Page 1

arXiv:math-ph/0512022v2 15 May 2006

The Guillemin–Sternberg conjecture for noncompact groups

and spaces

P. Hochs & N.P. Landsman

Radboud University Nijmegen

Institute for Mathematics, Astrophysics, and Particle Physics

Toernooiveld 1, 6525 ED NIJMEGEN, THE NETHERLANDS

hochs@math.ru.nl, landsman@math.ru.nl

February 7, 2008

Abstract

The Guillemin–Sternberg conjecture states that “quantisation commutes with reduction”

in a specific technical setting. So far, this conjecture has almost exclusively been stated

and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely

due to the use of SpincDirac operator techniques, has reached a high degree of perfection

under these compactness assumptions. In this paper we formulate an appropriate Guillemin–

Sternberg conjecture in the general case, under the main assumptions that the Lie group

action is proper and cocompact. This formulation is motivated by our interpretation of the

“quantisation commuates with reduction” phenomenon as a special case of the functoriality

of quantisation, and uses equivariant K-homology and the K-theory of the group C∗-algebra

C∗(G) in a crucial way. For example, the equivariant index - which in the compact case takes

values in the representation ring R(G) - is replaced by the analytic assembly map - which

takes values in K0(C∗(G)) - familiar from the Baum–Connes conjecture in noncommutative

geometry. Under the usual freeness assumption on the action, we prove our conjecture for all

Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe

it is valid for all unimodular Lie groups.

1

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CONTENTS2

Contents

1Introduction2

2 Assumptions and result

2.1Assumptions

2.2The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Reduction by Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

11

13

15

16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3Differential operators on vector bundles

3.1The homomorphism VN

3.2Spaces of L2-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4Multiplication of sections by functions . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

17

18

20

24

24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Dirac operators

4.1The isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

25

26

5 Abelian discrete groups

5.1 The assembly map for abelian discrete groups . . . . . . . . . . . . . . . . . . . . .

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

5.3The operator part of the assembly map

5.4Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

29

31

33

34

. . . . . . . . . . . . . . . . . . . . . . . .

6Example: action of Z2non R2n

6.1Prequantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3The case n = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

35

36

36

A Naturality of the assembly map

A.1 The statement

A.2 Integrals of families of operators

38

38

39

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1Introduction

In 1982 two fascinating conjectures appeared about group actions. Guillemin and Sternberg [26]

gave a precise mathematical formulation of Dirac’s idea that “quantisation commutes with reduc-

tion” [13], in which they defined the former as geometric quantisation. As it stood, their conjecture

- which they proved under special assumptions - only made sense for actions of compact Lie groups

on compact symplectic manifolds. These compactness assumptions were left in place throughout

all later refinements in the formulation of the conjecture and the ensuing proofs thereof under more

general assumptions [32, 49, 50, 51, 53, 69, 71]. Baum and Connes, on the other hand, formulated

a conjectural description of the K-theory of the reduced C∗-algebra C∗

group G in terms of its proper actions [4]. Here the emphasis was entirely on the noncompact case,

as the Baum–Connes conjecture is trivially satisfied for compact groups. The modern formulation

of the conjecture in terms of the equivariant K-homology of the classifying space for proper G-

actions was given in [5]. In this version it has now been proved for all (almost) connected groups

G [10] as well as for a large class of discrete groups (see [70]).

r(G) of a locally compact

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1INTRODUCTION3

Although the two conjectures in question have quite a lot in common - such as the central

role played by index theory and Dirac operators,1or their acute relevance to modern physics,

especially the quantisation of singular phase spaces (cf. [41, 43]) - the compact/noncompact divide

(and perhaps also the sociological division between the communities of symplectic geometry and

noncommutative geometry) seems to have precluded much “interaction” between them. As we

shall see, however, some of the ideas surrounding the Baum–Connes conjecture are precisely what

one needs to generalize the Guillemin–Sternberg conjecture to the locally compact case.

Being merely desirable in mathematics, such a generalization is actually crucial for physics.

As a case in point, we mention the problem of constructing Yang–Mills theory in dimension 4 -

this is one of the Clay Mathematics Institute Millennium Prize Problems - where the groups and

spaces in question are not just noncompact, but even infinite-dimensional! Also, the work of Dirac

that initiated the modern theory of constraints and reduction in mechanics and field theory was

originally motivated by the problem - still open - of quantising general relativity, where again the

groups and spaces are infinite-dimensional. From this perspective, the present work, in which we

attempt to push the Guillemin–Sternberg conjecture beyond the compactness barrier to the locally

compact situation, is merely an exiguous step in the right direction.

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

(G ? M,ω)?

?

Q

??

RC

??

G ? Q(M,ω)

?

RQ

??

(MG,ωG)?

Q

??Q(MG,ωG).

(1)

Here (M,ω) is a symplectic manifold carrying a strongly Hamiltonian action of a Lie group G (cf.

Section 2), with associated Marsden–Weinstein quotient (MG,ωG), i.e.

MG= Φ−1(0)/G,(2)

where Φ : 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 Reduction as outlined by Dirac [13] is

defined as Marsden–Weinstein reduction [1, 46, 47, 48]. However - and this explains both the

fascination and the mystery of the field of constrained quantisation - the mathematical meaning

of the remaining three arrows in the diagram is open to discussion: see [13, 18, 21, 29, 38, 66] for

various perspectives. In any case, Q stands for Quantization, RQ denotes Quantum Reduction,

and all authors seem to agree that, if at all possible, the arrows should be 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 for the 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 that Q(M,ω) - the

quantisation of the phase space (M,ω) as such - should be a Hilbert 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 has been selected to construct these data; see below. Provided that

G is compact, the quantum reduction operation RQthen consists in taking the G-invariant part

Q(M,ω)Gof Q(M,ω). Similarly, Q(MG,ωG) is a Hilbert space without any further dressing. Now

assume that M and G are both compact: in that case, the reduced space MGis compact as well,

so that Q(M,ω) and Q(MG,ωG) 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 numerical equality

dim(Q(M,ω)G) = dim(Q(MG,ωG)).(3)

1Here the motivating role played by two closely related papers on the discrete series representations of semisimple

Lie groups should be mentioned. It seems that Parthasarathy [55] influenced Bott’s formulation of the Guillemin–

Sternberg conjecture (see below), whereas Atiyah and Schmid [3] in part inspired the Baum–Connes conjecture (see

[12, 36]).

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1INTRODUCTION4

This equality becomes a meaningful conjecture once an explicit construction of the objects

G ? Q(M,ω) and Q(MG,ωG) as specified above has been prescribed.

berg [26] considered the case in which the symplectic manifold M is compact, prequantisable,

and equipped with a positive-definite complex polarization J ⊂ TCM left invariant by the given

strongly Hamiltonian G-action. Recall that a symplectic manifold (M,ω) is called prequantisable

when the cohomology class [ω]/2π in H2(M,R) is integral, i.e., lies in the image of H2(M,Z) under

the natural homomorphism H2(M,Z) → H2(M,R). In that case, there exists a line bundle Lω

over M whose first Chern class c1(Lω) maps to [ω]/2π under this homomorphism; Lωis called the

prequantisation line bundle over M. Under these circumstances, the quantisation operation Q may

be defined through geometric quantisation (cf. [25] for a recent and pertinent treatment): one picks

a connection ∇ on Lωwhose curvature is ω, and defines the Hilbert space Q(M,ω) as the space

Q(M,ω) = H0(M,Lω) of polarized sections of Lω(i.e. of sections annihilated by all ∇X, X ∈ J).

This Hilbert space carries a natural unitary representation of G determined by the classical data,

as polarized sections of Lω are mapped into each other by the lift of the G-action. Moreover, it

turns out that the reduced space MG- assumed to be a manifold - inherits all relevant structures

on M (except, of course, the G-action), so that it is quantisable as well, in the same fashion. Thus

(3) becomes, in self-explanatory notation,

Guillemin and Stern-

dim(H0(M,Lω)G) = dim(H0(MG,LωG)),(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 earlier attempts to generalize geomet-

ric quantisation - notably in a cohomological direction - and the associated Guillemin–Sternberg

conjecture culminated in an unpublished proposal by Raoul Bott to define quantisation in terms

of the (equivariant) index of a suitable Dirac operator. See, e.g., [64]. As this definition forms

the starting point of our 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 Spincstructures and the

associated Dirac operators).

The first step in Bott’s definition of quantisation is to canonically associate a Spincstructure

(P,∼=) to a given symplectic and prequantisable manifold (M,ω) [25, 49]. First, one picks an

almost complex structure J on M that is compatible with ω (in that ω(·,J·) is positive definite

and symmetric, i.e. a metric). This J canonically induces a Spincstructure PJ on TM [16, 25],

but this is not the right one to use here. The Spincstructure P needed to quantise M is the

one obtained by twisting PJ with the prequantisation line bundle Lω. This means (cf. [25], App.

D.2.7) that P = PJ×ker(π)U(Lω), where π : Spinc(n) → SO(n) is the usual covering projection.

We denote the associated SpincDirac operator by D /L

even-dimensional, any Dirac operator on M worth its name decomposes in the usual way as

?

D /+

M. See for example [16, 20]. Since M is

D / =

0D /−

0

?

,(5)

and we abuse notation in writing

index(D / ) = dimker(D /+) − dimker(D /−).(6)

When M is compact, the Dirac operators (D /L

finite-dimensional kernels, whose dimensions define the quantisation of (M,ω) as

M)±determined by the Spincstructure (P,∼=) have

Q(M,ω) = index(D /L

M) ∈ Z.(7)

This number turns out to be independent of the choice of the Spincstructure on M, as long as

it satisfies the above requirement, and is entirely determined by the cohomology class [ω] (this is

not true for the Spincstructure and the associated Dirac operator itself) [25]. If the symplectic

manifold (M,ω) is K¨ ahler and the line bundle L is ‘positive enough’, then the index of D /L

Mequals

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1INTRODUCTION5

the dimension of the space H0(M,L) of holomorphic sections of the prequantum line bundle. This

provides some justification for Bott’s definition of quantisation.

So far, quantisation just associates an integer to (M,ω). Bott’s definition of quantisation gains

in substance when a compact Lie group G acts on M in strongly Hamiltonian fashion. In that

case, the pertinent Spincstructure may be chosen to be G invariant, and consequently the spaces

ker(D /±) are finite-dimensional complex G modules; we denote their isomorphism classes by square

brackets. In this situation we write

G-index(D / ) = [ker(D /+)] − [ker(D /−)],(8)

which defines an element of the representation ring2R(G) of G. Thus the quantisation of (M,ω)

with associated G-action may be defined as

Q(G ? M,ω) = G-index(D /L

M) ∈ R(G). (9)

As before, this element only depends on [ω] (and on the G-action, of course). When G is trivial,

one may identify R(e) with Z through the isomorphism

[V ] − [W] ?→ 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 com-

pact. Namely, in diagram (1) the upper right corner is now construed as an element of R(G),

whereas the lower right corner lies in R(e)∼= Z; as in the original case, the geometric quantisation

of the reduced space (MG,ωG) is defined whenever that of (M,ω) is. The quantum reduction map

RQ: R(G) → Z is simply defined by

RQ: [V ] − [W] ?→ ([V ] − [W])G:= dim(VG) − dim(WG),(10)

where VGis the G-invariant part of V , etc. Thus the Guillemin–Sternberg conjecture in the setting

of Bott’s definiton of quantisation simply reads

?

G-index?D /L

M

??G

= index?D /LG

MG

?.(11)

In this form, the conjecture was proved by Meinrenken [49], who merely assumed that M and MG

are orbifolds.3Also see [25, 32, 53, 69, 71] for other proofs and further references. A step towards

a noncompact version of the Guillemin-Sternberg conjecture has been taken by Paradan in [53],

where he considers actions by compact groups on possibly noncompact manifolds. He proves that

in this setting quantization commutes with reduction under certain conditions, that are met if the

manifold in question is a coadjoint orbit of a semisimple Lie group, and the group acting on it is

a maximal compact subgroup. Our generalisation is more or less in an 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 for noncompact groups and manifolds. We

specify our precise assumptions in Section 2 below; for the moment we just mention that it seems

impossible to even formulate the conjecture unless we assume that the strongly Hamiltonian action

G ? (M,ω) is proper and cocompact (or G-compact, which means that M/G is compact). When

this is the case, we may pass from the compact to the noncompact case by making the following

replacements (or lack of these) in the formalism:

1. Symplectic reduction is unchanged.

2. The definition of the SpincDirac operator D /L

Massociated to (M,ω) is unchanged.

2R(G) is defined as the abelian group with one generator [L] for each finite-dimensional complex representation

L of G, and relations [L] = [M] when L and M are equivalent and [L] + [M] = [L ⊕ M]. The tensor product of

representations defines a ring structure on R(G).

3Even that assumption turned out to be unnecessary [50].

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1INTRODUCTION6

3. The quantisation of the reduced space (which is compact by our regularity assumptions) is

unchanged.

4. The representation ring R(G) is replaced by K0(C∗(G)),4i.e. the usual K0 group of the

group C∗-algebra of G.5

5. The equivariant index G-index(D /L

M) ∈ R(G) is replaced by µG

M([D /L

M]) ∈ K0(C∗(G)), where

µG

M: KG

0(M) → K0(C∗(G)) (12)

is the analytic assembly map [5, 52, 70], KG

group defined by G ? M [5, 31, 34, 70], and [D /L

Dirac operator D /L

M.

0(M) is the equivariant analytical K-homology

M] is the class in KG

0(M) defined by the Spinc

6. Accordingly, the quantisation of the unreduced space (G ? M,ω) is now given by

Q(G ? M,ω) = G-index(D /L

M) ∈ K0(C∗(G)),(13)

where

G-index(D /L

M) := µG

M([D /L

M])(14)

purely as a matter of notation.6

7. The map RQ: R(G) → Z given by (10) is replaced by the map

RQ=??

?

G

?

∗: K0(C∗(G)) → Z

(15)

functorially induced by map

L1(G) or f ∈ Cc(G) and extended to f ∈ C∗(G) by continuity).7Here we make the usual

identification of K0(C) with Z. Again, purely as a matter of notation we write this map as

x ?→ xG.

