of this example are worked out in Section 6, where we also give

some motivation for these formulae.

Example 2.4 Suppose .M1;!1/ is a compact symplectic manifold, K is a compactLie group, and let a proper Hamiltonian action of K on M1 be given. Let ˆ be themomentum map, and suppose K acts freely on ˆ�1.0/. Suppose Œ!� is an integralcohomology class. (These assumptions are made for example by Tian & Zhang[70].) Let � be a discrete group acting properly and freely on a symplectic manifold.M2;!2/, leaving !2 invariant. Suppose that M2=� is compact, and that there isan equivariant prequantum line bundle over M2. Then the direct product action ofK �� on M1 �M2 satisfies the assumptions of this section.


2.2. The Dirac operator

The prequantum connection r on L induces a differential operator

r W�k.M IL/!�kC1.M IL/;such that for all ˛ 2�k.M/ and s 2 C1.M;L/,

r.˛˝ s/D d˛˝ sC .�1/k˛^rs:Let �0;� be the projection

�0;� W��C.M IL/!�0;�.M IL/:

Composing the restriction to �0;q.M IL/ of the complexification of r with �0;�,we obtain a differential operator

N@L WD �0;� ır W�0;q.M IL/!�0;qC1.M IL/:Let

N@�L W�0;qC1.M IL/!�0;q.M IL/

be the formal adjoint of N@L with respect to the L2-inner product on compactlysupported forms.

Definition 2.5 (Dolbeault–Dirac operator) The Dolbeault–Dirac operator on M ,coupled to L via r is the differential operator

D/ LM WD N@LC N@�L W�0;�.M IL/!�0;�.M IL/:

This operator isG-equivariant by equivariance of the connection r (assumption6) and invariance of the almost complex structure J (assumption 7).

The Dolbeault–Dirac operator defines an unbounded symmetric operator on theHilbert space of L2-sections

HLM WD L2.M;

V0;�T �M ˝L/;

with respect to the Liouville measure dm on M . The metric onV0;�

T �M ˝ Lcomes from the given metric H on L and from the Riemannian metric B D !.�;J �/on TM . Because M is metrically complete, the closure of D/ LM is a self-adjointoperator on HL

M .16 So we can apply the functional calculus (see e.g. [59]) to definethe bounded operator FLM WD b

�D/ LM

�on HL

M , where b is a normalising function[31]:

16This follows from the connection between Dirac operators and Riemannian metrics as given forexample in [11], section VI.1, combined with Section 10.2 of [31]. See also [75] and page 96 of [20].


Definition 2.6 A smooth function b W R! R is called a normalising function if ithas the following three properties:

� b is odd;

� b.t/ > 0 for all t > 0;

� limt!˙1b.t/D˙1.

Let C0.M/ denote the C �-algebra of continuous functions on M that vanish atinfinity, and let B.HL

M / be the the C �-algebra of bounded operators on HLM . Let

�M W C0.M/! B.HLM /

be the representation of C0.M/ on HLM defined by multiplication of sections by

functions. Then the triple .HLM ;F

LM ;�M / defines a K-homology class�

D/ LM WD ŒHL

M ;FLM ;�M � 2KG0 .M/;

which is independent of b. See [5, 31, 53] for the definition of K-homology, and inparticular Theorem 10.6.5 in [31] for the claim that

�D/ LM

defines a K-homology

class.

Remark 2.7 The Dolbeault–Dirac operator has the same principal symbol as theSpinc Dirac operator associated to the almost complex structure J on M and theline bundle L (see e.g. [16], page 48), namely the Clifford action of T �M onV0;�

T �M ˝L. So the two operators define the same class in K-homology. (Thelinear path between the operators provides a homotopy.) Thus, if we consider theK-homology class ŒD/ LM �, we may take D/ LM to be either the Spinc Dirac operator orthe Dolbeault–Dirac operator.

2.3. Reduction by �

Because the subgroup � of G is discrete, the symplectic quotient M� of M by � isequal to the orbit manifold M=� . The symplectic form !� on M=� is determinedby

p�!� D !;with p WM !M=� the quotient map. The action of G on M descends to an actionof G=� on M=� , which satisfies the assumptions of Section 2.1 (with M , !, andG replaced by M=� , !� and G=� , respectively). The quotient L=� turns out tobe the total space of a prequantum line bundle over M=� . This is implied by thefollowing fact.


Proposition 2.8 LetH be a group acting properly and freely on a manifoldM . Letq WE!M be an H -vector bundle. Then the induced projection

qH WE=H !M=H

defines a vector bundle over M=H .Let C1.M;E/H be the space of H -invariant sections of E. The linear map

E W C1.M;E/H ! C1.M=H;E=H/; (31)

defined by E .s/.H �m/DH � s.m/;

is an isomorphism of C1.M/H Š C1.M=H/-modules.

Hence the quotient spaceL=� is a complex line bundle overM=� , and its spaceof sections is C1.M=�;L=�/Š C1.M;L/� .

The connection r on L is G-equivariant, so it defines a G=�-equivariantconnection r� on L=� as follows. Let p W M ! M=� be the quotient map. Itstangent map Tp W TM ! T .M=�/ induces an isomorphism of vector bundles overM=�

Tp� W .TM/=�! T .M=�/:

LeteTp� W C1.M=�;.TM/=�/

Š�! X.M=�/

be the isomorphism of C1.M=�/ modules induced by Tp� . Consider theisomorphism of C1.M/� Š C1.M=�/-modules

' W X.M/� TM��! C1.M=�;.TM/=�/

eTp�

��! X.M=�/:

Let v 2 X.M/� be a �-invariant vector field on M . Then the operator rv is �-equivariant, and hence maps �-invariant scetions of L to invariant sections. Thecovariant derivative r�

'.v/on C1.M=�;L=�/ is defined by the commutativity of

the following diagram:

C1.M=�;L=�/r�

'.v/ �� C1.M=�;L=�/

C1.M;L/� L Š

��

rv �� C1.M;L/� :

L Š��

A computation shows that r� satisfies the properties of a connection, and that itscurvature is �r��2 D 2�i!� :


Furthermore, the G-invariant Hermitian metric H on L descends to a G=�-invariant Hermitian metric H� on L=� , and the connection r� is Hermitian withrespect to this metric. Finally, the G-invariant almost complex structure J on TMinduces a G=�-invariant almost complex structure J � on T .M=�/ Š .TM/=� .The corresponding Riemannian metric is denoted by

B� D !�.�;J � �/:

From the Dirac operator D/ L=�M=�

on M=� associated to the almost complexstructure J � , coupled to the line bundle L=� via r� , we form the bounded operatorFL=�

M=�WD b

�D/ L=�M=�

�(where b is a normalising function) on the Hilbert space

HL=�

M=�WD L2

�M=�;

V0;�T �.M=�/˝L=�

�;

which is defined with respect to the metrics onV0;�

T �.M=�/ and L=� comingfrom those on

V0;�T �M and L respectively, and the measure dO on M=� defined

as follows.Let U � M be a fundamental domain for the �-action. That is, U is an open

subset, � �U is dense inM , and ifm is a point in U , and � 2 � is such that � �m 2 U ,then � D e. Then for all measurable functions f on M=� we defineZ

M=�

f .O/dO WDZU

p�f .m/dm; (32)

where p WM !M=� is the quotient map. If V is another fundamental domain, thesubsets � �U and � � V differ by a set of measure zero, so this definition does notdepend on the choice of the fundamental domain. An equivalent way of definingdO is to say that the dO-measure of a measurable subset A�M=� equals the dm-measure of the subset p�1.A/\U ofM . And since dm is the Liouville measure on.M;!/, the measure dO is precisely the Liouville measure on .M=�;!�/.

We then have the K-homology classhD/ L=�M=�

iWD ŒHL=�

M=�;F

L=�

M=�;�M=� � 2KG=�0 .M=�/:

2.4. The main result

In Subsection 3.1 we define a homomorphism

V� WKG0 .M/!KG=�0 .M=�/;


such that the following diagram commutes:

KG0 .M/�G

M ��

V�

��

K0.C�.G//

R.�/Q

��KG=�0 .M=�/

�G=�

M=�� K0.C�.G=�//:

(33)

Here �GM and �G=�

M=�are analytic assembly maps (see [5, 71, 53]), and the

homomorphismR.�/Q is defined in (20). In Appendix A, we sketch how to generalise

Valette’s proof in [53] of commutativity of diagram (33) (‘naturality of the assemblymap’) for discrete groups to the nondiscrete case.

The main step in our proof of Theorem 1.2 is the following:

Theorem 2.9 The homomorphism V� maps the K-homology class of the Diracoperator D/ LM to the K-homology class of the Dirac operator D/ L=�

M=�on the reduced

space M=�:

V��ŒD/ LM �

�D hD/ L=�M=�

i2KG=�0 .M=�/:

As we noted in the Introduction, Theorem 1.2 follows from Theorem 2.9,the naturality of the assembly map (diagram (33)) and the Guillemin–Sternbergconjecture for compact G and M .

