index theory on locally homogeneous spaces

K-Theory 4: 547-568, 1991. 547 �9 1991 Kluwer Academic Publishers. Printed in the Netherlands.

Index Theory on Locally Homogeneous Spaces

JEFFREY FOX* Department of Mathematics, University of Colorado, Boulder, and SUNY, Albany, U.S.A.

and

PETER HASKELL** Department of Mathematics, Virginia Polytechnic Institute and State University, 460 McBryde Hall, Blaeksburg, VA 24061, U.S.A.

(Received: December 1990)

Abstract. We describe how the equivariant K homology class of an invariant elliptic operator on a homogeneous space of a linear semisimple Lie group determines the L2-index of the associated operator on a finite volume locally homogeneous space. The machinery of equivariant K homology and of KK theory can be used to prove theorems about L2-indices. We give an application motivated by the problem of calculating multiplicities of subrepresentations of quasi-regular representations.

Key words. L2-index theory, locally homogeneous space, equivariant K homology.

O. Introduction

Let G be a linear, connected, semisimple Lie group, and let F be a torsion-free

discrete subgroup having finite covolume in G. (More generally one can assume

that G and the torsion-free F satisfy the assumptions o f Section 2.1 of [BG].)

Assume H is a compact subgroup of G. Let P be a properly supported, G-invariant,

elliptic pseudodifferential operator of nonnegative order between sections o f homo-

geneous bundles W~ and W2 over H\G. P descends to define an elliptic operator

P : L 2 ( W ~ / F ) - * L 2 ( W 2 / F ) over H\G/F . Denote by P* the Hilbert space adjoint

o f P. Al though P need not be Fredholm, the kernels o f P and P* are finite-

dimensional, and one can define the L2-index of P to be d i m ( k e r n e l ( P ) ) -

dim(kernel(P*)) (see [M]). The purpose o f our paper is to show how to recover information about such indices f rom the G-equivariant K homology of H\G.

Let KKa(Co(H\G), C) denote the G-equivariant K homology of the algebra Co(H\G) of cont inuous functions that vanish at infinity on H\G. This K homology

group shares several properties with the better known K homology groups of compact manifolds. First the G-invariant elliptic pseudodifferential operator 15: Z2(Wl) ---* L2(W2) defines an element [P] o f KKG(Co(H\G), C). Second such an

element is determined by the h o m o t o p y class (respecting the G-action) o f the

* Supported by the National Science Foundation under Grant No. DMS-8903472. ** Supported by the National Science Foundation under Grant No. DMS-8901436.

548 JEFFREY FOX AND PETER HASKELL

principal symbol of the pseudodifferential operator. Third if H1 and/-/2 are compact subgroups of G with Hi c / / 2 , then the map f : H~kG ~H2kG induced from the identity map on G determines a mapf,:KKG(Co(H~kG), C) ~KKa(Co(H2\G), C). (See, e.g., [K1], [K3], [B1] for discussions of these assertions.)

The connection between equivariant K homology of a homogeneous space and L 2 index theory on a locally homogeneous space is provided by the following three steps. First P has analytic index Inda(P) in the K theory of the group C* algebra, Ko(C*G) or KK(C, C'G). Inda(P) can be recovered from [if] eKKa(Co(HkG), C) (see [K2]).

Second kernel(P), respectively kernel(P*), is the kernel, resp. kernel of the adjoint, of a truly Fredholm operator Pd : L2(W~/F) ~ L2(W2/F) [M]. This operator is the restriction of P to the discrete spectrum parts L2(Wi/F) of L2(Wi/F)." This Fredholm operator determines a class [Pa] e KK(C, C). The isomorphism KK(C, C) = 2 identifies [Pal with L2-index(P).

Finally the discrete spectrum part L~(G/F) of L2(G/F) defines an element [F] of the K homology of C'G, KK(C*G, C). The Kasparov product In- da(P) | [F] equals [Pd] ~ KK(C, C). The construction of [F] and the calculation of this product are our contributions to the reasoning that leads from the equivariant K homology class [P], with its familiar properties, to the more elusive L2-index(P).