G: C∗(G) → C given by f ?→

?

Gf(g)dg (defined on f ∈

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 unimodular Lie group,

let (M,ω) be a symplectic manifold, and let G ? M be a proper strongly Hamiltonian action.

Suppose 0 is a regular value of the associated momentum map. Suppose that the action is cocompact

and admits an equivariant prequantum line bundle L. Assume there is an almost complex structure

J on M compatible with ω. Let D /L

Mbe the Dirac operator on M associated to J and coupled to

L, and let D /LG

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

LG. Then

??

In this paper we will prove:

G

?

∗◦ µG

M

?D /L

M

?= index?D /LG

MG

?.

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

A special case of the situation described in Subsection 2.1 is the case where G is a torsion-free

discrete group acting freely and cocompactly on M. Then Conjecture 1.1 follows from a result of

Pierrot ([57], Theor` eme 3.3.2).

4The use of K0(C∗(G)) instead of K0(C∗

conjecture; we will clarify this point in footnote 7 below.

5See [7, 22, 62, 72] for the general K-theory of C∗-algebras, see [14, 56] for group C∗-algebras, and see [5, 63, 70]

for the K-theory of group C∗-algebras.

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).

7This 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 that?

of G on C by the usual correspondence between nondegenerate representations of C∗(G) and continuous unitary

representations of G [14, 56].

r(G)) is actually quite unusual in the context of the Baum–Connes

Gis just the representation of C∗(G) corresponding to the trivial representation

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1INTRODUCTION7

Example 1.3. Suppose G is a semisimple Lie group with maximal compact subgroup K, 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. Let Oλ⊂ g∗be the coadjoint orbit of

G through the element λ ∈ t∗. Then, if iλ is a dominant integral weight, we would expect the

quantisation of Oλ, coupled to a suitable line bundle, to be the class in K0(C∗(G)) that corresponds

to the discrete series representation Hλwhose lowest K-type has highest weight iλ. In this case

Conjecture 1.1 reduces to the uninteresting equality

Q?(Oλ)G

However, we can try to generalise Conjecture 1.1 so that the reduction map RQis replaced by

a reduction map Rµ

Q, which amounts to taking the multiplicity of the discrete series representation

whose lowest K-type is the dominant weight iµ instead of the multiplicity of the trivial represen-

tation. Furthermore, we note that the symplectic reduction (Oλ)µof Oλat the value µ is a point

if λ = µ, and empty otherwise. Therefore, we would expect that

Q?(Oλ)µ?= δλµ

= Rµ

= Rµ

?= Q(∅) = 0 = RQ(Q(Oλ)).

Q(Hλ)

Q(Q(Oλ)),

where δλµis the Kronecker delta. The study of Conjecture 1.1 for semisimple groups is work in

progress.

The truth of our generalized Guillemin–Sternberg conjecture for a special class of noncompact

groups may be some justification for our specific formulation of the generalization, but in fact there

is a much deeper reason why the “quantisation commutes with reduction” issue should be stated

in precisely the way we have given. Namely, in the above formulation the Guillemin–Sternberg

conjecture is a special case of the (conjectural) functoriality of quantisation. This single claim

summarizes a research program, of which the first steps may be found in the papers [39, 40, 42]. In

summary, one may define a “classical” category C and a “quantum” category Q, and construe the

act of quantisation as a functor Q : C → Q. While the categories in question haven been rigorously

constructed in [39] and [30, 33], respectively,8the existence of the functor Q is so far hypothetical.

However, the picture that emerges from the cases where Q has been constructed should hold in

complete generality: deformation quantisation (in the C∗-algebraic sense first proposed by Rieffel

[60, 61]) is the object side of Q, whereas geometric quantisation (in the sense of Bott as explained

above) is the arrow side of Q. Moreover, in the setting of strongly Hamiltonian group actions

as considered above, the “quantisation commutes with reduction” principle is nothing but the

functoriality of quantisation (in cases where the functor has indeed been defined).9

Outline of the proof

We believe our generalized Guillemin–Sternberg conjecture to be true for all unimodular Lie groups

G, but for reasons of human frailty we are only able to prove it in this paper when G has a discrete

normal subgroup Γ, such that the quotient group K := G/Γ is compact.10This incorporates a

number of interesting examples. Our proof is based on:

1. The validity of the Guillemin–Sternberg conjecture in the compact case [32, 49, 51, 50, 53,

69, 71];

8The objects of C are integrable Poisson manifolds and its arrows are regular Weinstein dual pairs; see [39] for the

meaning of the qualifiers. The arrows are composed by a generalization of the symplectic reduction procedure, and

isomorphism of objects in C turns out to be the same as Morita equivalence of Poisson manifolds in the sense of Xu

[75]. The category Q is nothing but the Kasparov–Higson category KK, whose objects are separable C∗-algebras

and whose sets of arrows are Kasparov’s KK-groups, composed with Kasparov’s intersection product. See [7, 30, 33].

9Let G ? (M,ω) define the Weinstein dual pair pt ← M → g∗in the usual way [73], the arrow → being given by

the momentum map. Functoriality of quantisation means that Q(pt ← M → g∗) ×KKQ(g∗←֓ 0 → pt) = Q((pt ←

M → g∗) ×C(g∗←֓ 0 → pt)). This equality is exactly the same as (11). See [42].

10Such groups are automatically unimodular. The Guillemin–Sternberg conjecture may not hold in the non-

unimodular case; see [19].

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1INTRODUCTION8

2. Naturality of the assembly map for discrete groups [52];

3. Symplectic reduction in stages [38, 45, 48];

4. Quantum reduction in stages.

In this paper we show, among other things, that the Guillemin–Sternberg conjecture for discrete

(and possibly noncompact) groups G is a consequence of the second point. For G as specified in

the previous paragraph, naturality of the assembly map implies a K-equivariant version thereof.

The third and fourth items are used in an almost trivial way, namely in setting up the following

diagram, which provides an outline of our proof.

Preq(G ? M,ω)

[D /•

M]

??

R(Γ)

C

??

KG

0(M)

µG

M

??K0(C∗(G))

R(Γ)

Q

??

Preq(K ? MΓ,ωΓ)

?

D /•

MΓ

?

??

R(K)

C

??

KK

0(MΓ)

µK

MΓ??K0(C∗(K))

R(K)

Q

??

Preq((MΓ)K),(ωΓ)K)

?

D /•

MG

?

??K0

?(MΓ)K

?

index

??Z

(16)

Here the following notation is used. We write

K=G/Γ;(17)

MΓ

=M/Γ,(18)

as Γ is discrete (so that its associated momentum map ΦΓis identically zero, whence Φ−1

Furthermore, Preq(G ? M,ω) is defined relative to a given Hamiltonian action of G on a symplectic

manifold (M,ω), and consists of all possible prequantisations (L,∇,H) of this action. A necessary

condition for Preq(G ? M,ω) to be nonempty is that the cohomology class [ω]/2π ∈ H2(M,R) be

integral. (If the group G is compact, this condition is also sufficient.) We make this assumption.

Similarly, Preq(K ? MΓ,ωΓ) is defined given the K-action on MΓinduced by the G-action on M,

and Preq((MΓ)K,(ωΓ)K) is just the set of prequantisations of the symplectic manifold

?(MΓ)K,(ωΓ)K

this isomorphism is a special and almost trivial case of the theorem on symplectic reduction in

stages [38, 45]. The maps R(Γ)

CC

denote Marsden–Weinstein reduction (at zero) with

respect to the groups Γ and K, respectively. We define the quantum counterparts of these maps

by

Γ(0) = M).

?∼= (MG,ωG);(19)

and R(K)

R(Γ)

Q

R(K)

Q

:=(?

??

Γ)∗;

?

(20)

:=

K

∗. (21)

Here (?

C∗(G/Γ) given by

Γ)∗: K0(C∗(G)) → K0(C∗(K)) is the map functorially induced by the map?

??

initially defined on f ∈ Cc(G); see [23] for the continuity of this map.11Finally, the maps involving

the symbol [D /•] are defined by taking the K-homology class of the Dirac operator coupled to a

given prequantum line bundle, (as outlined above and as explained in detail in the main body of

the paper below). Thus the commutativity of the upper part of diagram (16) is the equality

Γ: C∗(G) →

Γf?(Γg) =

?

γ∈Γ

f(γg),(22)

µK

MΓ

?D /LΓ

MΓ

?= R(Γ)

Q

?µG

M[D /L

M]?,(23)

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

Page 9

1INTRODUCTION9

whereas commutativity of the lower part yields

indexD /(LΓ)K

(MΓ)K= R(K)

Q

?µK

MΓ

?D /LΓ

MΓ

??.(24)

It is easily shown that

?

K◦?

Γ=?

G,(25)

so that by functoriality of K0one has

R(K)

Q

◦ R(Γ)

Q= R(G)

Q, (26)

with R(G)

Q

= RQas in (15). Using (26) and

R(K)

C

◦ R(Γ)

C

= R(G)

C

:= RC, (27)

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

Preq(G ? M,ω)

RC

??

Q

??K0(C∗(G))

RQ

??

Preq(MG,ωG)

Q

??Z.

(28)

Clearly, commutativity of (28) is precisely the commutativity of diagram (1) with the post-modern

meaning we have given to its ingredients. Since diagram (28) commutes when the two inner

diagrams in diagram (16) commute, the latter would prove our generalized Guillemin–Sternberg

conjecture. Now the lower diagram commutes by the validity of the Guillemin–Sternberg conjecture

for compact K, whereas the upper diagram decomposes as

Preq(G ? M,ω)

[D /•

M]

??

R(Γ)

C

??

KG

0(M)

µG

M??

VΓ

??

K0(C∗(G))

R(Γ)

Q

??

Preq(K ? MΓ,ωΓ)

?

D /•

MΓ

?

??KK

0(MΓ)

µK

MΓ??K0(C∗(K)),

(29)

where VΓis a map defined in Subsection 3.1 (the V stands for Valette, who was the first to write

this map down in a more special context). Verifying the commutativity of 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 generalization of the nat-

urality of the assembly map for discrete groups as proved by Valette [52] to possibly nondiscrete

groups. This is dealt with in Appendix A. The commutativity of the left-hand inner diagram is

Theorem 2.9:

VΓ[D /L

M] =?D /LΓ

MΓ

?.(30)

The proof of this result occupies Sections 3 and 4.

In Section 3, we compute the image under the map VΓof a K-homology class associated to a

general equivariant, elliptic, symmetric, first order differential operator D on a Γ-vector bundle E

over a Γ-manifold M. If the action of Γ on M is free, as we assume, then the quotient space E/Γ

defines a vector bundle over M/Γ. The operator D induces an operator DΓon this quotient bundle.

It turns out that the homomorphism VΓmaps the class associated to D to the class associated to

DΓ.

In Section 4, we show that if D /L

Mis the Dirac operator on a symplectic manifold M, coupled

to a prequantum line bundle L, then the operator?D /L

the Dirac operator on the quotient M/Γ coupled to the line bundle L/Γ.

As an encore, in Section 5 we give an independent proof of our generalized Guillemin–Sternberg

conjecture for the case that G is discrete and abelian. This proof, based on a paper by Lusztig

M

?Γfrom the previous paragraph is precisely

Page 10

2ASSUMPTIONS AND RESULT10

[44] (see also [5], pp. 242–243) gives considerable insight in the situation. It is based on an explicit

computation of the image under µΓ

Mof a K-homology class [D] associated to a Γ-equivariant

elliptic differential operator D on a Γ-vector bundle E over a Γ-manifold M. Because in this case

C∗(Γ)∼= C(ˆΓ) (withˆΓ the unitary dual of Γ), this image corresponds to the formal difference of two

equivalence classes of vector bundles overˆΓ. These bundles are described as the kernel and cokernel

of a ‘field of operators’?Dα

Dα and the bundles Eα are constructed explicitly from D and E, respectively. The quantum

reduction of the class µΓ

trivial representation. Because D1is the operator DΓmentioned above, the Guillemin–Sternberg

conjecture follows from the computation in Section 4.

Finally, in Section 6 we check the discrete abelian case in an instructive explicit computation.

We will see that the quantisation of the action of Z2on R2corresponds to a certain line bundle

over the two-torus T2=ˆZ2. The quantum reduction of this K-theory class is its rank, the integer

1. This is also the quantisation of the reduced space T2= R2/Z2, as can be seen either directly or

by applying Atiyah-Singer for Dirac operators. Although this is the simplest example of Guillemin-

Sternberg for noncompact groups and spaces, the details are nontrivial and, in our opinion, well

worth spelling out.

?

α∈ˆΓon a ‘field of vector bundles’?Eα→ M/Γ?

M[D] is the index of the operator D1on E1→ M/Γ, where 1 ∈ˆΓ is the

α∈ˆΓ. The operators

Acknowledgements

This work is supported by n.w.o. through grant no. 616.062.384 for the second author’s ‘Pionier

project’ Quantization, noncommutative geometry, and symmetry.

The authors would like to thank Erik van den Ban, Rogier Bos, Siegfried Echterhoff, Gert

Heckman, Herv´ e Oyono-Oyono, John Roe, Elmar Schrohe, Alain Valette and Jan Wiegerinck for

useful suggestions at various stages of this work. The authors are also indebted to the referees for

several useful remarks.

2Assumptions and result

We now state the assumptions under which we will prove our generalised Guillemin–Sternberg con-

jecture, i.e. Theorem 1.2. These assumptions are mainly used in the proof of our key intermediate

result, Theorem 2.9, which is proved in Sections 3 and 4.

We first fix some notation and assumptions. If M is a manifold, then the spaces of vector fields

and differential forms on M are denoted by X(M) and Ω∗(M), respectively. The symbol ? denotes

contraction of differential forms by vector fields. If the manifold M is equipped with an almost

complex structure, then we have the space Ω0,∗(M) of differential forms on M of type (0,∗). Unless

stated otherwise, all manifolds, maps and actions are supposed to be C∞.