Remark 2.10 We will actually prove a stronger result than Theorem 2.9. Write

V��ŒHL

M ;FLM ;�M �

�D ŒH� ;F� ;�� :Then there is a unitary isomorphism

WH� !HL=�

M=�

that intertwines the pertinent representations of G=� and of C.M=�/, and theoperators F� and FL=�

M=�.

3. Differential operators on vector bundles

In this section, we will compute the image under the homomorphism VN indiagram (33) of a K-homology class associated to an equivariant elliptic first orderdifferential operator on a vector bundle over a smooth manifold. The result isCorollary 3.13. In Section 4 we will see that Theorem 2.9 is a special case ofCorollary 3.13.


Let G be a unimodular Lie group, and let N be a closed normal subgroup ofG. Let dg and dn be Haar measures on G and N respectively. Let M be a smoothmanifold on whichG acts properly, such that the action ofN onM is free. SupposeM=N is compact.17

3.1. The homomorphism VN

Let us briefly state the definition of the homomorphism

VN WKG0 .M/!KG=N0 .M=N/:

For details we refer to [53]. Let H be a Z2-graded Hilbert space carrying aunitary representation of G, F a G-equivariant bounded operator on H , and � arepresentation of C0.M/ in H that is G-equivariant in the sense that for all g 2 Gand f 2 C0.M/, one has g�.f /g�1 D �.g � f /. Suppose .H;F;�/ defines a K-homology cycle. Then

VN ŒH;F;�� WD ŒHN ;FN ;�N �;

with HN , FN and �N defined as follows.18

Consider the subspace Hc WD �.Cc.M//H � H and the sesquilinear form.�;�/N on Hc given by

.;�/N WDZN

.;n � �/Hdn;

for ;� 2Hc . This form turns out to be positive semidefinite. Consider the quotientspace of Hc by the kernel of this form, and complete this quotient in the innerproduct .�;�/N . This completion is HN .

Next, we use the fact that any K-homology class can be represented by a cyclewhose operator is properly supported:

Definition 3.1 The operator F is properly supported if for every f 2 Cc.M/ thereis an h 2 Cc.M/ such that �.h/F�.f /D F�.h/.

Suppose F is properly supported. Then it preserves Hc , and the restriction ofF toHc is bounded with respect to the form .�;�/N . Hence F jHc

induces a boundedoperator FN on HN by continuous extension. The representation � extends to themultiplier algebra Cb.M/ of C0.M/. The algebra C0.M=N/ can be embedded

17Compactness ofM=N is used in the proof of Lemma 3.3, but not in an essential way (see Footnote21).

18The construction below originated in Rieffel’s theory of induced representations of C�-algebras[60], which independently found its way into the Baum–Connes conjecture [5] and into the theory ofconstrained quantisation [38, 39].


into Cb.M/ via the isomorphism C.M=N/ Š C.M/N , and then we can use anargument similar to the one used in the definition of FN to show that � induces arepresentation �N of C0.M=N/ in HN .

3.2. Spaces of L2-sections

Now let q W E ! M be a G-vector bundle, equipped with a G-invariant metric.�;�/E . Let dm be a G-invariant measure on M , and let L2.M;E/ be the space ofsquare-integrable sections of E with respect to this measure. Let �M W C0.M/!B.L2.M;E// be the representation defined by multiplying sections by functions.Let L2.M;E/N be the Hilbert space constructed from L2.M;E/ as in the definitionof the homomorphism VN . We will show that L2.M;E/N is naturally isomorphic19

to the Hilbert space L2.M=N;E=N/ of square-integrable sections of the quotientvector bundle

qN WE=N !M=N

(see Proposition 2.8). The L2-inner product on sections of E=N is defined via themetric on E=N induced by the one on E, and the measure dO on M=N with theproperty that for all measurable sections20 ' WM=N !M and all f 2 Cc.M/,Z

M

f .m/dmDZM=N

ZN

f .n �'.O//dndO (34)

(see [8], Proposition 4b, p. 44). If N is discrete and dn is the counting measure,then the measure dO from (32) satisfies property (34).

Note that in this example, the space

L2c.M;E/ WD �.Cc.M//L2.M;E/

is the space of compactly supported L2-sections of E. Consider the linear map

W L2c.M;E/! L2.M=N;E=N/; (35)

defined by

.s/.Nm/ WDN �ZN

n � s.n�1m/dn;

for all s 2 L2c.M;E/ and m 2M . Because s is compactly supported and the actionis proper, the integrand is compactly supported for all m 2M .

19A natural isomorphism between Hilbert spaces is an isomorphism defined without choosing basesof the spaces in question.

20Measurable in the sense that the inverse image of any Borel measurable subset of M is Borelmeasurable in M=N .


Proposition 3.2 The map induces a natural G=N -equivariant unitary isomor-phism

W L2.M;E/N Š�! L2.M=N;E=N/: (36)

Proof: It follows from a lengthy but straightforward computation that the map isisometric, in the sense that for all s 2 L2c.M;E/,

k.s/kL2.M=N;E=N/ D kskN ;

where k � kN is the norm corresponding to the inner product .�;�/N . Furthermore, is surjective, see Lemma 3.3 below. By these two properties, induces a bijectivelinear map

W L2c.M;E/=K! L2.M=N;E=N/; (37)

where K is the space of sections s 2 L2c.M;E/ with kskN D 0. This mapis a norm preserving linear isomorphism from L2c.M;E/=K onto the completespace L2.M=N;E=N/. Hence the space L2c.M;E/=K is already complete, so thatL2.M;E/N D L2c.M;E/=K:

So (37) is actually a unitary isomorphism

W L2.M;E/N ! L2.M=N;E=N/:

The fact that N is a normal subgroup implies that this isomorphism intertwines thepertinent representations of G=N .

Lemma 3.3 The map in (35) is surjective.

Proof: Let � 2 L2.M=N;E=N/. We will construct a section s 2 L2c.M;E/ suchthat .s/D � , using the following diagram:

EpE ��

q

��

E=N

qN

��M

p ��M=N:

Here the horizontal maps are quotient maps and define principal fibre bundles, andthe vertical maps are vector bundle projections.

Let fUj g be an open cover of M=N that admits local trivialisations

j W p�1.Uj /Š�! Uj �N

�Nj W q�1N .Uj /

Š�! Uj �E0:


Here E0 is the typical fibre of E. BecauseM=N is compact, the cover fUj gmay besupposed to be finite. Via the isomorphism of vector bundles p�.E=N/ Š E, thetrivialisations �Nj induce local trivialisations of E:

�j W q�1.p�1.Uj //Š�! p�1.Uj /�E0:

And then, we can form trivialisations

Ej W p�1E .q�1

N .Uj //Š�! q�1

N .Uj /�N;by

p�1E .q�1

N .Uj //D q�1.p�1.Uj //Šp�1.Uj /�E0 via �jŠUj �N �E0 via jŠ q�1

N .Uj /�N via �Nj :

Here the symbol ‘Š’ indicates an N -equivariant diffeomorphism. It follows fromthe definition of the trivialisation �j that Ej composed with projection onto q�1

N .Uj /

equals pE , so that Ej is indeed an isomorphism of principal N -bundles.For every j , define the section sj 2 L2.M;E/ by

sj . �1j .O;n//D � Ej ��1.�.O/;n/

for all O 2 Uj and n 2N , and extended by zero outside p�1.Uj /. By compactness21

of M=N , there is a compact subset eC �M that intersects all N -orbits. Let K � Nbe a compact subset of dn-volume 1, and set C WD K �eC . Then for all m 2M , thevolume of the compact set

Vm WD fn 2N In�1m 2 C gis at least 1. Define the section Qs of E by

Qs.m/D P

j sj .m/ if m 2 C0 if m 62 C .

21 This is the only place where compactness of M=N is used (the covering fUj g may also belocally finite). And even here, this assumption is not essential: it follows from this proof that allcompactly supported L2-sections ofE=N are in the image of . Hence has dense image, so that theinduced map fromL2.M;E/N toL2.M=N;E=N/ is surjective. By small adaptations to the proofs ofPropositions 3.2 and 3.8, everything in this section still applies if M=N is noncompact. But becausewe assume that the quotient M=� is compact anyway, we will take the lazy option and suppose thatM=N is compact.


Then Qs 2 L2c.M;E/, and for all m 2M ,

.Qs/.Nm/DXj;

Nm2Uj

ZVm

pE�n � sj .n�1m/

�dn

DXj;

Nm2Uj

ZVm

pE�� Ej��1

.�.Nm/;n � j .n�1m//�dn;

where .Nm; j .n�1m// WD j .n�1m/. Now since pE ı� Ej��1 is projection onto

q�1N .Uj /, the latter integral equals

#fj INm 2 Uj gvol.Vm/�.Nm/:

Setting ˆ.m/ WD #fj INm 2 Uj gvol.Vm/ gives a measurable function ˆ on Mwhich is bounded below by 1 and N -invariant by invariance of dn. Hence

s WD 1

ˆQs;

is a section s 2 L2c.M;E/ for which .s/D � .

3.3. Differential operators

Let G and E ! M be as in Section 3.2. Let D W C1.M;E/ ! C1.M;E/ be aG-equivariant first order differential operator that is symmetric with respect to theL2-inner product on compactly supported sections. Then D defines an unboundedoperator on L2.M;E/. We assume that this operator has a self-adjoint extension,which we also denote by D.