With the above relationship established, the properties of K homology have the following implications for index theory on locally homogeneous spaces.

Let H be a compact subgroup of G and let K be a compact subgroup containing H. Let f : HkG ~K/G be induced from the identity map on G, and let f , be the induced map on equivariant K homology. Denote by I/~ the map defined in Remark 6.8 from KKG(Co(HkG), C) to Ko(C*G) satisfying IH([ff]) = Ind,(if). Let IK be the analogous map with K in place of H. Then IH([ff]) = Ix(f ,([if])).

Thus techniques being developed for calculating L2-indices on locally symmetric spaces (e.g. in [BaM], [Br], [BrS], [Mfi], [St]) are sufficient to calculate L2-indices on locally homogeneous spaces.

Let K be a maximal compact subgroup of a connected G, and assume G has a compact Cartan subgroup T ~ K. Let f : TkG ~KkG be induced from the identity on G. Suppose/~ is a Dolbeault operator on T\G with kernel the discrete series rc and with kernel(/~*) = (0). Suppose 152 is a Dirac operator on K\G with kernel rc and kernel(/9*)= (0). (The papers [L], [Pa], [ASch] and [Sch] provide a good starting point for the history of the theorems stating that all discrete series can be realized in these ways.) Our reasoning shows that Inda(s = Inda(/~2). F. Williams [W] has shown that for ~c taken from a broad class of discrete series, the multiplicity of z~ in L2(GkF) equals the L2-index of D2. Our reasoning shows that Williams' theorem implies that an index-theoretic version of Langlands' conjecture [L], mult(Tz, L2(GkF))=L2-index(Dl), is true for all ~z in the class considered by Williams. We thank the referee for informing us that this result also appears in manuscripts of Williams [Wl], [W2].

INDEX THEORY ON LOCALLY HOMOGENEOUS SPACES 549

Remark. As the previous application indicates, one source of interest in these L2-indices is their connection with representation theory. In [FH], techniques like those of the present paper yield multiplicity formulas for LZ-indices of Dirac operators associated to discrete series outside of Williams' class. Multiplicities of representations other than the discrete series enter these formulas. G = Spin(4, 1) is the only group considered in [FH], but results for other groups are in preparation.

Remarks. (1) It is a consequence of Proposition 3.15 that the L2-indices studied here share an important property with indices of elliptic operators on closed manifolds: they depend only on the principal symbol.

(2) If F is a cocompact subgroup of G, the results of this paper are true, by the same proofs, under much more general assumptions on G. However, these results for cocompact F are also consequences of the Atiyah-Singer index and L2-index theorems. On the other hand, for noncocompact F, our arguments require results like those of [BG] concerning the action of LI(G) on L~(G/F). Because we do not know of other noncocompact settings in which such results have been proven, we assume throughout this paper that G and the torsion-free F are those of [BG]. H always denotes a compact subgroup of G.

1. K K Theory, Products, and Cycles Arising from Completely Continuous Representations

In this section we recall the definition of KK theory and a method for calculating Kasparov products. We also describe some special cycles representing elements of KK(C*G, C). G. Kasparov [K3] developed KK theory, the equivariant version of which is described in [K1]. A Connes and G. Skandalis [CSk], [Sk] developed the connection approach to Kasparov products. The observation that this approach defines products in equivariant KK theory appears in [K1]. An exposition of KK theory and its products appears in [B1]. A superscript G indicates that we are discussing G-equivariant KK theory: G acts on all algebras and modules by automorphisms under continuity and compatibility assumptions stated in [K1]. All definitions given for KK G theory become definitions for non-equivariant KK theory by dropping all conditions involving G.