If a vector bundle E → M is given (which is supposed to be complex unless stated otherwise),

then the space of smooth sections of E is denoted by C∞(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 M with coefficients in E is denoted by Ω∗(M;E), and similarly we

have the space Ω0,∗(M;E) for almost complex manifolds. If F → M is another vector bundle, and

ϕ : E → F is a homomorphism of vector bundles, then composition with ϕ gives a homomorphism

of C∞(M)-modules

˜ ϕ : C∞(M,E) → C∞(M,F).

If a group G acts on M, and if E is a G-vector bundle over M, then we have the canonical

representation of G on C∞(M,E) given by (g ·s)(m) = g ·s(g−1m), for g ∈ G and s ∈ C∞(M,E).

A superscript ‘G’ denotes the subspace of G-invariant elements. Thus we obtain for example the

vector spaces C∞(M,E)G, Ω0,∗(M)G, etc. The Lie algebra of a Lie group is denoted by a lower

case Gothic letter, so that for example the group G has the Lie algebra g.

In the context of Hilbert spaces and Hilbert C∗-modules, we denote the spaces of bounded,

compact and finite-rank operators by B, K and F, respectively.

Page 11

2 ASSUMPTIONS AND RESULT11

The spaces of continuous functions, bounded continuous functions, continuous functions vanish-

ing at infinity and compactly supported continuous functions on a topological space X are denoted

by C(X), Cb(X), C0(X) and Cc(X), respectively.

2.1Assumptions

Let (M,ω) be a symplectic manifold, and let G be a Lie group. Suppose that G has a discrete,

normal subgroup Γ, such that the quotient group K := G/Γ is compact. For example,

• G = K, Γ = {e} with K a compact Lie group,

• G = Γ discrete,

• G = Rn, Γ = Znso that G/Γ is the torus Tn,

or direct products of these three examples. In fact, if G is connected, then the subgroup Γ must

be central, and G is the product of a compact group and a vector space.

The assumption that G/Γ 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 compact groups 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 ω.

3. The quotient space M/G is compact.

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

Φ : M → g∗,

that is equivariant with respect to the co-adjoint representation of G in g∗, such that for all

X ∈ g,

dΦX= −XM?ω.

Here ΦX is the function on M obtained by pairing Φ with X and XM is the vector field on

M induced by X.

5. The discrete subgroup Γ acts freely on M, and the whole group G acts freely on the level set

Φ−1(0).

6. The symplectic manifold (M,ω) admits a G-equivariant prequantisation. That is, there is a

G-equivariant complex line bundle

L → M,

equipped with a G-invariant Hermitian metric H, and a Hermitian connection ∇ with curva-

ture two-form ∇2= 2πiω, which is equivariant as an operator from C∞(M,L) to Ω1(M;L).

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

B(·,·) := ω(·,J·)

defines a Riemannian metric on M.

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

12Sometimes an action is called ‘Hamiltonian’ as opposed to ‘strongly Hamiltonian’ if it admits a momentum

map that is not necessarily equivariant or Poisson. We will not use this terminology; for us the word ‘Hamiltonian’

always means ‘strongly Hamiltonian’.

Page 12

2ASSUMPTIONS AND RESULT12

Ad 3. Because the quotient G/Γ is compact, compactness of M/G is equivalent to compactness

of M/Γ. The latter assumption is used in the proof of Lemma 3.3 (where Γ is replaced by a general

closed normal subgroup N), but it is not essential. Furthermore, in the definitions of K-homology

in [5, 52] it is assumed that the orbit spaces of the group actions involved are compact.13Finally,

compactness of M/Γ allows us to apply the Guillemin-Sternberg conjecture for compact groups

and spaces to this quotient.

Ad 4.The map Φ is called a momentum map of the action.

assumption that Φ is equivariant is equivalent to the assumpion that it is a Poisson map with

respect to the negative Lie-Poisson structure on g∗(see for example [38], Corollary III.1.2.5 or

[46], §11.6). Note that if G = Γ is discrete, then the action is automatically Hamiltonian. Indeed,

Lie(Γ) = {0}, so that the zero map is a momentum map.

Ad 5. Freeness of the action of Γ on M implies that the quotient M/Γ is smooth, and that a

Γ-vector bundle E → M induces a vector bundle E/Γ → M/Γ. And if G acts freely on Φ−1(0),

then it follows from de definition of momentum maps that 0 is a regular value of Φ (Smale’s

lemma [65]), so that the reduced space MGis a smooth manifold. If freeness is replaced by local

freeness in Assumption 5, then M/Γ and MGare orbifolds. In that case, a quotient vector bundle

E/Γ → M/Γ can be replaced by ‘the vector bundle over M/Γ whose space of smooth sections is

C∞(M,E)Γ(see Proposition 2.8).14

Ad 6. Example 2.3 below shows that it is not always obvious if this assumption is satisfied.

Equivariance of the connection ∇ implies that the Dolbeault–Dirac operator on M, coupled to L

via ∇ (Definition 2.5) is equivariant.

The Kostant formula

X ?→ −∇XM+ 2πiΦX

For connected groups, the

defines a representation of g in the space of smooth sections of L. In the literature on the Guillemin-

Sternberg conjecture, it is usually assumed that the action of G on L is such that the corresponding

representation of g in C∞(M,L) is given by the Kostant formula. Then, if the group G is connected,

the connection ∇ satisfies g∇vg−1= ∇g·vfor all g ∈ G and v ∈ X(M). This property is equivalent

to equivariance of the connection ∇ in the sense of assumption 6.

If the manifold M is simply connected and the group G is discrete, then Hawkins [27] gives a

procedure to lift the action of G on M to a projective action on the trivial line bundle over M, such

that a given connection is equivariant. Under a certain condition (integrality of a group cocycle),

this projective action is an actual action.

Ad 7. An equivalent assumption is that there exists a G-invariant Riemannian metric 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. In Section 3,

we will work under more general assumptions.

Remark 2.1. If the group G is compact, then some of these assumptions are always satisfied.

First of all, the action is automatically proper. Furthermore, if the cohomology class [ω] ∈ H2

is integral, then a prequantum line bundle exists, and the connection can be made equivariant by

averaging over G. Also, averaging over G makes any Riemannian metric G-invariant, so that

assumption 7 is also satisfied. And finally, since M/G is compact, so is M. In particular, M is

complete, so assumption 8 is satisfied.

dR(M)

Remark 2.2. If the action of G on M is (locally) free and Hamiltonian, and M/G is compact, then

G must be discrete. Indeed, if the action is locally free then by Smale’s lemma the momentum map

Φ is a submersion, and in particular an open mapping. And since it is G-equivariant, it induces

ΦG: M/G → g∗/Ad∗(G),

13K-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 on M. For in that case, G must be

discrete (see Remark 2.2).

Page 13

2ASSUMPTIONS AND RESULT13

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

ΦG(M/G) ⊂ g∗/Ad∗(G)

is a compact open subset. Because g∗/Ad∗(G) is connected, it must therefore be compact. This,

however, can only be the case (under our assumptions) when G is discrete.15Indeed, we have

Ad∗(G)∼= Ad∗(K) ⊂ GL(k∗)∼= GL(g∗).

So Ad∗(G) is compact, and g∗/Ad∗(G) cannot be compact, unless g∗= 0, i.e. G is discrete.

Example 2.3. Let M = C, with coordinate z = q + ip, and the standard complex structure. We

equip M with the symplectic form ω = dp ∧ dq.

Consider the group G = Γ = Z + iZ ⊂ C. We let it act on M by addition:

(k + il) · z = z + k + il,

for k,l ∈ Z, z ∈ C.

Consider the trivial line bundle L = M × C → M. We define an action of Z + iZ on L by

letting the elements 1,i ∈ Z + iZ act as follows:

1 · (z,w) = (z + 1,w);

i · (z,w) = (z + i,e−2πizw),

for z,w ∈ C. Define a Hermitian metric H on L by

H ((q + ip,w),(q + ip,w′)) = e2π(p−p2)w ¯ w′.

This metric is Z + iZ-invariant, and the connection

∇ = d + 2πipdz + π dp

is Hermitian, Z + iZ-invariant, and has curvature form 2πiω.

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,ω1) is a compact symplectic manifold, K is a compact Lie group,

and let a proper Hamiltonian action of K on M1 be given. Let Φ be the momentum map, and

suppose K acts freely on Φ−1(0). Suppose [ω] is an integral cohomology class. (These assumptions

are made for example by Tian & Zhang [69].) Let Γ be a discrete group acting properly and freely

on a symplectic manifold (M2,ω2), leaving ω2invariant. Suppose that M2/Γ is compact, and that

there is an equivariant prequantum line bundle over M2. Then the direct product action of K ×Γ

on M1× M2satisfies the assumptions of this section.

2.2 The Dirac operator

The prequantum connection ∇ on L induces a differential operator

∇ : Ωk(M;L) → Ωk+1(M;L),

such that for all α ∈ Ωk(M) and s ∈ C∞(M,L),

∇(α ⊗ s) = dα ⊗ s + (−1)kα ∧ ∇s.

Let π0,∗be the projection

π0,∗: Ω∗

C(M;L) → Ω0,∗(M;L).

15If G = K is a compact connected Lie group, then k∗/Ad∗(K) is a Weyl chamber.

Page 14

2ASSUMPTIONS AND RESULT14

Composing the restriction to Ω0,q(M;L) of the complexification of ∇ with π0,∗, we obtain a

differential operator

¯∂L:= π0,∗◦ ∇ : Ω0,q(M;L) → Ω0,q+1(M;L).

Let

¯∂∗

L: Ω0,q+1(M;L) → Ω0,q(M;L)

be the formal adjoint of¯∂Lwith respect to the L2-inner product on compactly supported forms.

Definition 2.5 (Dolbeault–Dirac operator). The Dolbeault–Dirac operator on M, coupled to

L via ∇ is the differential operator

D /L

M:=¯∂L+¯∂∗

L: Ω0,∗(M;L) → Ω0,∗(M;L).

This operator is G-equivariant by equivariance of the connection ∇ (assumption 6) and invari-

ance of the almost complex structure J (assumption 7).

The Dolbeault–Dirac operator defines an unbounded symmetric operator on the Hilbert space

of L2-sections

HL

with respect to the Liouville measure dm on M. The metric on?0,∗T∗M ⊗ L comes from the

given metric H on L and from the Riemannian metric B = ω(·,J·) on TM.

metrically complete, the closure of D /L

functional calculus (see e.g. [58]) to define the bounded operator FL

is a normalising function [31]:

M:= L2(M,?0,∗T∗M ⊗ L),

Because M is

Mis a self-adjoint operator on HL

M.16So we can apply the

M:= b?D /L

M

?on HL

M, where b

Definition 2.6. A smooth function b : R → R is called a normalising function if it has the

following three properties:

• b is odd;

• b(t) > 0 for all t > 0;

• limt→±∞b(t) = ±1.

Let C0(M) denote the C∗-algebra of continuous functions on M that vanish at infinity, and let

B(HL

M) be the the C∗-algebra of bounded operators on HL

M. Let

πM: C0(M) → B(HL

M)

be the representation of C0(M) on HL

the triple (HL

M,πM) defines a K-homology class

Mdefined by multiplication of sections by functions. Then

M,FL

?D /L

M

?:= [HL

M,FL

M,πM] ∈ KG

0(M),

which is independent of b. See [5, 31, 52] for the definition of K-homology, and in particular

Theorem 10.6.5 in [31] for the claim that?D /L

Remark 2.7. The Dolbeault–Dirac operator has the same principal symbol as the SpincDirac

operator associated to the almost complex structure J on M and the line bundle L (see e.g. [16],

page 48), namely the Clifford action of T∗M on?0,∗T∗M ⊗ L. So the two operators define the

same class in K-homology. (The linear path between the operators provides a homotopy.) Thus,

if we consider the K-homology class [D /L

or the Dolbeault–Dirac operator.

M

?defines a K-homology class.

M], we may take D /L

Mto be either the SpincDirac operator

16This follows from the connection between Dirac operators and Riemannian metrics as given for example in [11],

section VI.1, combined with Section 10.2 of [31]. See also [74] and page 96 of [20].

Page 15

2ASSUMPTIONS AND RESULT15

2.3Reduction by Γ

Because the subgroup Γ of G is discrete, the symplectic quotient MΓof M by Γ is equal to the

orbit manifold M/Γ. The symplectic form ωΓon M/Γ is determined by

p∗ωΓ= ω,

with p : M → M/Γ the quotient map. The action of G on M descends to an action of G/Γ on

M/Γ, which satisfies the assumptions of Section 2.1 (with M, ω, and G replaced by M/Γ, ωΓand

G/Γ, respectively). The quotient L/Γ turns out to be the total space of a prequantum line bundle

over M/Γ. This is implied by the following fact.

Proposition 2.8. Let H be a group acting properly and freely on a manifold M. Let q : E → M

be an H-vector bundle. Then the induced projection

qH: E/H → M/H

defines a vector bundle over M/H.

Let C∞(M,E)Hbe the space of H-invariant sections of E. The linear map

ψE: C∞(M,E)H→ C∞(M/H,E/H),(31)

defined by

ψE(s)(H · m) = H · s(m),

is an isomorphism of C∞(M)H ∼= C∞(M/H)-modules.

Hence the quotient space L/Γ is a complex line bundle over M/Γ, and its space of sections is

C∞(M/Γ,L/Γ)∼= C∞(M,L)Γ.

The connection ∇ on L is G-equivariant, so it defines a G/Γ-equivariant connection ∇Γon L/Γ

as follows. Let p : M → M/Γ be the quotient map. Its tangent map Tp : TM → T(M/Γ) induces

an isomorphism of vector bundles over M/Γ

TpΓ: (TM)/Γ → T(M/Γ).

Let

?

TpΓ: C∞(M/Γ,(TM)/Γ)

∼

=

− → X(M/Γ)

be the isomorphism of C∞(M/Γ) modules induced by TpΓ. Consider the isomorphism of C∞(M)Γ ∼= C∞(M/Γ)-

modules

ϕ : X(M)ΓψTM

− − − → C∞(M/Γ,(TM)/Γ)

?

TpΓ

− − → X(M/Γ).