FUNCTIONAL CALCULUS AND PROPERLY SUPPORTED OPERATORS

Applying the functional calculus to the self-adjoint extension of D, we define thebounded, self-adjoint operator b.D/ on L2.M;E/, for any bounded measurablefunction b on R.22 The operator b.D/ is G-equivariant because of the followingresult about functional calculus of unbounded operators, which follows directlyfrom the definition as given for example in [59], page 261.

Lemma 3.4 Let H be a Hilbert space, and let D � H be a dense subspace. Leta W D ! H be a self-adjoint operator. Let H 0 be another Hilbert space, and letT WH !H 0 be a unitary isomorphism. Let f be a measurable function on R. Then

Tf .a/T �1 D f .TaT �1/:22If D is elliptic and b is a normalising function , then

�L2.M;E/;b.D/;�M

�is an equivariant

K-homology cycle over M (see Theorem 10.6.5 in [31]).


We will later consider the case where�L2.M;E/;b.D/;�M

�is a K-homology

cycle, and apply the map VN to this cycle. It is therefore important to us that theoperator b.D/ is properly supported (Definition 3.1) for well-chosen functions b:

Proposition 3.5 If b is a bounded measurable function with compactly supported(distributional) Fourier transform Ob, then the operator b.D/ is properly supported.

The proof of this proposition is based on the following two facts, whose proofscan be found in [31], Section 10.3.

Proposition 3.6 If b is a bounded measurable function on R with compactlysupported Fourier transform, then for all s;t 2 C1

c .M;E/,�b.D/s;t

�L2.M;E/

D 1

2�

ZR

�ei�Ds;t

�L2.M;E/

Ob.�/d�:

This is Proposition 10.3.5. from [31]. By Stone’s theorem, the operator ei�D ischaracterised by the requirements that � 7! ei�D is a group homomorphism from R

to the unitary operators on L2.M;E/, and that for all s 2 C1c .M;E/,

@

@�

ˇ�D0

ei�Ds D iDs:

Lemma 3.7 Let s 2 C1c .M;E/, and let h 2 C1

c .M/ be equal to 1 on the supportof s. Let � 2R such that j�j< kŒD;�M .h/�k�1. Then

suppei�Ds � supph:

This follows from the proof of Proposition 10.3.1. from [31].Proof of Proposition 3.5: Suppose supp Ob � Œ�R;R�. Let f 2 Cc.M/, and chooseh 2 C1

c .M/ such that h equals 1 on the support of f , and that kŒD;�M .h/�k � 1R

.Let 1M be the constant function 1 on M . Then by Lemma 3.7,

�M .1M � h/ei�D�M .f /D 0; (38)

for all � 2��R;RŒ. Here we have extended the nondegenerate representation �M ofC0.M/ on L2.M;E/ to the multiplier algebra Cb.M/ of C0.M/. So by Proposition3.6, we have for all s;t 2 C1

c .M;E/,��M .1M � h/b.D/�M .f /s;t

�L2.M;E/

D �b.D/�M .f /s;�M .1M � Nh/t�L2.M;E/

D 1

2�

ZR

�ei�D�M .f /s;�M .1M � Nh/t

�L2.M;E/

Ob.�/d�

D 1

2�

Z R

�R��M .1M � h/ei�D�M .f /s;t

�L2.M;E/

Ob.�/d�D 0;


by (38). So �1��M .h/

�b.D/�M .f /D �M .1M � h/b.D/�M .f /D 0;

and hence b.D/ is properly supported.

THE IMAGE OF b.D/ UNDER VN

Now suppose that D is elliptic and that b is a normalising function with compactlysupported Fourier transform,23 so that b.D/ is the kind of operator that definesa K-homology class over M . Because b.D/ is properly supported, it preservesL2c.M;E/ and the construction used in the definition of the map VN applies to b.D/.The resulting operator b.D/N on L2.M;E/N is defined by commutativity of thefollowing diagram:

L2c.M;E/��

b.D/

��

L2.M;E/N

b.D/N

��L2c.M;E/

�� L2.M;E/N :

On the other hand, the operator D induces an unbounded operator onL2.M=N;E=N/, because it restricts to

QDN W C1.M;E/N ! C1.M;E/N ;

from which we obtain

DN WD �1EQDN E W C1.M=N;E=N/! C1.M=N;E=N/

(see Proposition 2.8). We regardDN as an unbounded operator onL2.M=N;E=N/.It is symmetric with respect to the L2-inner product, and hence essentially self-adjoint by [31], Corollary 10.2.6. We therefore have the bounded operator b.DN /

on L2.M=N;E=N/.Our claim is:

Proposition 3.8 The isomorphism from Proposition 3.2 intertwines the operatorsb.D/N and b.DN /:

L2.M;E/N� ��

b.D/N��

L2.M=N;E=N/

b.DN /

��L2.M;E/N

� �� L2.M=N;E=N/:

23If g is a smooth, even, compactly supported function on R, and f WD g � g is its convolutionsquare, then b.�/ WD RR

ei�x�1ix f .x/dx is such a function (see [31], Exercise 10.9.3).


We will prove this claim by reducing it to the commutativity of another diagram.This diagram involves the Hilbert space QL2.M=N;E=N/, which is defined as thecompletion of the space C1.M;E/N in the inner product

.�; / WDZM=N

��.'.O//; .'.O/

�EdO;

for any measurable section ' W M=N ! M . The map E from Proposition 2.8extends continuously to a unitary isomorphism

Q E W QL2.M=N;E=N/! L2.M=N;E=N/:

The unbounded operator QDN on QL2.M=N;E=N/ is essentially self-adjointbecause DN is, and because Q E intertwines the two operators. Hence we haveb. QDN / 2 B� QL2.M=N;E=N/�. We will deduce Proposition 3.8 from

Lemma 3.9 The following diagram commutes:

L2c.M;E/

RNn�

��

b.D/

��

QL2.M=N;E=N/b. QDN /

��L2c.M;E/

RNn�� QL2.M=N;E=N/;

where the mapRNn� is given by24

�RNn � .s/

�.Nm/D

ZN

n � s.n�1m/dn:

Proof: Step 1. Because the representation of N in L2.M;E/ is unitary, we have�RNn � .s/;t

�L2.M;E/

D�s;RNn � .t/

�L2.M;E/

for all s;t 2 L2c.M;E/.Step 2. By Proposition 3.10 below and equivariance of D, we have�R

Nn��ıD D QDN ı R

Nn�

on C1c .M;E/.

24Note that the space QL2.M=N;E=N/ can be realised as a space of sections of E.


Step 3. For all s 2 C1c .M;E/, we have by Proposition 3.10,

@

@�

ˇ�D0

RNn � ıei�DsD

ZN

@

@�ei�Dn � sdn

D iZN

n �DsdnD i QDN

RN n�.s/ (by Step 2)

D @

@�

ˇ�D0

ei�QDN R

Nn � .s/:

So by Stone’s theorem, RNn � ıei�D D ei� QDN ı RNn�

for all � 2R.

Step 4. For all s;t 2 C1c .M;E/,�

b. QDN /RNn � .s/;t�

L2.M;E/

D 1

2�

ZR

�ei�

QDN RNn � .s/;t

�L2.M;E/

Ob.�/d� (by Proposition 3.6)

D 1

2�

ZR

�RNn � ei�Ds;t

�L2.M;E/

Ob.�/d� (by Step 3)

D 1

2�

ZR

�ei�Ds;

RNn � .t/

�L2.M;E/

Ob.�/d� (by Step 1)

D �b. QD/s;RNn � .t/�L2.M;E/(by Proposition 3.6)

D �RNn � b. QD/s;t�L2.M;E/(by Step 1).

This completes the proof.

Proposition 3.10 Let M1 and M2 be manifolds, and suppose M2 is equipped witha measure dm2. Let E!M1 be a vector bundle, and let

D W C1.M1;E/! C1.M1;E/

be a differential operator. Let p1 W M1 �M2 ! M1 be projection onto the firstfactor, and consider the operator

p�1D W C1.M1 �M2;p

�1E/! C1.M1 �M2;p

�1E/;

defined by�p�1D

�p�1� D p�

1

�D�

�for all � 2 C1.M1;E/.


Then for all s 2 C1.M1 �M2;p�1E/ and all m1 2M1,

D

�ZM2

s.�;m2/dm2�.m1/D

ZM2

p�1Ds.m1;m2/dm2:

We now derive Proposition 3.8 from Lemma 3.9.Proof of Proposition 3.8: Consider the following cube:

L2c.M;E/

RNn� ��

b.D/

��

��

��

��

��

��

�QL2.M=N;E=N/b. QDN /

�� Q E

��

��

��

��

��

��

L2c.M;E/��

��

��

��

��

��

QL2.M=N;E=N/

��

Q E

��

��

��L2.M;E/N ��

b.D/N��

L2.M=N;E=N/

b.DN /

��L2.M;E/N

� �� L2.M=N;E=N/:

The rear square (with the operators b.D/ and b. QDN / in it) commutes by Lemma3.9. The left hand square (with the operators b.D/ and b.D/N ) commutes bydefinition of b.D/N , and the right hand square (with b. QDN / and b.DN /) commutesby Lemma 3.4. The top and bottom squares commute by definition of the map , sothat the front square commutes as well, which is Proposition 3.8.