DEFINITION 1.1. For C* algebras A and B, the set of Kasparov (A, B)-bimodules, g~(A, B) is the set of triples (E, F, ~0) where:

(1) E = E ~ (~E 1 is a countably generated Z/22-graded Hilbert B-module; (2) ~p is a G-equivariant homomorphism q~ : A --* ~(E) , the adjointable operators

on E; (we often omit ~0 from this notation, especially when A = C.) (3) F is a G-continuous degree-one element of A~ satisfying for each a s A

and each g e G:

(a) q~(a)(F 2 - I) s ~r the compact operators on E; (b) [~o(a), El e Y(E) ;


(c) ~o(a)(F - F*) ~ S((E); (d) ~o(a)(g(F) - F) ~ vY'(E).

DEFINITION 1.2. KKO(A, B) equals gO(A, B), the set of G-(co)cycles for (A, B), modulo the equivalence relation generated by homotopy.

Tensor Product of Modules. Suppose

(El, F1, (p) ~ g~ B) and (E2, F2, 0-) e gO(B, D).

Let E = E1 | E2 be the graded tensor product. The inner product on E is given by the formula

(xl | x2, Yl | Y2 )E = (X2, a( (Xl, y, )E~ )(Y2) )e2" (1.3)

For x e E l , there is an operator Tx ~ L~(E2, E) defined by T x ( y ) = x | Its adjoint Tx* satisfies T*(z | = o-((x, Z)EI)(y).

DEFINITION 1.4. Use the notation of the preceding paragraph. An operator F e ~,e(E) is called an F2-connection for E~ if for every x e El:

(1) T~ oF2-(-1)exeF2Fo Tx 6 oU(E2, E); ( 2 ) F2 o T~* - ( - 1 ) ~ x ~ 2 T * o F e ~ ( E , , E2) .

DEFINITION 1.5. Use the notation of the preceding paragraph. Assume A is separable. Let q / b e the map A ~ 5e(E) arising naturally from q~. (E, F, q~') is called a Kasparov product of (E~, F~, ~p) and (E2, F2, o-) if:

(1) F is an F2-connection for El; (2) (E, F, q~') e gO(A, D); (3) for each a s A ~0(a)[F~ | I, F]~o(a)* >~ 0 mod ~ ( E ) .

NOTATION 1.6. If (E, F, ~o') satisfies the conditions of Definition 1.5, we write (E, F, (p') = (E~, F1, (p) | (E2, F2, a).

T H E O R E M 1.7 [K1].

([(El, Fa, ~o)], [(E2, F2, o)]) ~ [(E,, F,, ~o) | (E2, F2, a)]

is a well-defined map

KKO(A, B) • KKO(B, D) ~ KKO(A, D).

DEFINITION 1.8. Kasparov [K1] defines a map

jO : KK~ B) ~ KK(C*(G, A), C*(G, B)).

Let (E, F, cp) E g~ B). Let C~(G, A), respectively C~(G, B), respectively C~(G, E), denote the continuous compactly supported A-valued, resp. B-valued, resp. E- valued, functions on G. We describe a structure of pre-Kasparov (C,.(G, A), C,.(G, B))-bimodule on C~(G, E). Then one defines j~ F, q~)) by completion. Let


e, el, e2 be elements of C(.(G, E), let f ~ Co(G, A), and let h e Co(G, B)

@1, e2)(g) = fc s l(el(s), e2(sg))e ds, (1.9)

(e . h)(g) = f~ e(s)(s . h(s-~g)) ds, (1.10)

( f " e)(g) = fG qo(f(s))(s " e(s lg)) ds. (1.11)

The operator ~ in jG((E,/7, q~)) is defined by

~(e)(g) = F(e(g)). (1.12)

Remark 1.13. Let zc be a unitary representation of G on a Hilbert space H. For each f e L I(G) there is an operator a n ( f ) e 5~ defined by

a~( f ) = fGf(g)~(g )dg. (1.14)

The map a~:L~(G)-*5~(H) extends to a map o-~: C * G ~ S Y ( H ) . cry(C'G) is contained in the norm closure of a~(LI(G)).