Let v ∈ X(M)Γbe a Γ-invariant vector field on M. Then the operator ∇v is Γ-equivariant, and

hence maps Γ-invariant scetions of L to invariant sections. The covariant derivative ∇Γ

C∞(M/Γ,L/Γ) is defined by the commutativity of the following diagram:

ϕ(v)on

C∞(M/Γ,L/Γ)

??

∇Γ

ϕ(v)??C∞(M/Γ,L/Γ)

C∞(M,L)Γ

ψL

∼

=

∇v

??C∞(M,L)Γ.

ψL

∼

=

??

A computation shows that ∇Γsatisfies the properties of a connection, and that its curvature is

?∇Γ?2= 2πiωΓ.

Furthermore, the G-invariant Hermitian metric H on L descends to a G/Γ-invariant Hermitian

metric HΓon L/Γ, and the connection ∇Γis Hermitian with respect to this metric. Finally, the

Page 16

2ASSUMPTIONS AND RESULT16

G-invariant almost complex structure J on TM induces a G/Γ-invariant almost complex structure

JΓon T(M/Γ)∼= (TM)/Γ. The corresponding Riemannian metric is denoted by

BΓ= ωΓ(·,JΓ·).

From the Dirac operator D /L/Γ

to the line bundle L/Γ via ∇Γ, we form the bounded operator FL/Γ

M/Γon M/Γ associated to the almost complex structure JΓ, coupled

M/Γ:= b

?

D /L/Γ

M/Γ

?

(where b is a

normalising function) on the Hilbert space

HL/Γ

M/Γ:= L2?

M/Γ,?0,∗T∗(M/Γ) ⊗ L/Γ

?

,

which is defined with respect to the metrics on?0,∗T∗(M/Γ) and L/Γ coming from those on

Let U ⊂ M be a fundamental domain for the Γ-action. That is, U is an open subset, Γ · U is

dense in M, and if m is a point in U, and γ ∈ Γ is such that γ · m ∈ U, then γ = e. Then for all

measurable functions f on M/Γ we define

?

M/Γ

?0,∗T∗M and L respectively, and the measure dO on M/Γ defined as follows.

f(O)dO :=

?

U

p∗f(m)dm,(32)

where p : M → M/Γ is the quotient map. If V is another fundamental domain, the subsets

Γ · U and Γ · V differ by a set of measure zero, so this definition does not depend on the choice

of the fundamental domain. An equivalent way of defining dO is to say that the dO-measure

of a measurable subset A ⊂ M/Γ equals the dm-measure of the subset p−1(A) ∩ U of M. And

since dm is the Liouville measure on (M,ω), the measure dO is precisely the Liouville measure on

(M/Γ,ωΓ).

We then have the K-homology class

?

D /L/Γ

M/Γ

?

:= [HL/Γ

M/Γ,FL/Γ

M/Γ,πM/Γ] ∈ KG/Γ

0

(M/Γ).

2.4 The main result

In Subsection 3.1 we define a homomorphism

VΓ: KG

0(M) → KG/Γ

0

(M/Γ),

such that the following diagram commutes:

KG

0(M)

µG

M

??

VΓ

??

K0(C∗(G))

R(Γ)

Q

??

KG/Γ

0

(M/Γ)

µG/Γ

M/Γ??K0(C∗(G/Γ)).

(33)

Here µG

defined in (20). In Appendix A, we sketch how to generalise Valette’s proof in [52] of commutativity

of diagram (33) (‘naturality of the assembly map’) for discrete groups to the nondiscrete case.

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

Mand µG/Γ

M/Γare analytic assembly maps (see [5, 70, 52]), and the homomorphism R(Γ)

Q

is

Theorem 2.9. The homomorphism VΓmaps the K-homology class of the Dirac operator D /L

the K-homology class of the Dirac operator D /L/Γ

M/Γon the reduced space M/Γ:

?

Mto

VΓ

[D /L

M]

?

=

?

D /L/Γ

M/Γ

?

∈ KG/Γ

0

(M/Γ).

Page 17

3DIFFERENTIAL OPERATORS ON VECTOR BUNDLES17

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–Sternberg conjecture for compact G and M.

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

?[HL

Then there is a unitary isomorphism

VΓ

M,FL

M,πM]?= [HΓ,FΓ,πΓ].

χ : HΓ→ HL/Γ

M/Γ

that intertwines the pertinent representations of G/Γ and of C(M/Γ), and the operators FΓand

FL/Γ

M/Γ.

3Differential operators on vector bundles

In this section, we will compute the image under the homomorphism VN in diagram (33) of a

K-homology class associated to an equivariant elliptic first order differential operator on a vector

bundle over a smooth manifold. The result is Corollary 3.13. In Section 4 we will see that Theorem

2.9 is a special case of Corollary 3.13.

Let G be a unimodular Lie group, and let N be a closed normal subgroup of G. Let dg and

dn be Haar measures on G and N respectively. Let M be a smooth manifold on which G acts

properly, such that the action of N on M is free. Suppose M/N is compact.17

3.1The homomorphism VN

Let us briefly state the definition of the homomorphism

VN: KG

0(M) → KG/N

0

(M/N).

For details we refer to [52]. Let H be a Z2-graded Hilbert space carrying a unitary representation

of G, F a G-equivariant bounded operator on H, and π a representation of C0(M) in H that is

G-equivariant in the sense that for all g ∈ G and f ∈ C0(M), one has gπ(f)g−1= π(g·f). Suppose

(H,F,π) defines a K-homology cycle. Then

VN[H,F,π] := [HN,FN,πN],

with HN, FN and πN defined as follows.18

Consider the subspace Hc:= π(Cc(M))H ⊂ H and the sesquilinear form (·,·)Non Hcgiven by

?

N

for ξ,η ∈ Hc. This form turns out to be positive semidefinite. Consider the quotient space of Hc

by the kernel of this form, and complete this quotient in the inner product (·,·)N. This completion

is HN.

Next, we use the fact that any K-homology class can be represented by a cycle whose operator

is properly supported:

(ξ,η)N:=(ξ,n · η)Hdn,

Definition 3.1. The operator F is properly supported if for every f ∈ Cc(M) there is an h ∈ Cc(M)

such that π(h)Fπ(f) = Fπ(h).

Suppose F is properly supported. Then it preserves Hc, and the restriction of F to Hc is

bounded with respect to the form (·,·)N. Hence F|Hcinduces a bounded operator FN on HN by

continuous extension. The representation π extends to the multiplier algebra Cb(M) of C0(M).

The algebra C0(M/N) can be embedded into Cb(M) via the isomorphism C(M/N)∼= C(M)N,

and then we can use an argument similar to the one used in the definition of FN to show that π

induces a representation πN of C0(M/N) in HN.

17Compactness of M/N is used in the proof of Lemma 3.3, but not in an essential way (see Footnote 21).

18The construction below originated in Rieffel’s theory of induced representations of C∗-algebras [59], which

independently found its way into the Baum–Connes conjecture [5] and into the theory of constrained quantisation

[37, 38].

Page 18

3DIFFERENTIAL OPERATORS ON VECTOR BUNDLES 18

3.2 Spaces of L2-sections

Now let q : 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 of square-integrable sections of E with

respect to this measure. Let πM: C0(M) → B(L2(M,E)) be the representation defined by multi-

plying sections by functions. Let L2(M,E)Nbe the Hilbert space constructed from L2(M,E) as in

the definition of 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 quotient vector bundle

qN: E/N → M/N

(see Proposition 2.8). The L2-inner product on sections of E/N is defined via the metric on E/N

induced by the one on E, and the measure dO on M/N with the property that for all measurable

sections20ϕ : M/N → M and all f ∈ Cc(M),

?

M

M/N

f(m)dm =

??

N

f(n · ϕ(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

L2

c(M,E) := π(Cc(M))L2(M,E)

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

χ : L2

c(M,E) → L2(M/N,E/N),(35)

defined by

χ(s)(Nm) := N ·

?

N

n · s(n−1m)dn,

for all s ∈ L2

integrand is compactly supported for all m ∈ M.

c(M,E) and m ∈ M. Because s is compactly supported and the action is proper, the

Proposition 3.2. The map χ induces a natural G/N-equivariant unitary isomorphism

χ : L2(M,E)N

∼

=

− → L2(M/N,E/N).(36)

Proof. It follows from a lengthy but straightforward computation that the map χ is isometric, in

the sense that for all s ∈ L2

c(M,E),

?χ(s)?L2(M/N,E/N)= ?s?N,

where ? · ?N is the norm corresponding to the inner product (·,·)N. Furthermore, χ is surjective,

see Lemma 3.3 below. By these two properties, χ induces a bijective linear map

χ : L2

c(M,E)/K → L2(M/N,E/N),(37)

where K is the space of sections s ∈ L2

linear isomorphism from L2

L2

So (37) is actually a unitary isomorphism

c(M,E) with ?s?N = 0. This map is a norm preserving

c(M,E)/K onto the complete space L2(M/N,E/N). Hence the space

c(M,E)/K is already complete, so that L2(M,E)N= L2

c(M,E)/K.

χ : L2(M,E)N→ L2(M/N,E/N).

The fact that N is a normal subgroup implies that this isomorphism intertwines the pertinent

representations of G/N.

?

19A natural isomorphism between Hilbert spaces is an isomorphism defined without choosing bases of the spaces

in question.

20Measurable in the sense that the inverse image of any Borel measurable subset of M is Borel measurable in

M/N.

Page 19

3DIFFERENTIAL OPERATORS ON VECTOR BUNDLES 19

Lemma 3.3. The map χ in (35) is surjective.

Proof. Let σ ∈ L2(M/N,E/N). We will construct a section s ∈ L2

using the following diagram:

c(M,E) such that χ(s) = σ,

E

pE??

q

??

E/N

qN

??

M

p??M/N.

Here the horizontal maps are quotient maps and define principal fibre bundles, and the vertical

maps are vector bundle projections.

Let {Uj} be an open cover of M/N that admits local trivialisations

τj: p−1(Uj)

∼

=

− → Uj× N

∼

=

− → Uj× E0.θN

j: q−1

N(Uj)

Here E0is the typical fibre of E. Because M/N is compact, the cover {Uj} may be supposed to

be finite. Via the isomorphism of vector bundles p∗(E/N)∼= E, the trivialisations θN

trivialisations of E:

θj: q−1(p−1(Uj))

jinduce local

∼

=

− → p−1(Uj) × E0.

And then, we can form trivialisations

τE

j: p−1

E(q−1

N(Uj))

∼

=

− → q−1

N(Uj) × N,

by

p−1

E(q−1

N(Uj)) = q−1(p−1(Uj))

∼= p−1(Uj) × E0

∼= Uj× N × E0

∼= q−1

via θj

via τj

via θN

N(Uj) × N

j.

Here the symbol ‘∼=’ indicates an N-equivariant diffeomorphism. It follows from the definition of

the trivialisation θjthat τE

an isomorphism of principal N-bundles.

For every j, define the section sj∈ L2(M,E) by

jcomposed with projection onto q−1

N(Uj) equals pE, so that τE

jis indeed

sj(τ−1

j

(O,n)) =?τE

j

?−1(σ(O),n)

for all O ∈ Uj and n ∈ N, and extended by zero outside p−1(Uj). By compactness21of M/N,

there is a compact subset? C ⊂ M that intersects all N-orbits. Let K ⊂ N be a compact subset of

Vm:= {n ∈ N;n−1m ∈ C}

dn-volume 1, and set C := K ·?C. Then for all m ∈ M, the volume of the compact set

is at least 1. Define the section ˜ s of E by

˜ s(m) =

? ?

jsj(m)if m ∈ C

if m ?∈ C.0

21This is the only place where compactness of M/N is used (the covering {Uj} may also be locally finite). And

even here, this assumption is not essential: it follows from this proof that all compactly supported L2-sections of

E/N are in the image of χ. Hence χ has dense image, so that the induced map from L2(M,E)Nto L2(M/N, E/N)

is surjective. By small adaptations to the proofs of Propositions 3.2 and 3.8, everything in this section still applies

if M/N is noncompact. But because we assume that the quotient M/Γ is compact anyway, we will take the lazy

option and suppose that M/N is compact.

Page 20

3DIFFERENTIAL OPERATORS ON VECTOR BUNDLES20

Then ˜ s ∈ L2

c(M,E), and for all m ∈ M,

χ(˜ s)(Nm) =

?

j,

Nm∈Uj

?

j,

Nm∈Uj

?

Vm

pE

?n · sj(n−1m)?dn

=

?

Vm

pE

??τE

j

?−1(σ(Nm),n · ψj(n−1m))?dn,

where (Nm,ψj(n−1m)) := τj(n−1m). Now since pE◦?τE

integral equals

j

?−1is projection onto q−1

N(Uj), the latter

#{j;Nm ∈ Uj}vol(Vm)σ(Nm).

Setting Φ(m) := #{j;Nm ∈ Uj}vol(Vm) gives a measurable function Φ on M which is bounded

below by 1 and N-invariant by invariance of dn. Hence

s :=1

Φ˜ s,

is a section s ∈ L2

c(M,E) for which χ(s) = σ.

?

3.3Differential operators

Let G and E → M be as in Section 3.2. Let D : C∞(M,E) → C∞(M,E) be a G-equivariant first

order differential operator that is symmetric with respect to the L2-inner product on compactly

supported sections. Then D defines an unbounded operator 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 the bounded, self-

adjoint operator b(D) on L2(M,E), for any bounded measurable function b on R.22The operator

b(D) is G-equivariant because of the following result about functional calculus of unbounded op-

erators, which follows directly from the definition as given for example in [58], page 261.

Lemma 3.4. Let H be a Hilbert space, and let D ⊂ H be a dense subspace. Let a : D → H

be a self-adjoint operator. Let H′be another Hilbert space, and let T : H → H′be a unitary

isomorphism. Let f be a measurable function on R. Then

Tf(a)T−1= f(TaT−1).

We will later consider the case where?L2(M,E),b(D),πM

the map VN to this cycle. It is therefore important to us that the operator b(D) is properly

supported (Definition 3.1) for well-chosen functions b:

?is a K-homology cycle, and apply

Proposition 3.5. If b is a bounded measurable function with compactly supported (distributional)

Fourier transformˆb, then the operator b(D) is properly supported.