3.4. Multiplication of sections by functions

Let G, M and E be as in Subsections 3.2 and 3.3. As before, let

�M W C0.M/! B.L2.M;E//

and�M=N W C0.M=N/! B.L2.M=N;E=N//

be the representations defined by multiplication of sections by functions. Let

.�M/N W C0.M=N/! B.L2.M;E/N /

be the representation obtained from �M by the procedure in Subsection 3.1.

Lemma 3.11 The isomorphism (36) intertwines the representations .�M/N and�M=N .


Proof: The representation .�M/N is induced by

�NM W C.M=N/! B.L2c.M;E//;��NM .f /s

�.m/D f .N �m/s.m/:

For all f 2 C.M=N/, s 2 L2c.M;E/ and m 2M , we have

��NM .f /s

�.N �m/DN �

ZN

n �f .N �n�1m/s.n�1 �m/dn

DN �f .N �m/ZN

n � s.n�1 �m/dnD .�M=N .f /.s//.N �m/:

3.5. Conclusion

Let G, M , E, D, DN , �M and �M=N be as in Sections 3.2 – 3.4. Supposethat the vector bundle E carries a Z2-grading with respect to which the operatorD is odd. Suppose D is elliptic and essentially self-adjoint as an unboundedoperator on L2.M;E/.25. Let b be a normalising function with compactly supportedFourier transform. Then Proposition 3.2, Proposition 3.8 and Lemma 3.11 may besummarised as follows.

Theorem 3.12 Let .L2.M;E/N ;b.D/N ;.�M/N / be the triple obtained from.L2.M;E/;b.D/;�M / by the procedure of Subsection 3.1. Then there is a unitaryisomorphism

W L2.M;E/N ! L2.M=N;E=N/

that intertwines the representations of G=N , the operators b.D/N and b.DN /, andthe representations .�M/N and �M=N .

Corollary 3.13 The image of the class

ŒD� WDhL2.M;E/;b.D/;�M

i2KG0 .M/

under the homomorphism VN defined in Subsection 3.1 is

VN ŒD�DhL2.M=N;E=N/;b.DN /;�M=N

iDW �DN

2KG=N0 .M=N/:

25This is the case if M is complete and D is a Dirac operator on M , see footnote 16.


Remark 3.14 If the action ofG onM happens to be free, then Corollary 3.13 allowsus to restate the Guillemin–Sternberg conjecture 1.1 without using techniques fromnoncommutative geometry. Indeed, for free actions we have

RQ ı�GMhD/ LM

iD �feg

M=GıVG

hD/ LM

iby naturality of �

D index�D/ LM

�G by Corollary 3.13

D dim�

ker�D/ LM

�C�G � dim�

ker�D/ LM

��G 2 Z:

Here the kernels of�D/ LM

�˙ are taken in the spaces of smooth, not necessarily L2,sections of

V0;�T �M ˝ L. Note that even though these kernels may be infinite-

dimensional, their G-invariant parts are not, because they are the kernels of the

elliptic operators��D/ LM

�˙�G on the compact manifold M=G. So Conjecture 1.1becomes

indexD/ LG

MGD dim

�ker�D/ LM

�C�G � dim�

ker�D/ LM

��G:

Unfortunately, in our situation this argument would only apply to discrete groups(see Remark 2.2).

4. Dirac operators

In this section, we make the assumptions stated in Subsection 2.1. In particular, � isa normal discrete subgroup of G. The goal of this section is to prove that Theorem2.9 is a special case of Corollary 3.13:

Proposition 4.1 Consider the Dolbeault–Dirac operator D/ LM on �0;�.M IL/, andthe induced operator

�D/ LM

�� on C1�M=�;�V0;�T �M ˝L�=��. There is an

isomorphism

„ W�0;�.M=�IL=�/! C1�M=�;�V0;�T �M ˝L�=��

that is isometric with respect to theL2-inner product and intertwines the Dolbeault–Dirac operator D/ L=�

M=�on �0;�.M=�IL=�/ and the operator

�D/ LM

�� .

Consequently, „ induces a unitary isomorphism between the correspondingL2-spaces, which by Lemma 3.4 intertwines the bounded operators obtained fromD/ L=�M=�

and�D/ LM

�� using a normalising function with compactly supported Fouriertransform. Hence Theorem 2.9 follows, as

V��D/ LM

�D h�D/ LM ��i by Corollary 3.13

D �D/ L=�M=�

by Proposition 4.1.


4.1. The isomorphism

The isomorphism of C1.M=�/-modules„ in Proposition 4.1 is defined as follows.By discreteness of � , the quotient map p W M ! M=� induces the vector bundlehomomorphism V

Tp� WVT �.M=�/!VT �M: (39)

Because Tp intertwines the almost complex structures on TM and T .M=�/, thehomomorphism (39) inducesV0;�

Tp� WV0;�T �.M=�/!V0;�

T �M: (40)

Composition with the quotient map T �M ! .T �M/=� turns (39) and (40) intoisomorphisms �V

Tp�� W VT �.M=�/! �VT �M

�=�I�V0;�

Tp�� W V0;�T �.M=�/! �V0;�

T �M�=�:

On the spaces of smooth sections of the vector bundles in question, the isomor-phisms (41) and (41) induce isomorphisms of C1.M=�/-modules

‰ W��.M=�/!C1�M=�;.VT �M/=��I

‰0;� W�0;�.M=�/!C1�M=�;�V0;�T �M

�=��:

Now the isomorphism „ is defined as

„ W�0;�.M=�IL=�/Š�0;�.M=�/˝C1.M=�/ C

1.M=�;L=�/‰0;�˝1C 1.M=�;L=�/��!

C1�M=�;�V0;�T �M

�=��˝C1.M=�/ C

1.M=�;L=�/Š C1�M=�;�V0;�

T �M ˝L�=��:It is isometric by definition of the measure dO on M=� and the metrics on thevector bundles involved. Therefore, it remains to prove that „ intertwines theoperators D/ L=�

M=�and

�D/ LM

�� .

4.2. Proof of Proposition 4.1

THE CONNECTIONS

Recall the isomorphism of C1.M/� Š C1.M=�/-modules E W C1.M;E/� !C1.M=�;E=�/ defined by (31), with H D � , for any �-vector bundle E over M .


Also consider the pullback p� of differential forms onM=� to invariant differentialforms on M . It defines an isomorphism of C1.M=�/Š C1.M/� -modules

p� W��.M=�/!��.M/� :


��.M IL/� r ��

Š��

��.M IL/�Š��

��.M/�˝C1.M/� C1.M;L/� ��.M/�˝C1.M/� C

1.M;L/�

��.M=�/˝C1.M=�/ C1.M=�;L=�/

p�˝ �1L

Š��

��.M=�/˝C1.M=�/C1.M=�;L=�/

p�˝ �1L

Š��

��.M=�IL=�/ r��

Š��

��.M=�IL=�/:Š��

The proof of this lemma is a matter of writing out definitions.By definition of the almost complex structure on T .M=�/, we have

p��0;q.M=�/�D�0;q.M/�

for all q. Therefore, Lemma 4.2 implies that the following diagram commutes:

�0;�.M IL/� N@L �� 0;�.M IL/�

�0;�.M=�IL=�/N@�

L ��

p�˝ �1L

Š��

�0;�.M=�IL=�/;p�˝ �1

LŠ��

(41)

with N@L and N@�L as in Subsection 2.2.

THE DIRAC OPERATORS

By definition of the measure dO on M=� and the metrics B� on T .M=�/ and H�

on L=� , the isomorphism

p�˝ �1L W�0;�.M=�IL=�/!�0;�.M IL/�

is isometric with respect to the inner product on �0;�.M=�IL=�/ defined by

.˛˝ �;ˇ˝ /DZM=�

B�.˛;ˇ/H�.�; /dO; (42)


for all ˛;ˇ 2 �0;�.M=�/ and �; 2 C1.M=�;L=�/, and the inner product on�0;�.M IL/� defined by

.�˝ s;˝ t /DZU

B.�;/H.s;t/dm; (43)

for all �; 2�0;�.M/� and s;t 2 C1.M;L/� . (Recall that U �M is a fundamentaldomain for the �-action.)

The Dolbeault–Dirac operators on M and M=� are defined by

D/ LM D N@LC N@�LI

D/ L=�M=�D N@�LC

�N@�L��:Here the formal adjoint

�N@�L�� is defined with respect to the inner product (42). Theformal adjoint N@�

L is defined byZM

.B˝H/�N@�L�;�

�dmD

ZM

.B˝H/��; N@L��dm;for all �;� 2 �0;�.M IL/, � with compact support. But this is actually the same asthe formal adjoint of N@L with respect to the inner product (43):

Lemma 4.3 Let � be a discrete group, acting properly and freely on a manifoldM ,equipped with a �-invariant measure dm. Suppose M=� is compact. Let E !M

be a �-vector bundle, equipped with a �-invariant metric h�;�i. Let

D W C1.M;E/! C1.M;E/

be a �-equivariant differential operator. Let

D� W C1.M;E/! C1.M;E/

be the operator such that for all s;t 2 C1.M;E/, t with compact support,ZM

hD�s;tidmDZM

hs;Dtidm:

Let U �M be a fundamental domain for the �-action. Then the restriction ofD� to C1.M;E/� satisfiesZ

U

hD�s;tidmDZU

hs;Dtidm;

for all s;t 2 C1.M;E/� .


