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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

2Assumptions and result

2.1Assumptions

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

2.3Reduction 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 . . . . . . . . . . . . . . . . . . . . . . . . .

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

17

17

18

20

24

24

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

4 Dirac operators

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

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

25

25

26

5Abelian discrete groups

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

5.2The 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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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].