The proof of this proposition is based on the following two facts, whose proofs can be found in

[31], Section 10.3.

Proposition 3.6. If b is a bounded measurable function on R with compactly supported Fourier

transform, then for all s,t ∈ C∞

?

R

22If D is elliptic and b is a normalising function , then?L2(M,E),b(D), πM

over M (see Theorem 10.6.5 in [31]).

c(M,E),

?b(D)s,t?

L2(M,E)=

1

2π

?eiλDs,t?

L2(M,E)ˆb(λ)dλ.

?is an equivariant K-homology cycle

Page 21

3DIFFERENTIAL OPERATORS ON VECTOR BUNDLES21

This is Proposition 10.3.5. from [31]. By Stone’s theorem, the operator eiλDis characterised

by the requirements that λ ?→ eiλDis a group homomorphism from R to the unitary operators on

L2(M,E), and that for all s ∈ C∞

????

Lemma 3.7. Let s ∈ C∞

such that |λ| < ?[D,πM(h)]?−1. Then

c(M,E),

∂

∂λ

λ=0

eiλDs = iDs.

c(M,E), and let h ∈ C∞

c(M) be equal to 1 on the support of s. Let λ ∈ R

suppeiλDs ⊂ supph.

This follows from the proof of Proposition 10.3.1. from [31].

Proof of Proposition 3.5. Suppose suppˆb ⊂ [−R,R]. Let f ∈ Cc(M), and choose h ∈ C∞

such that h equals 1 on the support of f, and that ?[D,πM(h)]? ≤

function 1 on M. Then by Lemma 3.7,

c(M)

1

R. Let 1M be the constant

πM(1M− h)eiλDπM(f) = 0,(38)

for all λ ∈] − R,R[. Here we have extended the nondegenerate representation πM of C0(M) on

L2(M,E) to the multiplier algebra Cb(M) of C0(M). So by Proposition 3.6, we have for all

s,t ∈ C∞

c(M,E),

?πM(1M− h)b(D)πM(f)s,t?

=

2π

R

?R

−R

= 0,

L2(M,E)=?b(D)πM(f)s,πM(1M−¯h)t?

1

?eiλDπM(f)s,πM(1M−¯h)t?

1

2π

L2(M,E)

?

L2(M,E)ˆb(λ)dλ

=

?πM(1M− h)eiλDπM(f)s,t?

L2(M,E)ˆb(λ)dλ

by (38). So

?1 − πM(h)?b(D)πM(f) = πM(1M− h)b(D)πM(f) = 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 compactly supported

Fourier transform,23so that b(D) is the kind of operator that defines a K-homology class over

M. Because b(D) is properly supported, it preserves L2

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 the following diagram:

c(M,E) and the construction used in the

L2

c(M,E)

?? ??

b(D)

??

L2(M,E)N

b(D)N

??

L2

c(M,E)

?? ??L2(M,E)N.

On the other hand, the operator D induces an unbounded operator on L2(M/N,E/N), because

it restricts to

˜DN: C∞(M,E)N→ C∞(M,E)N,

23If g is a smooth, even, compactly supported function on R, and f := g ∗ g is its convolution square, then

b(λ) :=?

R

eiλx−1

ix

f(x)dx is such a function (see [31], Exercise 10.9.3).

Page 22

3DIFFERENTIAL OPERATORS ON VECTOR BUNDLES 22

from which we obtain

DN:= ψ−1

E

˜DNψE: C∞(M/N,E/N) → C∞(M/N,E/N)

(see Proposition 2.8). We regard DNas an unbounded operator on L2(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 χ from Proposition 3.2 intertwines the operators b(D)N and

b(DN):

χ

L2(M,E)N

??

b(D)N

??

L2(M/N,E/N)

b(DN)

??

L2(M,E)N

χ

??L2(M/N,E/N).

We will prove this claim by reducing it to the commutativity of another diagram. This dia-

gram involves the Hilbert space˜L2(M/N,E/N), which is defined as the completion of the space

C∞(M,E)Nin the inner product

?

M/N

(σ,τ) :=

?σ(ϕ(O)),τ(ϕ(O)?

EdO,

for any measurable section ϕ : M/N → M. The map ψEfrom Proposition 2.8 extends continuously

to a unitary isomorphism

˜ψE:˜L2(M/N,E/N) → L2(M/N,E/N).

The unbounded operator˜DNon˜L2(M/N,E/N) is essentially self-adjoint because DNis, and

because˜ψE intertwines the two operators. Hence we have b(˜DN) ∈ B?˜L2(M/N,E/N)?. We will

deduce Proposition 3.8 from

Lemma 3.9. The following diagram commutes:

L2

c(M,E)

?

Nn·??

b(D)

??

˜L2(M/N,E/N)

b(˜ DN)

??

L2

c(M,E)

?

Nn·?? ˜L2(M/N,E/N),

where the map?

Nn· is given by24

??

Nn · (s)

?

(Nm) =

?

N

n · s(n−1m)dn.

Proof. Step 1. Because the representation of N in L2(M,E) is unitary, we have

??

for all s,t ∈ L2

c(M,E).

Nn · (s),t

?

L2(M,E)=

?

s,?

Nn · (t)

?

L2(M,E)

Step 2. By Proposition 3.10 below and equivariance of D, we have

??

Nn·

?

◦ D =˜DN◦?

Nn·

24Note that the space˜L2(M/N,E/N) can be realised as a space of sections of E.

Page 23

3DIFFERENTIAL OPERATORS ON VECTOR BUNDLES23

on C∞

Step 3. For all s ∈ C∞

c(M,E).

c(M,E), we have by Proposition 3.10,

∂

∂λ

λ=0

????

?

Nn · ◦eiλDs =

?

N

?

∂

∂λeiλDn · sdn

= i

= i˜DN?

=

∂λ

N

n · Dsdn

Nn·(s)

eiλ˜ DN?

(by Step 2)

∂

????

λ=0

Nn · (s).

So by Stone’s theorem,

?

Nn · ◦eiλD= eiλ˜ DN◦?

Nn·

for all λ ∈ R.

Step 4. For all s,t ∈ C∞

?b(˜DN)?

c(M,E),

Nn · (s),t?

L2(M,E)=

1

2π

1

2π

1

2π

?

?

?

R

?eiλ˜ DN?

??

?eiλDs,?

Nn · (t)?

Nn · (s),t?

L2(M,E)ˆb(λ)dλ(by Proposition 3.6)

=

R

Nn · eiλDs,t?

Nn · (t)?

L2(M,E)ˆb(λ)dλ(by Step 3)

=

=?b(˜D)s,?

=??

R

L2(M,E)ˆb(λ)dλ (by Step 1)

L2(M,E)

(by Proposition 3.6)

Nn · b(˜D)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 with a measure

dm2. Let E → M1be a vector bundle, and let

D : C∞(M1,E) → C∞(M1,E)

be a differential operator. Let p1: M1×M2→ M1be projection onto the first factor, and consider

the operator

p∗

defined by?p∗

Then for all s ∈ C∞(M1× M2,p∗

??

M2

We now derive Proposition 3.8 from Lemma 3.9.

1D : C∞(M1× M2,p∗

1σ = p∗

1

1E) → C∞(M1× M2,p∗

1E),

1D?p∗

?Dσ?for all σ ∈ C∞(M1,E).

1E) and all m1∈ M1,

?

Ds(·,m2)dm2

(m1) =

?

M2

p∗

1Ds(m1,m2)dm2.

Proof of Proposition 3.8. Consider the following cube:

L2

c(M,E)

?

Nn·??

b(D)

??

?????

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

˜L2(M/N,E/N)

b(˜ DN)

??

˜ ψE

?

?

?

???

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

L2

c(M,E)

??

?????

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

˜L2(M/N,E/N)

?

?

?

?

?

?

?

?

?

?

˜ψE

???

???

??

???

?

L2(M,E)N

χ

??

b(D)N

??

L2(M/N,E/N)

b(DN)

??

L2(M,E)N

χ

??L2(M/N,E/N).

Page 24

3DIFFERENTIAL OPERATORS ON VECTOR BUNDLES24

The rear square (with the operators b(D) and b(˜DN) in it) commutes by Lemma 3.9. The left hand

square (with the operators b(D) and b(D)N) commutes by definition of b(D)N, and the right hand

square (with b(˜DN) and b(DN)) commutes by Lemma 3.4. The top and bottom squares commute

by definition of the map χ, so that the front square commutes as well, which is Proposition 3.8. ?

3.4Multiplication of sections by functions

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

πM: C0(M) → B(L2(M,E))

and

πM/N: C0(M/N) → B(L2(M/N,E/N))

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

(πM)N: C0(M/N) → B(L2(M,E)N)

be the representation obtained from πM by the procedure in Subsection 3.1.

Lemma 3.11. The isomorphism (36) intertwines the representations (πM)Nand πM/N.

Proof. The representation (πM)Nis induced by

πN

?πN

M: C(M/N) → B(L2

M(f)s?(m) = f(N · m)s(m).

c(M,E) and m ∈ M, we have

?

c(M,E)),

For all f ∈ C(M/N), s ∈ L2

χ?πN

M(f)s?(N · m) = N ·

N

n · f(N · n−1m)s(n−1· m)dn

?

N

=?πM/N(f)χ(s)?(N · m).

= N · f(N · m)

n · s(n−1· m)dn

?

3.5Conclusion

Let G, M, E, D, DN, πM and πM/Nbe as in Sections 3.2 – 3.4. Suppose that the vector bundle

E carries a Z2-grading with respect to which the operator D is odd. Suppose D is elliptic and

essentially self-adjoint as an unbounded operator on L2(M,E).25. Let b be a normalising function

with compactly supported Fourier transform. Then Proposition 3.2, Proposition 3.8 and Lemma

3.11 may be summarised as follows.

Theorem 3.12. Let (L2(M,E)N,b(D)N,(πM)N) be the triple obtained from

(L2(M,E),b(D),πM) by the procedure of Subsection 3.1. Then there is a unitary isomorphism

χ : L2(M,E)N→ L2(M/N,E/N)

that intertwines the representations of G/N, the operators b(D)N and b(DN), and the representa-

tions (πM)Nand πM/N.

Corollary 3.13. The image of the class

[D] :=

?

L2(M,E),b(D),πM

?

∈ KG

0(M)

under the homomorphism VN defined in Subsection 3.1 is

?

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

VN[D] =L2(M/N,E/N),b(DN),πM/N

?

=:?DN?∈ KG/N

0

(M/N).

Page 25

4DIRAC OPERATORS25

Remark 3.14. If the action of G on M happens to be free, then Corollary 3.13 allows us to restate

the Guillemin–Sternberg conjecture 1.1 without using techniques from noncommutative geometry.

Indeed, for free actions we have

?

= index?D /L

= dim

M

RQ◦ µG

M

D /L

M

?

= µ{e}

M/G◦ VG

?

?G

D /L

M

?

by naturality of µ

M

by Corollary 3.13

?+?G

?

ker?D /L

− dim

?

ker?D /L

M

?−?G

∈ Z.

Here the kernels of

?0,∗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

manifold M/G. So Conjecture 1.1 becomes

?D /L

M

?±are taken in the spaces of smooth, not necessarily L2, sections of

??D /L

M

?±?G

on the compact

indexD /LG

MG= dim

?

ker?D /L

M

?+?G

− dim

?

ker?D /L

M

?−?G

.

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

2.2).

4Dirac operators

In this section, we make the assumptions stated in Subsection 2.1. In particular, Γ is a normal

discrete subgroup of G. The goal of this section is to prove that Theorem 2.9 is a special case of

Corollary 3.13:

Proposition 4.1. Consider the Dolbeault–Dirac operator D /L

operator?D /L

Ξ : Ω0,∗(M/Γ;L/Γ) → C∞?

that is isometric with respect to the L2-inner product and intertwines the Dolbeault–Dirac operator

D /L/Γ

M

Mon Ω0,∗(M;L), and the induced

. There is an isomorphism

M

?Γon C∞?

M/Γ,??0,∗T∗M ⊗ L?/Γ

?

M/Γ,??0,∗T∗M ⊗ L?/Γ

?

M/Γon Ω0,∗(M/Γ;L/Γ) and the operator?D /L

Consequently, Ξ induces a unitary isomorphism between the corresponding L2-spaces, which by

Lemma 3.4 intertwines the bounded operators obtained from D /L/Γ

function with compactly supported Fourier transform. Hence Theorem 2.9 follows, as

??D /L

=?D /L/Γ

4.1The isomorphism

?Γ.

M/Γand?D /L

M

?Γusing a normalising

VΓ

M

??

=

??D /L

M

?Γ?

?

by Corollary 3.13

M/Γ

by Proposition 4.1.

The isomorphism of C∞(M/Γ)-modules Ξ in Proposition 4.1 is defined as follows. The quotient

map p : M → M/Γ induces the vector bundle homomorphism

?Tp∗:?T∗(M/Γ) →?T∗M.

Because Tp intertwines the almost complex structures on TM and T(M/Γ), the homomorphism

(39) induces

?0,∗Tp∗:?0,∗T∗(M/Γ) →?0,∗T∗M.

(39)

(40)

Page 26

4DIRAC OPERATORS26

Composition with the quotient map T∗M → (T∗M)/Γ turns (39) and (40) into isomorphisms

??Tp∗?Γ:

??0,∗Tp∗?Γ:

?T∗(M/Γ) →??T∗M?/Γ;

?0,∗T∗(M/Γ) →??0,∗T∗M?/Γ.

(41)

(42)

On the spaces of smooth sections of the vector bundles in question, the isomorphisms (41) and

(42) induce isomorphisms of C∞(M/Γ)-modules

Ψ : Ω∗(M/Γ) → C∞?M/Γ,(?T∗M)/Γ?;(43)

Ψ0,∗: Ω0,∗(M/Γ) → C∞?

M/Γ,??0,∗T∗M?/Γ

?

. (44)

Now the isomorphism Ξ is defined as

Ξ : Ω0,∗(M/Γ;L/Γ)∼=

Ω0,∗(M/Γ) ⊗C∞(M/Γ)C∞(M/Γ,L/Γ)

C∞?