Proof: We will show that for all s 2 C1.M;E/� , and all t in a dense subspace ofC1.M;E/� , we have Z

U

hD�s;tidmDZU

hs;Dtidm:

Let be a section of E, with compact support in U . Define the section t ofE by extending the restriction jU �-invariantly to M . The space of all sections tobtained in this way is dense in C1.M;E/� with respect to the topology inducedby the inner product

.s;t/ WDZU

hs;tidm:

Then for all s 2 C1.M;E/� ,ZU

hD�s;tidmDZM

hD�s; idmDZM

hs;D idmDZU

hs;Dtidm:

We conclude that p� ˝ �1L is an isometric isomorphism with respect to the

inner products used to define the adjoints N@�L and

�N@�L��. Hence the commutativityof diagram (41) implies

Corollary 4.4 The following diagram commutes:

�0;�.M IL/� D=LM �� 0;�.M IL/�

�0;�.M=�IL=�/D=

L=�

M=��

p�˝ �1L

Š��

�0;�.M=�IL=�/:p�˝ �1

LŠ��

Remark 4.5 Corollary 4.4 shows that for discrete groups a much stronger statementthan the Guillemin-Sternberg conjecture holds. Indeed, by Remark 3.14 theGuillemin-Sternberg conjecture states that the restriction of the operator D=LM to�0;�.M IL/� is related to the operatorD=L=�

M=�by the fact that their indices are equal

(as operators on smooth, not necessarily L2, sections). But these operators are infact more strongly related: they are intertwined by an isometric isomorphism.

END OF THE PROOF OF PROPOSITION 4.1

The last step in the proof of Proposition 4.1 is a decomposition of the isomorphism

p� W��.M=�/!��.M/� :



��.M=�/ p�

Š ��

‰ Š��

��.M/�

^T �M

Š��

C1.M=�;.VT �M/=�/;

where ‰ is the isomorphism (41).

The proof of this lemma is a short and straightforward computation.

Proof of Proposition 4.1: Together with Lemma 4.6 and the definition of theoperator�D/ LM

�� W C1�M=�;�V0;�T �M ˝L�=��! C1�M=�;�V0;�

T �M ˝L�=��;Corollary 4.4 implies that the following diagram commutes:

�0;�.M IL/�Š

^0;�T �M˝ L

��

D=LM �� 0;�.M IL/�

Š ^0;�T �M

˝ L

��C1�M=�;�V0;�

T �M ˝L�=��D=LM

��

�� C1�M=�;�V0;�T �M ˝L�=��

�0;�.M=�IL=�/Š„D‰0;�˝1��

D=L=�

M=� �� 0;�.M=�IL=�/:Š„D‰0;�˝1��

Indeed, the outside diagram commutes by Corollary 4.4 and Lemma 4.6, and theupper square commutes by definition of

�D/ LM

�� . Hence the lower square commutesas well, which is Proposition 4.1.

5. Abelian discrete groups

In this section, we consider the situation of Section 2, with the additional assumptionthat G D � is an abelian discrete group. Then the Guillemin–Sternberg conjecturecan be proved directly, without using naturality of the assembly map (33). Thisproof is based on Proposition 4.1, and the description of the assembly map in thisspecial case given by Baum, Connes and Higson [5], Example 3.11 (which in turn isbased on Lusztig [45]). We will first explain this example in a little more detail thangiven in [5], and then show how it implies Theorem 1.2 for abelian discrete groups.


5.1. The assembly map for abelian discrete groups

The proof of the Guillemin–Sternberg conjecture for discrete abelian groups isbased on the following result:

Proposition 5.1 LetM , E,D andD� be as in Subsection 3.5. Suppose thatG D �is abelian and discrete. Using the normalising function b.x/D xp

1Cx2, we form the

operator F WD b.D/, so that we have the class�L2.M;E/;F

2K�0 .M/:

Then26

R.�/Q ı��M

�L2.M;E/;F

D indexD� :

In view of Proposition 4.1, Proposition 5.1 implies our Guillemin–Sternbergconjecture (i.e. Theorem 1.2) for discrete abelian groups.

KERNELS OF OPERATORS AS VECTOR BUNDLES

Using Example 3.11 from [5], we can explicitly compute

ŒE ;FE � WD ��M�L2.M;E/;F

2KK0.C;C �.�//: (44)

Note that since � is discrete, its unitary dual O� is compact. And because � isabelian, all irreducible unitary representations are of the form

U˛ W �! U.1/;

for ˛ 2 O� . Fourier transform defines an isomorphism C �.�/Š C0. O�/. Therefore,

KK0.C;C�.�//ŠK0.C �.�//ŠK0.C0. O�//ŠK0. O�/:

Because O� is compact, the image of ŒE ;FE � in K0. O�/ is the difference of theisomorphism classes of two vector bundles over O� . These two vector bundles can bedetermined as follows. For all ˛ 2 O� , we define the Hilbert space H˛ as the spaceof all measurable sections s˛ of E (modulo equality almost everywhere), such thatfor all � 2 � ,

� � s˛ D U˛.�/�1s˛;and such that the norm

ks˛k2˛ D hs˛;s˛i˛ (45)

26Recall that we abuse notation by writing indexD� WD dimker�D�

�C � dimker�D�

��.


is finite, where the inner product h�;�i˛ is defined by

hs˛;t˛i˛ WDZM=�

�s˛.'.O//;t˛.'.O//

�EdO;

where ' is any measurable section of the principal fibre bundle M ! M=� . Thespace H˛ is isomorphic to the space of L2-sections of the vector bundle E˛, where

E˛ WDE=.� � e U�1˛ .�/e/!M=�:

Let HD˛ be the dense subspace

HD˛ WD fs˛ 2H˛ \C1.M;E/IDs˛ 2H˛g �H˛: (46)

Because the operator D is �-equivariant, it restricts to an unbounded operator

D˛ WHD˛ !H˛

on H˛. It is essentially self-adjoint by [31], Corollary 10.2.6., and hence inducesthe bounded operator

F˛ WD D˛p1CD2

˛

2 B.H˛/: (47)

The grading on E induces a grading on H˛ with respect to which D˛ and F˛ areodd. The operators F˛ are elliptic pseudo-differential operators:

Lemma 5.2 LetD be an elliptic, first order differential operator on a vector bundleE ! M , and suppose D defines an essentially self-adjoint operator on L2.M;E/with respect to some measure on M and metric on E. Then the operator F WD

Dp1CD2

is an elliptic pseudo-differential operator.

Proof: This result seems well known and is easily derived from a number of relatedresults in the literature. It is, in any case, sufficient to show that .1CD2/� 1

2 isa pseudo- differential operator. The following proof was communicated to us byElmar Schrohe.27

According to [6], a bounded operator A W L2.Rn/ ! L2.Rn/ is a pseudo-differential operator on Rn iff all iterated commutators with xj (as a multiplicationoperator) and @xj

are bounded operators. This immediately yields the lemma forM DRn (cf. [6], Theorem 4.2). To extend this result to the manifold case, we recall

27An independent proof was suggested to us by John Roe, who mentioned that in the caseat hand the functional calculus for (pseudo) differential operators developed in [69] for compactmanifolds may be extended to the noncompact case. A third proof may be constructed using heatkernel techniques, as in the unpublished Diplomarbeit of Hanno Sahlmann (Rainer Verch, privatecommunication).


that an operator A W C1.M/ ! D0.M/ on a manifold M is a pseudo-differentialoperator when for each choice of smooth functions f , g with support in a singlecoordinate neighbourhood, fAg is a pseudo-differential operator on Rn. (Here onehas to admit nonconnected coordinate neighbourhoods.)

Now write .1CD2/� 12 as a Dunford integral (cf. [17], pp. 556–577), as follows:

.1CD2/� 12 D

IC

dz

2�i.1C z/� 1

2 .z�D2/�1:

To compute the commutators of f .1 C D2/� 12g with xj and @xj

, one may takethese inside the contour integral. Boundedness of all iterated commutators theneasily follows, using the fact that f and g have compact support.

The same argument, with the exponent �12

replaced by 12

, shows that .1CD2/12

is a pseudo-differential operator, and ellipticity of .1CD2/� 12 follows.

Consider the field of Hilbert spaces

.H˛/˛2 O� ! O�:In the next subsection, we will give this field the structure of a continuous fieldof Hilbert spaces by specifying its space of continuous sections C. O�;.H˛/˛2 O�/.Consider the subfields �

kerDC�˛2 O� ! O�I

.kerD�/˛2 O� ! O�:

These are indeed well-defined subfields of .H˛/˛2 O� because kerD˙ D kerF ˙ bythe elliptic regularity theorem.