LEMMA 1.15. Assume ~z is a unitary representation of G on a Hilbert space H. Assume that for every a e C'G, a~(a) e S ( H ) . Then

[Oo Ool ) defines an element of g(C*G, C).

Pro@ Check the conditions in the definition of g(C*G, C).

2. Homogeneous and Locally Homogeneous Bundles

In this section we set notation for homogeneous and locally homogeneous vector bundles. We recall some ideas involved in constructing Kasparov bimodules of sections of such bundles. Let H be a compact subgroup of G, and let W be a finite-dimensional right unitary H-representation. Let W be the associated vector bundle W x , G over H\G. The projection W--> H \ G is an equivariant map of right G-spaces. This structure determines a right G-action on the sections of this bundle. There is a C~(G)-valued inner product on sections of W defined for sl, s2 ~ C~(W) by

s2 )(g) = [ ((s I " g)(x), s2 (s))wx dx. (2.1) (s~, dH \a


There is a right action of C~(G) on C~(W) defined for s e C~(W) and f 6 C~(G) by

(s " f)(x) = fo (s " g)(x)f(g) dg. (2.2)

LEMMA 2.3 [K2]. Completing C~(W) with respect to the norm associated with the above inner product, one forms a Hilbert C'G-module. I f W is graded, the Hilbert C'G-module inherits a grading.

NOTATION 2.4. We use Ew to denote the Hilbert C 'G-module of the preceding lemma.

Remark 2.5. It is often convenient to view sections of W as W-valued functions on G f : G--+ W satisfying the H-invariance property f(hg) = f ( g ) - h - i. From this point of view the C'G-valued inner product and right C'G-act ion on E w arise by completion from the following. For fl and f2 smooth compactly supported H- invariant W-valued functions on G and for f ~ C~(G),

( ,f l ,A}(g) = fa (f '(sg-I) ' f2(s)}w ds

= ft. (J](s)'f2(sg)>w ds, (2.6)

= fofl (gs - ~)f(s) ds. (2.7) ( f , ' f)(g)

Remark 2.8. Let s I , s2 ~ C~(W), and letf l and f2 be the corresponding H-invariant elements of Cc~(G, W). Because the representation of H on W is unitary,

fu (sl(x)'s2(X)}wxdx =

up to a constant associated with the Haar measure of H. (For any given H one can arrange for this constant to equal 1.) Using superscript H to denote the subspace of H-invariant elements, we see that our identification of C~(W) with C~(G, W) ~ extends to an identification of the L 2 sections of W, Lz(w), with L2(G, W) *~.

Remark 2.9. An analogous construction identifies smooth compactly supported, respectively L 2, sections of the bundle W/F--+ H/G\F with the H-invariant smooth compactly supported, respectively L 2, W-valued functions on G/F.

Remark 2.10. Except where stated otherwise, we view sections of W, respectively of W/F, as H-invariant W-valued functions on G, resp. G/F.

DEFINITION 2.11. Define a map

Q : c ~ ( w ) | C~(G/F) -~ C~(W/F)


by

~)(x) = jGf(g)~(g-~x) dg. (2.12) Q(f |

LEMMA 2.13. Q(f | 4) has the required H-invariance. Proof

f~.f(g)~(g lhx) dg = tf(hs)~(s-'x ) ds Q_(f | ~)(hx)

= f a ( f ( s ) ' h - ' ) ~ ( s - ' x ) d s = ( f a f ( s ) ~ ( s - ' x ) d s ) ' h - '

Suppose f e C~(G, W) ~, i.e. f represents an element of C2(W). One can define

a map

Qs: L2(G/F) --* L2(W/F)

by

Q:(~) = Q( f | 3). (2.14)

Remark 2.15. Note that Qcis represented by convolution with a smooth compactly supported W-valued function on G, which happens to be H-invariant, followed by the identification of H-invariant W-valued functions on G/F with sections of W/F.