Ψ0,∗⊗1C∞(M/Γ,L/Γ)

− − − − − − − − − − − − − →

M/Γ,??0,∗T∗M?/Γ

?

⊗C∞(M/Γ)C∞(M/Γ,L/Γ)

∼= C∞?

M/Γ,??0,∗T∗M ⊗ L?/Γ

?

.

It is isometric by definition of the measure dO on M/Γ and the metrics on the vector bundles

involved. Therefore, it remains to prove that Ξ intertwines the operators D /L/Γ

M/Γand?D /L

M

?Γ.

4.2 Proof of Proposition 4.1

The connections

Recall the isomorphism of C∞(M)Γ ∼= C∞(M/Γ)-modules ψE : C∞(M,E)Γ→ C∞(M/Γ,E/Γ)

defined by (31), with H = Γ, for any Γ-vector bundle E over M. Also consider the pullback p∗

of differential forms on M/Γ to invariant differential forms on M. It defines an isomorphism of

C∞(M/Γ)∼= C∞(M)Γ-modules

p∗: Ω∗(M/Γ) → Ω∗(M)Γ.

Lemma 4.2. The following diagram commutes:

Ω∗(M;L)Γ

∇

??

∼

=

??

Ω∗(M;L)Γ

∼

=

??

Ω∗(M)Γ⊗C∞(M)Γ C∞(M,L)Γ

Ω∗(M)Γ⊗C∞(M)Γ C∞(M,L)Γ

Ω∗(M/Γ) ⊗C∞(M/Γ)C∞(M/Γ,L/Γ)

??

p∗⊗ψ−1

L

∼

=

??

Ω∗(M/Γ) ⊗C∞(M/Γ)C∞(M/Γ,L/Γ)

??

p∗⊗ψ−1

L

∼

=

??

Ω∗(M/Γ;L/Γ)

∇Γ

??

∼

=

Ω∗(M/Γ;L/Γ).

∼

=

The proof of this lemma is a matter of writing out definitions.

By definition of the almost complex structure on T(M/Γ), we have

p∗?Ω0,q(M/Γ)?= Ω0,q(M)Γ

Page 27

4DIRAC OPERATORS27

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

Ω0,∗(M;L)Γ

¯∂L

??Ω0,∗(M;L)Γ

Ω0,∗(M/Γ;L/Γ)

¯∂Γ

L??

p∗⊗ψ−1

L

∼

=

??

Ω0,∗(M/Γ;L/Γ),

p∗⊗ψ−1

L

∼

=

??

(45)

with¯∂Land¯∂Γ

Las in Subsection 2.2.

The Dirac operators

By definition of the measure dO on M/Γ and the metrics BΓon T(M/Γ) and HΓon L/Γ, the

isomorphism

p∗⊗ ψ−1

is isometric with respect to the inner product on Ω0,∗(M/Γ;L/Γ) defined by

?

M/Γ

L: Ω0,∗(M/Γ;L/Γ) → Ω0,∗(M;L)Γ

(α ⊗ σ,β ⊗ τ) =

BΓ(α,β)HΓ(σ,τ)dO,(46)

for all α,β ∈ Ω0,∗(M/Γ) and σ,τ ∈ C∞(M/Γ,L/Γ), and the inner product on Ω0,∗(M;L)Γdefined

by

?

U

for all ζ,ξ ∈ Ω0,∗(M)Γand s,t ∈ C∞(M,L)Γ. (Recall that U ⊂ M is a fundamental domain for

the Γ-action.)

The Dolbeault–Dirac operators on M and M/Γ are defined by

(ζ ⊗ s,ξ ⊗ t) =

B(ζ,ξ)H(s,t)dm,(47)

D /L

M=¯∂L+¯∂∗

D /L/Γ

L;

M/Γ=¯∂Γ

L+?¯∂Γ

L

?∗.

Here the formal adjoint?¯∂Γ

¯∂∗

L

?∗is defined with respect to the inner product (46). The formal adjoint

(B ⊗ H)?¯∂∗

for all η,θ ∈ Ω0,∗(M;L), θ with compact support. But this is actually the same as the formal

adjoint of¯∂Lwith respect to the inner product (47):

Lis defined by

?

M

Lη,θ?dm =

?

M

(B ⊗ H)?η,¯∂Lθ?dm,

Lemma 4.3. Let Γ be a discrete group, acting properly and freely on a manifold M, equipped with

a Γ-invariant measure dm. Suppose M/Γ is compact. Let E → M be a Γ-vector bundle, equipped

with a Γ-invariant metric ?·,·?. Let

D : C∞(M,E) → C∞(M,E)

be a Γ-equivariant differential operator. Let

D∗: C∞(M,E) → C∞(M,E)

be the operator such that for all s,t ∈ C∞(M,E), t with compact support,

?

M

?D∗s,t?dm =

?

M

?s,Dt?dm.

Let U ⊂ M be a fundamental domain for the Γ-action. Then the restriction of D∗to C∞(M,E)Γ

satisfies

?

U

for all s,t ∈ C∞(M,E)Γ.

?D∗s,t?dm =

?

U

?s,Dt?dm,

Page 28

4DIRAC OPERATORS28

Proof. We will show that for all s ∈ C∞(M,E)Γ, and all t in a dense subspace of C∞(M,E)Γ, we

have

?

U

Let τ be a section of E, with compact support in U. Define the section t of E by extending

the restriction τ|U Γ-invariantly to M. The space of all sections t obtained in this way is dense in

C∞(M,E)Γwith respect to the topology induced by the inner product

?

U

?D∗s,t?dm =

?

U

?s,Dt?dm.

(s,t) :=

?s,t?dm.

Then for all s ∈ C∞(M,E)Γ,

?

U

?D∗s,t?dm =

?

M

?D∗s,τ?dm =

?

M

?s,Dτ?dm =

?

U

?s,Dt?dm.

?

We conclude that p∗⊗ψ−1

to define the adjoints¯∂∗

Lis an isometric isomorphism with respect to the inner products used

Land?¯∂Γ

Corollary 4.4. The following diagram commutes:

L

?∗. Hence the commutativity of diagram (45) implies

Ω0,∗(M;L)Γ

D /L

M

??Ω0,∗(M;L)Γ

Ω0,∗(M/Γ;L/Γ)

D /L/Γ

M/Γ??

p∗⊗ψ−1

L

∼

=

??

Ω0,∗(M/Γ;L/Γ).

p∗⊗ψ−1

L

∼

=

??

Remark 4.5. Corollary 4.4 shows that for discrete groups a much stronger statement than the

Guillemin-Sternberg conjecture holds. Indeed, by Remark 3.14 the Guillemin-Sternberg conjecture

states that the restriction of the operator D /L

fact that their indices are equal (as operators on smooth, not necessarily L2, sections). But these

operators are in fact more strongly related: they are intertwined by an isometric isomorphism.

Mto Ω0,∗(M;L)Γis related to the operator D /L/Γ

M/Γby the

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∗: Ω∗(M/Γ) → Ω∗(M)Γ.

Lemma 4.6. The following diagram commutes:

Ω∗(M/Γ)

p∗

∼

=

∼

=

??

Ψ

∼

=

??

Ω∗(M)Γ

ψ∧T∗M

???????????????

C∞(M/Γ,(?T∗M)/Γ),

where Ψ is the isomorphism (43).

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 the operator

?Γ: C∞?

?D /L

M

M/Γ,??0,∗T∗M ⊗ L?/Γ

?

→ C∞?

M/Γ,??0,∗T∗M ⊗ L?/Γ

?

,

Page 29

5ABELIAN DISCRETE GROUPS29

Corollary 4.4 implies that the following diagram commutes:

Ω0,∗(M;L)Γ

∼

=

ψ∧0,∗T∗M⊗ψL??

C∞?

Ξ=Ψ0,∗⊗1

D /L

M

??Ω0,∗(M;L)Γ

∼

=

ψ∧0,∗T∗M⊗ψL??

??C∞?

Ξ=Ψ0,∗⊗1

M/Γ,??0,∗T∗M ⊗ L?/Γ

∼

=

??

D /L

M

?Γ

M/Γ,??0,∗T∗M ⊗ L?/Γ

∼

=

?

Ω0,∗(M/Γ;L/Γ)

??

D /L/Γ

M/Γ

??Ω0,∗(M/Γ;L/Γ).

??

Indeed, the outside diagram commutes by Corollary 4.4 and Lemma 4.6, and the upper square

commutes by definition of?D /L

4.1.

M

?Γ. Hence the lower square commutes as well, which is Proposition

?

5Abelian discrete groups

In this section, we consider the situation of Section 2, with the additional assumption that G = Γ

is an abelian discrete group. Then the Guillemin–Sternberg conjecture can be proved directly,

without using naturality of the assembly map (33). This proof is based on Proposition 4.1, and

the description of the assembly map in this special case given by Baum, Connes and Higson [5],

Example 3.11 (which in turn is based on Lusztig [44]). We will first explain this example in a

little more detail than given in [5], and then show how it implies Theorem 1.2 for abelian discrete

groups.

5.1The assembly map for abelian discrete groups

The proof of the Guillemin–Sternberg conjecture for discrete abelian groups is based on the fol-

lowing result:

Proposition 5.1. Let M, E, D and DΓbe as in Subsection 3.5. Suppose that G = Γ is abelian

and discrete. Using the normalising function b(x) =

that we have the class

?L2(M,E),F?∈ KΓ

Then26

R(Γ)

M

x

√1+x2, we form the operator F := b(D), so

0(M).

Q◦ µΓ

?L2(M,E),F?= indexDΓ.

In view of Proposition 4.1, Proposition 5.1 implies our Guillemin–Sternberg conjecture (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

[E,FE] := µΓ

M

?L2(M,E),F?∈ KK0(C,C∗(Γ)).(48)

Note that since Γ is discrete, its unitary dualˆΓ is compact. And because Γ is abelian, all irreducible

unitary representations are of the form

Uα: Γ → U(1),

for α ∈ˆΓ. Fourier transform defines an isomorphism C∗(Γ)∼= C0(ˆΓ). Therefore,

KK0(C,C∗(Γ))∼= K0(C∗(Γ))∼= K0(C0(ˆΓ))∼= K0(ˆΓ).

26Recall that we abuse notation by writing indexDΓ:= dimker?DΓ?+− dimker?DΓ?−.

Page 30

5ABELIAN DISCRETE GROUPS30

BecauseˆΓ is compact, the image of [E,FE] in K0(ˆΓ) is the difference of the isomorphism classes of

two vector bundles overˆΓ. These two vector bundles can be determined as follows. For all α ∈ˆΓ,

we define the Hilbert space Hαas the space of all measurable sections sαof E (modulo equality

almost everywhere), such that for all γ ∈ Γ,

γ · sα= Uα(γ)−1sα,

and such that the norm

?sα?2

α= ?sα,sα?α

(49)

is finite, where the inner product ?·,·?αis defined by

?sα,tα?α:=

?

M/Γ

?sα(ϕ(O)),tα(ϕ(O))?

EdO,

where ϕ is any measurable section of the principal fibre bundle M → M/Γ. The space Hα is

isomorphic to the space of L2-sections of the vector bundle Eα, where

Eα:= E/(γ · e ∼ U−1

α(γ)e) → M/Γ.

Let HD

αbe the dense subspace

HD

α:= {sα∈ Hα∩ C∞(M,E);Dsα∈ Hα} ⊂ Hα. (50)

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

Dα: HD

α→ Hα

on Hα. It is essentially self-adjoint by [31], Corollary 10.2.6., and hence induces the bounded

operator

Dα

?1 + D2

The grading on E induces a grading on Hα with respect to which Dα and Fα are odd. The

operators Fαare elliptic pseudo-differential operators:

Fα:=

α

∈ B(Hα). (51)

Lemma 5.2. Let D be an elliptic, first order differential operator on a vector bundle E → 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 :=

D

√1+D2is an elliptic pseudo-differential operator.

Proof. This result seems well known and is easily derived from a number of related results in the

literature. It is, in any case, sufficient to show that (1 + D2)−1

The following proof was communicated to us by Elmar Schrohe.27

According to [6], a bounded operator A : L2(Rn) → L2(Rn) is a pseudo-differential operator

on Rniff all iterated commutators with xj (as a multiplication operator) and ∂xjare bounded

operators. This immediately yields the lemma for M = Rn(cf. [6], Theorem 4.2). To extend this

result to the manifold case, we recall that an operator A : C∞(M) → D′(M) on a manifold M is a

pseudo-differential operator when for each choice of smooth functions f, g with support in a single

coordinate neighbourhood, fAg is a pseudo-differential operator on Rn. (Here one has to admit

nonconnected coordinate neighbourhoods.)

Now write (1 + D2)−1

?

C

27An independent proof was suggested to us by John Roe, who mentioned that in the case at hand the functional

calculus for (pseudo) differential operators developed in [68] for compact manifolds may be extended to the non-

compact case. A third proof may be constructed using heat kernel techniques, as in the unpublished Diplomarbeit

of Hanno Sahlmann (Rainer Verch, private communication).

2 is a pseudo- differential operator.

2 as a Dunford integral (cf. [17], pp. 556–577), as follows:

(1 + D2)−1

2=

dz

2πi(1 + z)−1

2(z − D2)−1.

Page 31

5ABELIAN DISCRETE GROUPS 31

To compute the commutators of f(1 + D2)−1

contour integral. Boundedness of all iterated commutators then easily follows, using the fact that

f and g have compact support.

The same argument, with the exponent −1

differential operator, and ellipticity of (1 + D2)−1

2g with xj and ∂xj, one may take these inside the

2replaced by1

2 follows.

2, shows that (1 + D2)

1

2 is a pseudo-

?

Consider the field of Hilbert spaces

(Hα)α∈ˆΓ→ˆΓ.

In the next subsection, we will give this field the structure of a continuous field of Hilbert spaces

by specifying its space of continuous sections C(ˆΓ,(Hα)α∈ˆΓ). Consider the subfields

?kerD+

?kerD−

These are indeed well-defined subfields of (Hα)α∈ˆΓbecause kerD±

larity theorem.

Suppose that (kerDα)α∈ˆΓand (cokerDα)α∈ˆΓare vector bundles overˆΓ in the relative topology.