Suppose that .kerD˛/˛2 O� and .cokerD˛/˛2 O� are vector bundles over O� in therelative topology. As in the proof of the Atiyah–Jänich Theorem (cf. [73]), theoperator D can always be replaced by another operator in such a way that the classŒL2.M;E/;F � does not change, and that these ‘fields of vector spaces’ are indeedvector bundles (see also [36]). Then:

Proposition 5.3 The image of the class�L2.M;E/;F

2 K�0 .M/ under theassembly map ��M is

��M�L2.M;E/;F

D ��kerDC�˛2 O�

� �.kerD�/˛2 O�

2K0. O�/:Proposition 5.3 wil be proved in the next two subsections.

5.2. The Hilbert C �-module part of the assembly map

In this subsection we determine the Hilbert C �.�/ Š C0. O�/-module E in (44)(Proposition 5.7).


A UNITARY ISOMORPHISM

Let d˛ be the measure on O� corresponding to the counting measure on � via theFourier transform. Consider the Hilbert space

H WDZ ˚

O�H˛d˛:

That is, H consists of the measurable maps

s W O�! .H˛/˛2 O� ; ˛ 7! s˛;

such that s˛ 2H˛ for all ˛, and

ksk2H D hs;siH WDZ

O�ks˛k2˛d˛ <1:

Define the linear map V WH ! L2.M;E/ by

.Vs/.m/ WDZ

O�s˛.m/d˛:

Lemma 5.4 The map V is a unitary isomorphism, with inverse�V �1�

�˛.m/D

X�2�

� � �.��1m/U˛.�/; (48)

for all � 2 Cc.M;E/� L2.M;E/.Remark 5.5 It follows from unitarity of V that Vs is indeed an L2-section of E forall s 2H . Conversely, a direct computation shows that for all � 2 L2.M;E/, ˛ 2 O�and � 2 � , one has

� � �V �1��˛D U˛.�/�1

�V �1�

�˛;

so that V �1� lies in H:

Sketch of proof of Lemma 5.4: The proof is based on the observations that for all˛ 2 O� , X

�2�U˛.�/D ı1.˛/; (49)

where ı1 2 D0. O�/ is the ı-distribution at the trivial representation 1 2 O� , and thatfor all � 2 � , Z

O�U˛.�/d˛ D ı�e; (50)

the Kronecker delta of � and the identity element. Using these facts, one can easilyverify that V is an isometry and that (48) is indeed the inverse of V .


The representation �H of � in H corresponding to the standard representation of �in L2.M;E/ via the isomorphism V is given by

.�H .�/s/˛ D U˛.�/�1s˛:

This follows directly from the definitions of the space H˛ and the map V .

FOURIER TRANSFORM

The Hilbert C �.�/-module E is the closure of the space Cc.M;E/ in the norm

k�k2E WD k� 7! h�;� � �iL2.M;E/kC�.�/:

The C �.�/-module structure of E is defined by

f � � DX�2�

f .�/� � �;

for all f 2 Cc.�/ and � 2 Cc.M;E/. The isomorphism V induces an isomorphismof the Hilbert C �.�/-module E with the closure EH of V �1.Cc.M;E//�H in thenorm

ksk2EHWD k� 7! hVs;� �VsiL2.M;E/kC�.�/ D k� 7! hs;�H .�/siHkC�.�/;

by unitarity of V . The C �.�/-module structure on EH corresponding to the one onE via V is given by

f � s DX�2�

f .�/�H .�/s; (51)

for all f 2 Cc.�/ and s 2 V �1.Cc.M;E//.Next, we use the isomorphism C0. O�/Š C �.�/ defined by the Fourier transform

7! O , whereO .�/D

ZO� .˛/U˛.�/d˛

for all 2 Cc. O�/. Because of (49) and (50), the inverse Fourier transform is givenby f 7! Of , where for f 2 Cc.�/, one has

Of .˛/DX�2�

f .�/U˛.�/�1:

So by Fourier transform, the Hilbert C �.�/-module EH corresponds to theHilbert C0. O�/-module OEH , which is the closure of the space V �1.Cc.M;E// in


the norm

ksk2OEHD��˛ 7!X

�2�hs;�H .�/siHU˛.�/�1

��C0. O�/

D sup˛2 O�

ˇX�2�hs;�H .�/siHU˛.�/�1

ˇ:

CONTINUOUS SECTIONS

Using the following lemma, we will describe the Hilbert C0. O�/-module OEH as thespace of continuous sections of a continuous field of Hilbert spaces.

Lemma 5.6 For all s;t 2 V �1.Cc.M;E//,X�2�hs;�H .�/tiHU˛.�/�1 D hs˛;t˛i˛:

Proof: Let ' be a measurable section of the principal fibre bundle M ! M=� .Then X

�2�hs;�H .�/tiHU˛.�/�1

DX�2�

�ZO�

ZM=�

�sˇ .'.O//;Uˇ .�/�1tˇ .'.O//

�EdOdˇ

�U˛.�/

�1

DZM=�

�s˛.'.O//;t˛.'.O//

�EdO

Dhs˛;t˛i˛:

by (49).

We conclude from (52) and Lemma 5.6 that OEH is the closure of V �1.Cc.M;E//in the norm

ksk2OEHD sup˛2 O�ks˛k2˛:

Therefore, it makes sense to define the space C. O�;�H˛�˛2 O�/ of continuous sectionsof the field of Hilbert spaces .H˛/˛2 O� as the C0. O�/-module OEH (cf. [15, 68]). Thenour construction implies

Proposition 5.7 The Hilbert C �.�/-module E is isomorphic to the Hilbert C0. O�/-module C. O�;.H˛/˛2 O�/.


Let us verify explicitly that the representations of C0. O�/ in OEH and inC. O�;.H˛/˛2 O�/ are indeed intertwined by the isomorphism induced by V and theFourier transform: for all f 2 Cc.�/ and all s 2 V �1.Cc.M;E//, we have

.f � s/˛ DX�2�

f .�/.�H .�/s/˛ by (51)

DX�2�

f .�/U˛.�/�1s˛

D Of .˛/s˛:

5.3. The operator part of the assembly map

Proposition 5.8 The operator F OEHon the Hilbert C0. O�/ module OEH D

C. O�;.H˛/˛2 O�/, induced by F 2 B.L2.M;E//, equals F OEHD �

F˛�˛2 O� , where the

‘field of operators’�F˛�˛2 O� is given by��

F˛�˛2 O�s

�ˇD Fˇ sˇ ;

for all ˇ 2 O� and all s 2 V �1.Cc.M;E// (and extended continuously to generals 2 OEH ). Here F˛ is the operator (47).

Proof: For s 2 V �1.Cc.M;E//, we have VF OEHs D FVs. So it is sufficient to

prove that for such s, one has FVs.m/D R O�F˛s˛.m/d˛, for all m 2M .Let HD � H be the space of s 2 H such that Vs 2 C1

c .M;E/, and s˛ 2 HD˛

for all ˛ 2 O� (see (46)). By Proposition 3.10, we have DVs.m/D R O�Ds˛.m/d˛ forall s 2HD andm 2M . Because of Lemma 3.4 this proves the proposition, becauseHD is dense in H .

Note that�F˛

�˛2 O� is initially defined on the subspace V �1.Cc.M; E//

of C. O�;.H˛/˛2 O�/. But since the unitary operator V intertwines�F˛�˛2 O� and

the bounded operator F on L2.M;E/, the field of operators�F˛�˛2 O� is bounded

in the norm k � k OEH, so that it extends continuously to a bounded operator on

C. O�;.H˛/˛2 O�/.Proof of Proposition 5.3: We have seen (cf. Propositions 5.7 and 5.8) that

��M�L2.M;E/;F

D �C. O�;.H˛/˛2 O�/;.F˛/˛2 O�

2KK0.C;C0. O�//:

The image of this class in K0.C0. O�// is the formal difference of projective C0. O�/-modules �

ker��F C�

˛2 O�� ker

�.F �/

˛2 O��: (52)


By compactness of M=� and the elliptic regularity theorem, the kernels of F C andF � are equal to the kernels of DC and D�, respectively. If we suppose that thekernels of DC and D� define vector bundles over O� , then by Lemma 5.9 below, theclass (52) equals h

C. O�;�kerDC�˛2 O�/

i� �C. O�;.kerD�/

˛2 O�/:

Under the isomorphism K0.C0. O�//ŠK0. O�/, the latter class corresponds to��kerDC�

˛2 O�� .kerD�/

˛2 O� 2K0. O�/:

Lemma 5.9 Let H be a continuous field of Hilbert spaces over a topological spaceX , and let � be its space of continuous sections. Let H0 be a subset of H such thatfor all x 2X , H0

x WDHx \H0 is a linear subspace of Hx . Set

�0 WD fs 2�Is.x/ 2H0x for all x 2Xg:

Let s W X ! H0 be a section. Then s is continuous in the subspace topology ofH0 in H if and only if s 2�0.Proof: Let s W X ! H be a section. Then s is a continuous section of H0 in thesubspace topology if and only if s is a continuous section of H and s.x/ 2 H0

x forall x. The topology on H is defined in such a way that s is continuous if and only ifs 2� [15, 68].