LEMMA 2.16. Qr is an integral operator on G/F with smooth kernel k07,)5)= E.~v f(xTy- t). Here ~, ~ ~ G/F, and x and y are arbitrarily chosen preimages of 2 and S inG.

Proof Q/-(~)(YO = Saf(s)~(s-12) ds. Let ~ be the pullback of ~ under G ~G/F. Then for any x in the preimage of if,

faf(s)r '2) ds=fJ(s)g(s-'x)ds=fJ(s)g((x l s ) - l )ds

= jGf(xy- ')~(y) dy.

Identify a fundamental domain in G with G/F. Then

fGf(xy--1)~(y)dy:iG[F(./~Ff(x])Y 1))~(-~) d. ~-

Remark 2.17. This proof uses the often useful technique of identifying functions on G/F with F-invariant functions on G.

3. Invariant and Locally Invariant Pseudodifferential Operators

In this section we recall some results and techniques from the theory of G-invariant pseudodifferential operators. In particular for an order zero, G-invariant, properly supported, elliptic pseudodifferential operator ~ on a homogeneous space H\G, we recall the definition of Inda(~) as an element of KK(C, C'G). Such a ~ descends


to define an operator on H\G/F. The restriction of this operator to the discrete spectrum defines a Fredholm operator. We define a cycle for KK(C, C) arising from this Fredholm operator. Along the way we point out how G-invariant first-order elliptic differential operators on H\G fit into this framework. This section is based on [CM], [K2], and [M], which state much of this material explicitly. We follow [K2] in stating some results for G-compact smooth manifolds X on which G acts properly, rather than just for homogeneous spaces. We let W~ and W2 denote G-vector bundles (with finite-dimensional fibers) over X. Generalizing the discussion of Section 2, one can construct Hilbert C'G-modules of sections of Wi, which are denoted Ew~ (see [K2]). For simplicity we assume that pseudodifferential operators are of type p = 1 and 6 -- 0.

THEOREM 3.1 [K2]. Assume ~ is a G-&variant, properly supported, pseudodifferential operator from sections of W1 to sections of W2. I f ~ is of negative order, it extends to define a compact operator from Ew~ to Ew2. I f ~ is elliptic of order zero, it defines a Fredholm operator from Ewl to Ew2. Thus it defines an element Ind,(~) of Ko(C*G) which depends only on the homotopy class (respecting the G-action) of the

principal symbol of ~.

COROLLARY 3.2. Let ~ be a G-&variant, properly supported, elliptic pseudodifferential operator of order zero from sections of W1 to sections of W2. Let P be an operator of the same kind whose principal symbol a f is in the same homotopy class (respecting the G-action) as the principal symbol of ~ and satisfies a ~a * - I = 0 and a * ~ p - I = 0 for all ~ T*X having [~[) some specified constant. Let if' be an operator of the same kind as fi and with principal symbol a p. = or*. Then

( ~ DEFINITION 3.3.

= C'G).

One is frequently interested in differential operators 15 of positive order, and i5(1 + 15"15)-1/2 need not be properly supported. We remedy this as follows.

Let 15 be a G-invariant, first-order, elliptic differential operator mapping C~(W~) ~C~( W 2) . Using Hilbert space closures and adjoints, one can form 15(1 + 15"15)-1/2. Multiply the distribution kernel of 15(1 + 15"15)-1/2 by a properly supported kernel function that is identically one in a neighborhood of the diagonal. Now apply Kasparov's averaging procedure to the operator defined by the resulting kernel. The finite propagation speed arguments of [BauDT] show that the resulting operator is elliptic pseudodifferential of order zero. By construction it is G-invariant

and properly supported.

Remark 3.4. We often given the name Ind,(15) to the class in KK(C, C'G) constructed from this order zero operator as in Corollary 3.2 and definition 3.3. Ind,(15) is independent of the choices used in its construction.