As in the proof of the Atiyah–J¨ anich Theorem (cf. [72]), the operator D can always be replaced

by another operator in such a way that the class [L2(M,E),F] does not change, and that these

‘fields of vector spaces’ are indeed vector bundles (see also [35]). Then:

Proposition 5.3. The image of the class?L2(M,E),F?∈ KΓ

is

µΓ

M

α

α∈ˆΓ

Proposition 5.3 wil be proved in the next two subsections.

α

?

?

α∈ˆΓ→ˆΓ;

α∈ˆΓ→ˆΓ.

α

α= kerF±

αby the elliptic regu-

0(M) under the assembly map µΓ

??kerD−

M

?L2(M,E),F?=

??kerD+

?

?

−

α

?

α∈ˆΓ

?

∈ K0(ˆΓ).

5.2The Hilbert C∗-module part of the assembly map

In this subsection we determine the Hilbert C∗(Γ)∼= C0(ˆΓ)-module E in (48) (Proposition 5.7).

A unitary isomorphism

Let dα be the measure onˆΓ corresponding to the counting measure on Γ via the Fourier transform.

Consider the Hilbert space

?⊕

ˆΓ

That is, H consists of the measurable maps

H :=Hαdα.

s :ˆΓ → (Hα)α∈ˆΓ,α ?→ sα,

such that sα∈ Hαfor all α, and

?s?2

H= ?s,s?H:=

?

ˆΓ

?sα?2

αdα < ∞.

Define the linear map V : H → L2(M,E) by

(V s)(m) :=

?

ˆΓ

sα(m)dα.

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

?V−1σ?

α(m) =

?

γ∈Γ

γ · σ(γ−1m)Uα(γ), (52)

for all σ ∈ Cc(M,E) ⊂ L2(M,E).

Page 32

5ABELIAN DISCRETE GROUPS 32

Remark 5.5. It follows from unitarity of V that V s is indeed an L2-section of E for all s ∈ H.

Conversely, a direct computation shows that for all σ ∈ L2(M,E), α ∈ˆΓ and γ ∈ Γ, one has

γ ·?V−1σ?

so that V−1σ lies in H.

Sketch of proof of Lemma 5.4. The proof is based on the observations that for all α ∈ˆΓ,

?

γ∈Γ

where δ1∈ D′(ˆΓ) is the δ-distribution at the trivial representation 1 ∈ˆΓ, and that for all γ ∈ Γ,

?

ˆΓ

the Kronecker delta of γ and the identity element. Using these facts, one can easily verify that V

is an isometry and that (52) is indeed the inverse of V .

The representation πHof Γ in H corresponding to the standard representation of Γ in L2(M,E)

via the isomorphism V is given by

α= Uα(γ)−1?V−1σ?

α,

Uα(γ) = δ1(α), (53)

Uα(γ)dα = δγe,(54)

?

(πH(γ)s)α= Uα(γ)−1sα.

This follows directly from the definitions of the space Hαand the map V .

Fourier transform

The Hilbert C∗(Γ)-module E is the closure of the space Cc(M,E) in the norm

?σ?2

E:= ?γ ?→ ?σ,γ · σ?L2(M,E)?C∗(Γ).

The C∗(Γ)-module structure of E is defined by

f · σ =

?

γ∈Γ

f(γ)γ · σ,

for all f ∈ Cc(Γ) and σ ∈ Cc(M,E). The isomorphism V induces an isomorphism of the Hilbert

C∗(Γ)-module E with the closure EH of V−1(Cc(M,E)) ⊂ H in the norm

?s?2

EH:= ?γ ?→ ?V s,γ · V s?L2(M,E)?C∗(Γ)= ?γ ?→ ?s,πH(γ)s?H?C∗(Γ),

by unitarity of V . The C∗(Γ)-module structure on EH corresponding to the one on E via V is

given by

f · s =f(γ)πH(γ)s,

?

γ∈Γ

(55)

for all f ∈ Cc(Γ) and s ∈ V−1(Cc(M,E)).

Next, we use the isomorphism C0(ˆΓ)∼= C∗(Γ) defined by the Fourier transform ψ ?→ˆψ, where

?

ˆΓ

for all ψ ∈ Cc(ˆΓ). Because of (53) and (54), the inverse Fourier transform is given by f ?→ˆf,

where for f ∈ Cc(Γ), one has

ˆf(α) =f(γ)Uα(γ)−1.

ˆψ(γ) =ψ(α)Uα(γ)dα

?

γ∈Γ

So by Fourier transform, the Hilbert C∗(Γ)-module EHcorresponds to the Hilbert C0(ˆΓ)-module

ˆEH, which is the closure of the space V−1(Cc(M,E)) in the norm

???α ?→

= sup

α∈ˆΓ

γ∈Γ

?s?2

ˆEH=

?

γ∈Γ

?

?s,πH(γ)s?HUα(γ)−1???

?s,πH(γ)s?HUα(γ)−1???.

C0(ˆΓ)

???

(56)

Page 33

5ABELIAN DISCRETE GROUPS33

Continuous sections

Using the following lemma, we will describe the Hilbert C0(ˆΓ)-moduleˆEHas the space of continuous

sections of a continuous field of Hilbert spaces.

Lemma 5.6. For all s,t ∈ V−1(Cc(M,E)),

?

γ∈Γ

?s,πH(γ)t?HUα(γ)−1= ?sα,tα?α.

Proof. Let ϕ be a measurable section of the principal fibre bundle M → M/Γ. Then

??

ˆΓ

M/Γ

?

M/Γ

= ?sα,tα?α.

?

γ∈Γ

?s,πH(γ)t?HUα(γ)−1=

?

γ∈Γ

?

?sβ(ϕ(O)),Uβ(γ)−1tβ(ϕ(O))?

?sα(ϕ(O)),tα(ϕ(O))?

EdOdβ

?

Uα(γ)−1

=

EdO

by (53).

?

We conclude from (56) and Lemma 5.6 thatˆEH is the closure of V−1(Cc(M,E)) in the norm

?s?2

ˆEH= sup

α∈ˆΓ

?sα?2

α.

Therefore, it makes sense to define the space C(ˆΓ,?Hα

Hilbert spaces (Hα)α∈ˆΓas the C0(ˆΓ)-moduleˆEH(cf. [15, 67]). Then our construction implies

Proposition 5.7. The Hilbert C∗(Γ)-module E is isomorphic to the Hilbert C0(ˆΓ)-module C(ˆΓ,(Hα)α∈ˆΓ).

Let us verify explicitly that the representations of C0(ˆΓ) inˆEHand in C(ˆΓ,(Hα)α∈ˆΓ) are indeed

intertwined by the isomorphism induced by V and the Fourier transform: for all f ∈ Cc(Γ) and

all s ∈ V−1(Cc(M,E)), we have

?

γ∈Γ

=f(γ)Uα(γ)−1sα

?

α∈ˆΓ) of continuous sections of the field of

(f · s)α=

f(γ)(πH(γ)s)α

by (55)

?

γ∈Γ

=ˆf(α)sα.

5.3The operator part of the assembly map

Proposition 5.8. The operator FˆEHon the Hilbert C0(ˆΓ) moduleˆEH= C(ˆΓ,(Hα)α∈ˆΓ), induced

by F ∈ B(L2(M,E)), equals FˆEH=?Fα

??Fα

for all β ∈ˆΓ and all s ∈ V−1(Cc(M,E)) (and extended continuously to general s ∈ˆEH). Here Fα

is the operator (51).

?

α∈ˆΓ, where the ‘field of operators’?Fα

?

?

α∈ˆΓis given by

α∈ˆΓs

?

β= Fβsβ,

Proof. For s ∈ V−1(Cc(M,E)), we have V FˆEHs = FV s. So it is sufficient to prove that for such

s, one has FV s(m) =?

Let HD⊂ H be the space of s ∈ H such that V s ∈ C∞

(see (50)). By Proposition 3.10, we have DV s(m) =?

Because of Lemma 3.4 this proves the proposition, because HDis dense in H.

ˆΓFαsα(m)dα, for all m ∈ M.

c(M,E), and sα∈ HD

ˆΓDsα(m)dα for all s ∈ HDand m ∈ M.

αfor all α ∈ˆΓ

Page 34

5ABELIAN DISCRETE GROUPS34

Note that?Fα

field of operators?Fα

bounded operator on C(ˆΓ,(Hα)α∈ˆΓ).

Proof of Proposition 5.3. We have seen (cf. Propositions 5.7 and 5.8) that

?

?

α∈ˆΓis initially defined on the subspace V−1(Cc(M,E)) of C(ˆΓ,(Hα)α∈ˆΓ). But

since the unitary operator V intertwines?Fα

α∈ˆΓis bounded in the norm ? · ?ˆEH, so that it extends continuously to a

?

α∈ˆΓand the bounded operator F on L2(M,E), the

?

?

µΓ

M

?L2(M,E),F?=C(ˆΓ,(Hα)α∈ˆΓ),(Fα)α∈ˆΓ

?

∈ KK0(C,C0(ˆΓ)).

The image of this class in K0(C0(ˆΓ)) is the formal difference of projective C0(ˆΓ)-modules

?

By compactness of M/Γ and the elliptic regularity theorem, the kernels of F+

to the kernels of D+

vector bundles overˆΓ, then by Lemma 5.9 below, the class (57) equals

?

Under the isomorphism K0(C0(ˆΓ))∼= K0(ˆΓ), the latter class corresponds to

??kerD+

ker

??F+

α

?

α∈ˆΓ

??

−

?

ker

??F−

α

?

α∈ˆΓ

??

.(57)

αand F−

αand D−

αare equal

αdefine

αand D−

α, respectively. If we suppose that the kernels of D+

C(ˆΓ,?kerD+

α

?

α∈ˆΓ)

?

−

?

C(ˆΓ,?kerD−

α

?

α∈ˆΓ)

?

.

α

?

α∈ˆΓ

?

−

??kerD−

α

?

α∈ˆΓ

?

∈ K0(ˆΓ).

?

Lemma 5.9. Let H be a continuous field of Hilbert spaces over a topological space X, and let ∆

be its space of continuous sections. Let H′be a subset of H such that for all x ∈ X, H′

is a linear subspace of Hx. Set

x:= Hx∩H′

∆′:= {s ∈ ∆;s(x) ∈ H′

xfor all x ∈ X}.

Let s : X → H′be a section. Then s is continuous in the subspace topology of H′in H if and

only if s ∈ ∆′.

Proof. Let s : X → H be a section. Then s is a continuous section of H′in the subspace topology

if and only if s is a continuous section of H and s(x) ∈ H′

in such a way that s is continuous if and only if s ∈ ∆ [15, 67].

xfor all x. The topology on H is defined

?

5.4Reduction

We will now describe the reduction map R(Γ)

Q: K0(C∗(Γ)) → Z, and prove Proposition 5.1.

Lemma 5.10. Let Γ be an abelian discrete group, and let i : {1} ֒→ˆΓ be the inclusion of the

trivial representation. The following diagram commutes:

K0(C∗(Γ))

R(Γ)

Q

??

∼

=

??

K0(C)

∼

=

??

K0(ˆΓ)

i∗

??K0({1}).

That is,

R(Γ)

Q([E]) = dimE1= rank(E) ∈ Z,

for all vector bundles E →ˆΓ.

Page 35

6EXAMPLE: ACTION OF Z2NON R2N

35

The proof is a straightforward verification.

End of proof of Proposition 5.1. From Lemma 5.10 and Proposition 5.3, we conclude that

?L2(M,E),F?= [kerD+

The Hilbert space H1is isomorphic to L2(M/Γ,E/Γ), and this isomorphism intertwines D1and

DΓ. Hence Proposition 5.1 follows.

R(Γ)

Q◦ µΓ

M

1] − [kerD−

1] = indexD1∈ Z.

?

6Example: action of Z2non R2n

Let M be the manifold M = T∗Rn ∼= R2n ∼= Cn. An element of M is denoted by (q,p) :=

(q1,p1,...,qn,pn), where qj,pj∈ R, or by q + ip = z := (z1,...,zn), where zj= qj+ ipj∈ C. We

equip M with the standard symplectic form ω :=?n

Let Γ be the group Γ = Z2n ∼= Z+iZ. The action of Γ on M by addition is denoted by α. Our

aim is to find a prequantisation for this action and the corresponding Dirac operator for general

n, and the quantisation of this action for n = 1. This is less trivial than it may seem because of

the coupling of the standard Dirac operator to the prequantum line bundle, which precludes the

use of the standard formulae. We will then see, just as in Section 5, that the reduction of this

quantisation is the quantisation of the reduced space T2.

j=1dpj∧ dqj.

6.1 Prequantisation

Let L := M × C → M be the trivial line bundle. Inspired by the construction of line bundles on

tori with a given Chern class (see e.g. [24], pp. 307–317), we lift the action of Γ on M to an action

of Γ on L (still called α), by setting

ej· (z,w) = (z + ej,w);

iej· (z,w) = (z + iej,e−2πizjw).

Here z ∈ M, w ∈ C, and

ej:= (0,...,0,1,0,...,0) ∈ Zn,

the 1 being in the jthplace. The corresponding representation of Γ in the space of smooth sections

of L is denoted by ρ:

(ρk+ils)(z) = αk+ils(z − k − il),

for k,l ∈ Znand z ∈ M. Define the metric H on L by

H ((z,w),(z,w′)) = h(z)w ¯ w′,

where z ∈ M, w,w′∈ C, and h ∈ C∞(M) is defined by

h(q + ip) := e2π?

j(pj−p2

j).

Let ∇ be the connection on L defined by

∇ := d + 2πi

n

?

j=1

pjdzj+ π dpj.

Proposition 6.1. The triple (L,H,∇) defines an equivariant prequantisation for (M,ω).