5.4. Reduction

We will now describe the reduction map R.�/Q W K0.C �.�// ! Z; and prove

Proposition 5.1.

Lemma 5.10 Let � be an abelian discrete group, and let i W f1g ,! O� be theinclusion of the trivial representation. The following diagram commutes:

K0.C�.�//

R.�/Q ��

Š��

K0.C/

Š��

K0. O�/ i� �� K0.f1g/:That is,

R.�/Q .ŒE�/D dimE1 D rank.E/ 2 Z;

for all vector bundles E! O� .


The proof is a straightforward verification.End of proof of Proposition 5.1.

End of proof of Proposition 5.1: From Lemma 5.10 and Proposition 5.3, we con-clude that

R.�/Q ı��M

�L2.M;E/;F

D ŒkerDC1 �� ŒkerD�

1 �D indexD1 2 Z:

The Hilbert space H1 is isomorphic to L2.M=�;E=�/, and this isomorphismintertwines D1 and D� . Hence Proposition 5.1 follows.

6. Example: action of Z2n on R2n

Let M be the manifold M D T �Rn Š R2n Š Cn. An element of M is denotedby .q;p/ WD .q1;p1;:::;qn;pn/, where qj ;pj 2 R, or by qC ip D z WD .z1;:::;zn/,where zj D qj C ipj 2 C. We equip M with the standard symplectic form ! WDPnjD1dpj ^ dqj :

Let � be the group � D Z2n Š ZC iZ: The action of � on M by additionis denoted by ˛. Our aim is to find a prequantisation for this action and thecorresponding Dirac operator for general n, and the quantisation of this actionfor n D 1. This is less trivial than it may seem because of the coupling of thestandard Dirac operator to the prequantum line bundle, which precludes the use ofthe standard formulae. We will then see, just as in Section 5, that the reduction ofthis quantisation is the quantisation of the reduced space T2.

6.1. Prequantisation

Let L WD M �C ! M be the trivial line bundle. Inspired by the construction ofline bundles on tori with a given Chern class (see e.g. [24], pp. 307–317), we liftthe action of � on M to an action of � on L (still called ˛), by setting

ej � .z;w/D .zC ej ;w/Iiej � .z;w/D .zC iej ;e�2�izjw/:

Here z 2M , w 2C, and

ej WD .0;:::;0;1;0;:::;0/ 2 Zn;

the 1 being in the j th place. The corresponding representation of � in the space ofsmooth sections of L is denoted by �:

.�kCils/.z/D ˛kCils.z� k� i l/;


for k;l 2 Zn and z 2M . Define the metric H on L by

H..z;w/;.z;w0//D h.z/w Nw0;

where z 2M , w;w0 2C, and h 2 C1.M/ is defined by

h.qC ip/ WD e2�P

j .pj �p2j/:

Let r be the connection on L defined by

r WD d C 2�inXjD1

pj dzj C�dpj :

Proposition 6.1 The triple .L;H;r/ defines an equivariant prequantisation for.M;!/.

The proof of this proposition is a set of tedious computations. Because ofthe term 2�i

PnjD1pj dqj in the expression for the connection r, it has the

right curvature form. The terms �2�PnjD1pj dpj and �dpj do not change the

curvature, and have been added to make r equivariant. At the same time, the lattertwo terms ensure that there is a �-invariant metric (namelyH ) with respect to whichr is Hermitian.

As we mentioned in Subsection 2.1, there is a procedure in [27] to lift theaction of Z2n on R2n to a projective action on L that leaves the connection (forexample) r 0 WD d C 2�iPj pj dqj invariant. This projective action turns out tobe an actual action in this case, and preserves the standard metric on L. We thusobtain prequantisation of this action that looks much simpler than the one given inthis section. However, we found our formulas to be more suitable to compute thekernel of the associated Dirac operator.

6.2. The Dirac operator

In this subsection, we compute the Dolbeault–Dirac operatorD/ onM , coupled toL.To compute the quantisation of the action we are considering, we need to computethe kernels of

D/ C WDD/ j�0;even.M/ID/ � WDD/ j�0;odd.M/:

This is not easy to do in general. But for n D 1, these kernels are computed inSubsection 6.3.


In our expression for the Dirac operator, we will use multi-indices

l D .l1;:::;lq/� f1;:::;ng;where q 2 f0;:::;ng and l1 < �� < lq . We will write d Nzl WD d Nzl1 ^ ::: ^ d Nzlq . Ifl D ;, we set d Nzl WD 1M , the constant function 1 on M . Note that fd Nzlgl�f1;:::;ng isa C1.M/-basis of �0;�.M IL/.

Given l � f1;:::;ng and j 2 f1;:::;ng, we define

"jl WD .�1/#fr2f1;:::;qgIlr<j g;

plus one if an even number of lr is smaller than j , and minus one if the numberof such lr is odd. From the definition of the Dolbeault–Dirac operator one thendeduces:

Proposition 6.2 For all l � f1;:::;ng and all f 2 C1.M/, we have

D/�fd Nzl�DX

j2l"jl

��2 @f@zjC .i� � 4�ipj /f

�d Nzlnfj g

CX

1�j�n;j 62l

"jl

�@f

@ Nzj Ci�

2f

�d Nzl[fj g:

(53)

6.3. The case nD 1We now consider the case where nD 1. That is, M D C and � D ZC iZ. We canthen explicitly compute the quantisation of the action. This will allow us to illustrateour noncompact Guillemin–Sternberg conjecture by computing the four corners indiagram (1).

If nD 1, Proposition 6.2 reduces to

Corollary 6.3 The Dirac operator on C, coupled to L, is given by

D/ .f1Cf2d Nz/D�@f1

@ Nz Ci�

2f1

�d Nz� 2@f2

@zC .i� � 4�ip/f2:

That is to say, with respect to the C1.M/-basis f1M ;d Nzg of �0;�.M IL/, the Diracoperator D/ has the matrix form (5), where

D/ C D @

@ Nz Ci�

2I

D/ � D�2 @@zC i� � 4�ip:


In this case, the kernels of D/ C and D/ � can be determined explicitly:

Proposition 6.4 The kernel of D/ C consists of the sections s of L given by

s.z/D e�i� Nz=2'.z/;

where ' is a holomorphic function.The kernel of D/ � is isomorphic to the space of smooth sections t of L given by

t .z/D ei�z=2C�jzj2��z2=2 .z/;

where is a holomorphic function.

The unitary dual of the group Z C iZ D Z2 is the torus T2. Therefore, byProposition 5.3, the quantisation of the action of Z C iZ on C is the class inKK.C;C �.Z2// that corresponds to the class��

kerD/ C.˛;ˇ/

�.˛;ˇ/2T2

�h�

kerD/ �.˛;ˇ/

�.˛;ˇ/2T2

iinK0.T2/. It will turn out that the kernels ofD/ C

.˛;ˇ/andD/ �

.˛;ˇ/ indeed define vectorbundles over T2. Let us compute these kernels.

Proposition 6.5 Let �;� 2R. Define the section s�� 2 C1.M;L/ by

s��.z/D ei�ze��pXk2Z

e��k2

e�k.�Ci�C2�/e2�ikz:

Set ˛ WD ei� and ˇ WD ei�. Then kerD/ C.˛;ˇ/DCs��.

Remark 6.6 For all �;� 2R, we have

s�C2�;� D e�Ci�C3�s��Is�;�C2� D s��:

Hence the vector space Cs�� C1.M;L/ is invariant under � 7! � C 2� and� 7! �C2� . This is in agreement with the fact that Cs�� is the kernel ofD/ C

.ei�;ei�/.

Sketch of proof of Proposition 6.5: Let �;�2R, and s 2 C1.M;L/DC1.C;C/.Suppose s is in the kernel of D/ C

.˛;ˇ/. Let ' be the holomorphic function from

Proposition 6.4, and write

Q'.z/ WD e�i�ze�i�z=2'.z/DXk2Z

ak e2�ikz

(note that for all z 2C one has Q'.zC1/D Q'.z/). Then ak D e��k2

e�k.�Ci�C2�/a0,which gives the desired result.

Proposition 6.7 The kernel of D/ �.˛;ˇ/ is trivial for all .˛;ˇ/ 2 T2.


Sketch of proof: Let �;� 2 R and let t d Nz 2 �0;1.M IL/ D C1.M;L/d Nz.Suppose that t d Nz 2 kerD/ �

.ei�;ei�/. Let be the holomorphic function from

Proposition 6.4, and write

Q .z/ WD e�. Nz2Ci Nz/=2�i� Nz .z/DXk2Z

ck e2�ik Nz

(note that for all z 2 C one has Q .zC 1/ D Q .z/). Then ck D e�k2

ek.��i��2�/c0,which implies that c0 D 0.

We conclude:

Proposition 6.8 The quantisation of the action of Z2 on C is the class in K0.T2/

defined by the vector bundle28

.Cs��/.ei�;ei�/2T2 ! T2:

By Lemma 5.10, we now find that the reduction of the quantisation of the actionof Z2 on R2 is the one-dimensional vector space C � s0;0 � C1.M;L/, where

s0;0.z/D e��pXk2Z

e��k2

e�2�ke2�ikz:

As we saw in Section 5.1, this is precisely the index of D/ L=Z2

T2 . Schematically, wetherefore have29

Z2 ˚ R2� Q ��

�

RC

��

.Cs��/.ei�;ei�/2T2

�

RQ

��T2 � Q �� C � s0;0:

Remark 6.9 The fact that the geometric quantisation of the torus T2 is one-dimensional can alternatively be deduced from the Atiyah-Singer index theoremfor Dirac operators. Indeed, let D/ L=Z

2

T2 be the Dirac operator on the torus, coupledto the quotient line bundle L=Z2. Then by Atiyah-Singer, in the form stated forexample in [25] on page 117, one has

Q.T2/D indexD/ L=Z2

T2 DZ

T2

ech1.L=Z2/

DZ

T2

dp^ dqD 1;

28By Remark 6.6, this is indeed a well-defined vector bundle.29Note that it is a coincidence that the two-torus appears twice in this diagram: in this example

M=� D T2 D O� .


the symplectic volume of the torus, i.e. the volume determined by the Liouvillemeasure.

A. Naturality of the assembly map

Our “quantisation commutes with reduction” result is partly a consequence of thenaturality of the assembly map. For discrete groups, this naturality is explainedin detail by Valette [53]. We need to generalise “one half” of this naturality (theepimorphism case) to non-discrete groups.

A.1. The statement

Let G be a locally compact unimodular group acting properly on a locally compactHausdorff space X . We consider a closed normal subgroup30 N of G, and supposethat G and N are equipped with Haar measures dg and dn, respectively, where dnis assumed to be left-invariant. We suppose that X=G is compact.

The version of naturality of the assembly map that we will need is the following.

Theorem A.1 The homomorphism VN defined in Subsection 3.1 makes the follow-ing diagram commute:

KG0 .X/�G

X ��

VN

��

K0.C�.G//

R.N /Q

��KG=N0 .X=N/

�G=N

X=N�� K0.C�.G=N//:

Here �GX and �G=NX=N

are analytic assembly maps as defined in e.g. [5, 71, 53],and the map

R.N/Q D �RN�� (54)

is functorially induced by the mapRNW C �.G/! C �.G=N/ given on f 2 Cc.G/

by [23] RN.f / WNg 7!

ZN

f .ng/dn: (55)

To prove Theorem A.1, one can simply copy the proof for discrete groups inValette [53], replacing discrete groups by possibly non-discrete ones and sums byintegrals. In places where Valette uses the fact that a finite sum of compact operators

30Although for our purpose it is enough to consider discrete normal subgroups of G, we haveto work in the nondiscrete setting anyway, since G is not necessarily discrete. We therefore allownondiscrete subgroups N .


is again compact, one uses Lemma A.2. This lemma states that in some cases, theintegral over a compact set of a continuous family of compact operators is compact.

Another difference between the discrete and the nondiscrete cases is thedefinition of the map ‰ on page 110 of [53]. In the nondiscrete case this map isdefined as follows. If 2Hc , we will write N WD C ker.�;�/N for its class in HN .Then for all 2Hc , we have N 2HN;c . Define the linear map

‰ WHc ˝Cc.G/ Cc.G=N/!HN

by

‰Œ˝'�DZG=N

'.Ng�1/Ng � Nd.Ng/;

where d.Ng/ is the Haar measure on G=N corresponding to the Haar measuresdn and dg on N and G, respectively. To prove that the extension ‰ W E ˝C�.G/

C �.G=N/ ! QE is surjective, one can use a sequence of compactly supportedcontinuous functions on G=N that converges to the distribution ıNe 2 D0.G=N/with respect to the measure d.Ng/.

A.2. Integrals of families of operators

Lemma A.2 Let E be a Hilbert C �-module, and let F.E/ and K.E/ D F.E/ bethe algebras of finite-rank and compact operators on E , respectively. Let .M;�/ bea compact Borel space with finite measure. Suppose M is metrisable. Let ˛;ˇ WM ! B.E/ be continous, and let T 2K.E/ be a compact operator. Define the map� W M ! K.E/ by �.m/ D ˛.m/Tˇ.m/. Then the integral

RM �.m/dm defines a

compact operator on E .

We will prove this lemma in several steps. For continuous maps WM ! B.E/,we will use the norm

k k1 WD supm2Mk .m/kB.E/:

Lemma A.3 Let .M;�/ be a compact Borel space with finite measure, and let Ebe a Hilbert C �-module. Let � W M ! K.E/ be a continuous map. Suppose that� is ‘uniformly compact’, in the sense that there exists a sequence

��j�1jD1 WM !

F.E/ such that k�j � �k1 tends to zero as j !1. Suppose furthermore that forevery j 2 N, there is a sequence

��kj�1kD1 W M ! F.E/ of simple functions (i.e.

measurable functions having finitely many values), such that for all " > 0 there isan n 2N such that for all j;k n, k�kj ��j k< ". Then the integral

RM �.m/d�.m/

defines a compact operator on E .


Proof: For all j;k 2 N, the integralRM �

kj .m/d�.m/ is a finite sum of finite-rank

operators, and hence a finite-rank operator itself. And because k�jj � �k1 ! 0 asj tends to1, we have Z

M

�jj .m/d�.m/!

ZM

�.m/d�.m/

in B.E/. HenceRM�.m/d�.m/ is a compact operator.

Lemma A.4 In the situation of Lemma A.2, the conditions of Lemma A.3 aresatisfied.

Proof: Choose a sequence .Tj /1jD1 in F.E/ that converges to T . For m 2M , set

�j .m/D ˛.m/Tjˇ.m/:Then

k�j ��k1 � k˛k1kTj �T kB.E/kˇk1! 0

as j !1. Note that ˛ and ˇ are continuous functions on a compact space, so theirsup-norms are bounded.

Choose sequences of simple functions ˛k;ˇk W M ! B.E/ such that k˛k �˛k1 ! 0 and kˇk � ˇk1 ! 0 as j goes to 1 (see Lemma A.5 below). For allj;k 2N, set

�kj .m/ WD ˛k.m/Tjˇk.m/;for m 2M . Note that

k�kj ��j k1 D supm2Mk˛k.m/Tjˇk.m/�˛.m/Tjˇ.m/k

D supm2M

�k˛k.m/Tjˇk.m/�˛k.m/Tjˇ.m/k

Ck˛k.m/Tjˇ.m/�˛.m/Tjˇ.m/k�

� k˛kk1kTj kkˇk �ˇk1Ck˛k �˛k1kTj kkˇk1:The sequences k 7! k˛kk1 and j 7! kTj k are bounded, since ˛k! ˛ and Tj ! T .Hence, because the sequences k˛k �˛k1 and kˇk �ˇk1 tend to zero, we see thatk�kj � �j k can be made smaller than any " > 0 for k large enough, uniformly in j .

Lemma A.5 Let .M;�/ be a metrisable compact Borel space with metric dM , letY be a normed vector space, and let ˛ WM ! Y be a continuous map.

Then there exists a sequence of simple maps ˛k W M ! Y , such that thesequence

�k˛�˛kk1�1kD1 goes to zero as k goes to infinity.


Proof: For every k 2 N, choose a finite covering QUk D f QV 1k ;:::; QV nk

kg of M by

balls of radius 1k

. From each QUk , we construct a partition Uk D fV 1k ;:::;V nk

kg of M ,

by setting V 1kWD QV 1

k, and V j

kWD QV j

knSj�1

iD1 QV ik . for j D 2;:::;nk . Note that thesets V j

kare Borel-measurable. For all k 2N and j 2 f1;:::;nkg, choose an element

mj

k2 V j

k. Define the simple map ˛k WM ! Y by

˛k.m/ WD ˛.mjk/ if m 2 V j

k.

Note that, because ˛ is continuous (and uniformly continuous because M iscompact), for every " > 0 there is a k" 2N such that for all m;n 2M ,

dM .m;n/ <1

k") k˛.m/�˛.n/kY < ":

Hence for all " > 0, all k > k", and all m 2M (say m 2 V jk

),

k˛.m/�˛k.m/kY D k˛.m/�˛.mjk/kY < ":So k˛�˛kk1 indeed goes to zero.

Acknowledgments: This work is supported by N.W.O. through grant no.616.062.384 for the second author’s ‘Pionier project’ Quantization, noncommuta-tive geometry, and symmetry.

The authors would like to thank Erik van den Ban, Rogier Bos, SiegfriedEchterhoff, Gert Heckman, Hervé Oyono-Oyono, John Roe, Elmar Schrohe, AlainValette and Jan Wiegerinck for useful suggestions at various stages of this work.The authors are also indebted to the referees for several useful remarks.

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P. HOCHS [email protected]

Radboud University NijmegenInstitute for Mathematics, Astrophysics, and Particle PhysicsToernooiveld 1, 6525 ED NIJMEGENTHE NETHERLANDS

N. P. LANDSMAN [email protected]

Radboud University NijmegenInstitute for Mathematics, Astrophysics, and Particle PhysicsToernooiveld 1, 6525 ED NIJMEGENTHE NETHERLANDS

Received: December, 2005

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