The proof of this proposition is a set of tedious computations. Because of the term 2πi?n

and π dpjdo not change the curvature, and have been added to make ∇ equivariant. At the same

time, the latter two terms ensure that there is a Γ-invariant metric (namely H) with respect to

which ∇ is Hermitian.

j=1pjdqj

j=1pjdpj

in the expression for the connection ∇, it has the right curvature form. The terms −2π?n

Page 36

6EXAMPLE: ACTION OF Z2NON R2N

36

As we mentioned in Subsection 2.1, there is a procedure in [27] to lift the action of Z2non

R2nto a projective action on L that leaves the connection (for example) ∇′:= d + 2πi?

invariant. This projective action turns out to be an actual action in this case, and preserves the

standard metric on L. We thus obtain prequantisation of this action that looks much simpler than

the one given in this section. However, we found our formulas to be more suitable to compute the

kernel of the associated Dirac operator.

jpjdqj

6.2The Dirac operator

In this subsection, we compute the Dolbeault–Dirac operator D / on M, coupled to L. To compute

the quantisation of the action we are considering, we need to compute the kernels of

D /+:= D / |Ω0,even(M);

D /−:= D / |Ω0,odd(M).

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

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

l = (l1,...,lq) ⊂ {1,...,n},

where q ∈ {0,...,n} and l1 < ··· < lq. We will write d¯ zl:= d¯ zl1∧ ... ∧ d¯ zlq. If l = ∅, we

set d¯ zl:= 1M, the constant function 1 on M. Note that {d¯ zl}l⊂{1,...,n}is a C∞(M)-basis of

Ω0,∗(M;L).

Given l ⊂ {1,...,n} and j ∈ {1,...,n}, we define

εjl:= (−1)#{r∈{1,...,q};lr<j},

plus one if an even number of lris smaller than j, and minus one if the number of such lris odd.

From the definition of the Dolbeault–Dirac operator one then deduces:

Proposition 6.2. For all l ⊂ {1,...,n} and all f ∈ C∞(M), we have

D /?fd¯ zl?=

?

j∈l

εjl

?

−2∂f

∂zj

+ (iπ − 4πipj)f

?

d¯ zl\{j}

+

?

1≤j≤n,

j?∈l

εjl

?∂f

∂¯ zj

+iπ

2f

?

d¯ zl∪{j}.

(58)

6.3The case n = 1

We now consider the case where n = 1. That is, M = C and Γ = Z + iZ. We can then explicitly

compute the quantisation of the action. This will allow us to illustrate our noncompact Guillemin–

Sternberg conjecture by computing the four corners in diagram (1).

If n = 1, Proposition 6.2 reduces to

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

D / (f1+ f2d¯ z) =

?∂f1

∂¯ z

+iπ

2f1

?

d¯ z − 2∂f2

∂z

+ (iπ − 4πip)f2.

That is to say, with respect to the C∞(M)-basis {1M,d¯ z} of Ω0,∗(M;L), the Dirac operator D / has

the matrix form (5), where

D /+=

∂

∂¯ z+iπ

2;

D /−= −2∂

∂z+ iπ − 4πip.

Page 37

6EXAMPLE: ACTION OF Z2NON R2N

37

In this case, the kernels of D /+and D /−can be determined explicitly:

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

s(z) = e−iπ¯ z/2ϕ(z),

where ϕ is a holomorphic function.

The kernel of D /−is isomorphic to the space of smooth sections t of L given by

t(z) = eiπz/2+π|z|2−πz2/2ψ(z),

where ψ is a holomorphic function.

The unitary dual of the group Z + iZ = Z2is the torus T2. Therefore, by Proposition 5.3, the

quantisation of the action of Z + iZ on C is the class in KK(C,C∗(Z2)) that corresponds to the

class

??

in K0(T2). It will turn out that the kernels of D /+

T2. Let us compute these kernels.

kerD /+

(α,β)

?

(α,β)∈T2

?

−

??

kerD /−

(α,β)

?

(α,β)∈T2

?

(α,β)and D /−

(α,β)indeed define vector bundles over

Proposition 6.5. Let λ,µ ∈ R. Define the section sλµ∈ C∞(M,L) by

sλµ(z) = eiλze−πp?

k∈Z

e−πk2e−k(λ+iµ+2π)e2πikz.

Set α := eiλand β := eiµ. Then kerD /+

(α,β)= Csλµ.

Remark 6.6. For all λ,µ ∈ R, we have

sλ+2π,µ= eλ+iµ+3πsλµ;

sλ,µ+2π= sλµ.

Hence the vector space Csλµ⊂ C∞(M,L) is invariant under λ ?→ λ+2π and µ ?→ µ+2π. This is

in agreement with the fact that Csλµis the kernel of D /+

(eiλ,eiµ).

Sketch of proof of Proposition 6.5. Let λ,µ ∈ R, and s ∈ C∞(M,L) = C∞(C,C). Suppose s is

in the kernel of D /+

(α,β). Let ϕ be the holomorphic function from Proposition 6.4, and write

˜ ϕ(z) := e−iλze−iπz/2ϕ(z) =

?

k∈Z

ake2πikz

(note that for all z ∈ C one has ˜ ϕ(z + 1) = ˜ ϕ(z)). Then ak= e−πk2e−k(λ+iµ+2π)a0, which gives

the desired result.

?

Proposition 6.7. The kernel of D /−

(α,β)is trivial for all (α,β) ∈ T2.

Sketch of proof. Let λ,µ ∈ R and let td¯ z ∈ Ω0,1(M;L) = C∞(M,L)d¯ z. Suppose that td¯ z ∈

kerD /−

(eiλ,eiµ). Let ψ be the holomorphic function from Proposition 6.4, and write

˜ψ(z) := eπ(¯ z2+i¯ z)/2−iλ¯ zψ(z) =

?

k∈Z

cke2πik¯ z

(note that for all z ∈ C one has˜ψ(z + 1) =˜ψ(z)). Then ck= eπk2ek(λ−iµ−2π)c0, which implies

that c0= 0.

?

We conclude:

Page 38

ANATURALITY OF THE ASSEMBLY MAP38

Proposition 6.8. The quantisation of the action of Z2on C is the class in K0(T2) defined by the

vector bundle28

(Csλµ)(eiλ,eiµ)∈T2→ T2.

By Lemma 5.10, we now find that the reduction of the quantisation of the action of Z2on R2

is the one-dimensional vector space C · s0,0⊂ C∞(M,L), where

s0,0(z) = e−πp?

k∈Z

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

e−πk2e−2πke2πikz.

T2

. Schematically, we therefore have29

Z2? R2?

?

Q

??

RC

??

(Csλµ)(eiλ,eiµ)∈T2

?

RQ

??

T2?

Q

??C · s0,0.

Remark 6.9. The fact that the geometric quantisation of the torus T2is one-dimensional can

alternatively be deduced from the Atiyah-Singer index theorem for Dirac operators. Indeed, let

D /L/Z2

T2

be the Dirac operator on the torus, coupled to the quotient line bundle L/Z2. Then by

Atiyah-Singer, in the form stated for example in [25] on page 117, one has

Q(T2) = indexD /L/Z2

T2

=

?

?

T2ech1(L/Z2)

=

T2dp ∧ dq

= 1,

the symplectic volume of the torus, i.e. the volume determined by the Liouville measure.

ANaturality of the assembly map

Our “quantisation commutes with reduction” result is partly a consequence of the naturality of

the assembly map. For discrete groups, this naturality is explained in detail by Valette [52]. We

need to generalise “one half” of this naturality (the epimorphism case) to non-discrete groups.

A.1The statement

Let G be a locally compact unimodular group acting properly on a locally compact Hausdorff space

X. We consider a closed normal subgroup30N of G, and suppose that G and N are equipped with

Haar measures dg and dn, respectively. 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 following diagram

commute:

µG

X

KG

0(X)

??

VN

??

K0(C∗(G))

R(N)

Q

??

KG/N

0

(X/N)

µG/N

X/N??K0(C∗(G/N)).

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/Γ = T2=ˆΓ.

30Although for our purpose it is enough to consider discrete normal subgroups of G, we have to work in the

nondiscrete setting anyway, since G is not necessarily discrete. We therefore allow nondiscrete subgroups N.

Page 39

ANATURALITY OF THE ASSEMBLY MAP39

Here µG

Xand µG/N

X/Nare analytic assembly maps as defined in e.g. [5, 70, 52], and the map

R(N)

Q

=??

N

?

∗

(59)

is functorially induced by the map?

N: C∗(G) → C∗(G/N) given on f ∈ Cc(G) by [23]

?

N

?

N(f) : Ng ?→

f(ng)dn.(60)

To prove Theorem A.1, one can simply copy the proof for discrete groups in Valette [52],

replacing discrete groups by possibly non-discrete ones and sums by integrals. In places where

Valette uses the fact that a finite sum of compact operators is again compact, one uses Lemma

A.2. This lemma states that in some cases, the integral over a compact set of a continuous family

of compact operators is compact.

Another difference between the discrete and the nondiscrete cases is the definition of the map

Ψ on page 110 of [52]. In the nondiscrete case this map is defined as follows. If ξ ∈ Hc, we will

write ξN:= ξ + ker(·,·)N for its class in HN. Then for all ξ ∈ Hc, we have ξN∈ HN,c. Define the

linear map

Ψ : Hc⊗Cc(G)Cc(G/N) → HN

by

Ψ[ξ ⊗ ϕ] =

?

G/N

ϕ(Ng−1)Ng · ξNd(Ng),

where d(Ng) is the Haar measure on G/N corresponding to the Haar measures dn and dg on N

and G, respectively. To prove that the extension Ψ : E ⊗C∗(G)C∗(G/N) →˜E is surjective, one

can use a sequence of compactly supported continuous functions on G/N that converges to the

distribution δNe∈ D′(G/N) with respect to the measure d(Ng).

A.2Integrals of families of operators

Lemma A.2. Let E be a Hilbert C∗-module, and let F(E) and K(E) = F(E) be the algebras of

finite-rank and compact operators on E, respectively. Let (M,µ) be a compact Borel space with

finite measure. Suppose M is metrisable. Let α,β : M → B(E) be continous, and let T ∈ K(E)

be a compact operator. Define the map φ : M → K(E) by φ(m) = α(m)Tβ(m). Then the integral

?

We will prove this lemma in several steps. For continuous maps ψ : M → B(E), we will use the

norm

?ψ?∞:= sup

Mφ(m)dm defines a compact operator on E.

m∈M?ψ(m)?B(E).

Lemma A.3. Let (M,µ) be a compact Borel space with finite measure, and let E be a Hilbert

C∗-module. Let φ : M → K(E) be a continuous map. Suppose that φ is ‘uniformly compact’, in

the sense that there exists a sequence?φj

j → ∞. Suppose furthermore that for every j ∈ N, there is a sequence

simple functions (i.e. having finitely many values), such that for all ε > 0 there is an n ∈ N such

that for all j,k ≥ n, ?φk

on E.

Proof. For all j,k ∈ N, the integral?

hence a finite-rank operator itself. And because ?φj

?∞

j=1: M → F(E) such that ?φj− φ?∞tends to zero as

?φk

j

?∞

k=1: M → F(E) of

j− φj? < ε. Then the integral?

Mφ(m)dµ(m) defines a compact operator

Mφk

j(m)dµ(m) is a finite sum of finite-rank operators, and

j− φ?∞→ 0 as j tends to ∞, we have

?

M

?

M

φj

j(m)dµ(m) →

φ(m)dµ(m)

in B(E). Hence?

Mφ(m)dµ(m) is a compact operator.

?

Page 40

ANATURALITY OF THE ASSEMBLY MAP 40

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

Proof. Choose a sequence (Tj)∞

j=1in F(E) that converges to T. For m ∈ M, set

φj(m) = α(m)Tjβ(m)

Then

?φj− φ?∞≤ ?α?∞?Tj− T?B(E)?β?∞→ 0

as j → ∞. Note that α and β are continuous functions on a compact space, so their sup-norms

are bounded.

Choose sequences of simple functions αk,βk: M → B(E) such that ?αk− α?∞ → 0 and

?βk− β?∞→ 0 as j goes to ∞ (see Lemma A.5 below). For all j,k ∈ N, set

φk

j(m) := αk(m)Tjβk(m),

for m ∈ M. Note that

?φk

j− φj?∞= sup

m∈M?αk(m)Tjβk(m) − α(m)Tjβ(m)?

= sup

m∈M

+ ?αk(m)Tjβ(m) − α(m)Tjβ(m)?

?

?αk(m)Tjβk(m) − αk(m)Tjβ(m)?

?

≤ ?αk?∞?Tj??βk− β?∞+ ?αk− α?∞?Tj??β?∞.

The sequences k ?→ ?αk?∞and j ?→ ?Tj? are bounded, since αk→ α and Tj→ T. Hence, because

the sequences ?αk−α?∞and ?βk−β?∞tend to zero, we see that ?φk

than any ε > 0 for k large enough, uniformly in j.

j−φj? can be made smaller

?

Lemma A.5. Let (M,µ) be a metrisable compact Borel space with metric dM, let Y be a normed

vector space, and let α : M → Y be a continuous map.

Then there exists a sequence of simple maps αk: M → Y such that?αk?−1(V ) is measurable

for all V ⊂ Y , and such that the sequence??α − αk?∞

Proof. For every k ∈ N, choose a finite covering˜Uk= {˜V1

From each˜Uk, we construct a partition Uk = {V1

Vj

and j ∈ {1,...,nk}, choose an element mj

?∞

k=1goes to zero as k goes to infinity.

k,...,˜Vnk

k} of M, by setting V1

kare Borel-measurable. For all k ∈ N

k. Define the simple map αk: M → Y by

k} of M by balls of radius

1

k.

k,...,Vnk

k:=˜V1

k, and

k:=˜Vj

k\?j−1

i=1˜Vi

k. for j = 2,...,nk. Note that the sets Vj

k∈ Vj

αk(m) := α(mj

k)if m ∈ Vj

k.

Note that, because α is continuous (and uniformly continuous because M is compact), for every

ε > 0 there is a kε∈ N such that for all m,n ∈ M,

dM(m,n) <

1

kε

⇒?α(m) − α(n)?Y < ε.

Hence for all ε > 0, all k > kε, and all m ∈ M (say m ∈ Vj

k),

?α(m) − αk(m)?Y = ?α(m) − α(mj

k)?Y < ε.

So ?α − αk?∞indeed goes to zero.

?

Page 41

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