Remark 3.5. Kasparov [K2] shows that for first order/~ as above, one need not introduce proper support to define -Ind,(/5) ~ Ko(C*G). Kasparov also outlines a construction for elliptic differential operators of order greater than one.

We now describe some aspects of descending these G-invariant, properly supported, pseudodifferential operators from homogeneous spaces H\G to locally homogeneous spaces H\G/F. First L2(G/F) decomposes into a direct sum of irreducible unitary representations, called the discrete part, and a direct sum of direct integrals of unitary representations, called the continuous part. Following [M] we denote this decomposition

L2(G/F) = L2(G/F) | L2(G/F). (3.6)

Recall that the L 2 sections of a bundle W/F-- ,H\G/F can be written as H-.invariant W-valued L 2 functions on G/F. The decomposition above leads to a decomposition of the space of sections into a discrete and a continuous part.

Here

c~(w/r) = L~(W/F) | L~(w/F). (3.7)

= fG/F

After a change

/ . ~-1 (Y) ]c, (fl'f2>Ew(S)ffz(S ly) ds dy

of variables, one sees that the two inner products agree.

L~(W/F) = {H-invariant elements of W | L~(G/F)},

and L~(W/F) is defined similarly.

PROPOSITION 3.8. Let G act on L2(G/F) by (g .~ ) ( y )=4(g - l y ) . Let c r : C * G ~ ( L 2 ( G / F ) ) arise from this representation as in remark 1.13. Then Ew | Le(G/F) ~- Le(W/F).

Proof The isomorphism is the extension to completions of the map Q of definition 2.11. Using approximations to delta functions, one can check that Q has dense range in L2(W/F). To complete the proof it suffices to check that for

f l , f2 E C~(W) and ~.~, 42 ~ C~(G/F), (Q(f l | ~1), O(f2 | ~2))L2(W/r~ equals (fl | 41 ,f2 | 42)ew | L 2(~/V~. The former inner product equals

fo./v(fGf,(g)~l(g-lx)dg, fj2(h)~2(h-'x)dh) dx. The latter inner product equals

<4,, o'(<f,, f2 >e~)(~.2) >L 2(a/r}

t ~-l(y)(a((f,,f2)ew)(gz)(y)) dy jG /F


Remark 3.9. Because the action of C*G on L2(G/F) respects the decomposition of L2(G/F) into subrepresentations, the tensor product of the preceding proposition does also. In fact we have

Ew | aL~(G/F) ~L~(W/F) and Ew | LZ(G/F) ~L~Z(W/F).

Remark 3.10. Let P be a G-invariant, properly supported, pseudodifferential operator of order zero (or less) from L 2 sections of W~ to L 2 sections of W2. (Wi = Wi • H G). We use P also to denote the associated operator mapping Ew~ to kw2. This /3 commutes with the C*G actions on Ewc By Proposition 3.8 and Remark 3.9, P | defines a bounded operator from L 2 sections of Wl/F to L 2 sections of W2/F, and P | is block diagonal with respect to the decomposition L2(Wi/F) = L~(Wi/F) �9 L~(Wi/F).

DEFINITION 3.11. P also defines a "descended" operator P : L2(WI/F) ~L2(W2/ F) as follows. For any go e L2(WI/F), let q5 denote its pullback to a F-invariant section of Wl. Because /~(~) is F-invariant and L 2 on fundamental domains, it defines an element ~ of L2(W2/F). Define P by P(go) = ~.

LEMMA 3.12. In the notation of the preceding two paragraphs P | I = P. Proof. For r E L2(G/F) let ~ denote its pullback to G. Borrowing notation

from Definition 2.11 and the proof of Proposition 3.8, let ~0 e Lz(w1/I ") satisfy for

s e G/F go(2) = fcf(s)~(s 1s ds.

The ~ appearing in Definition 3.11 satisfies

~(x )= f f(s)~(s-lx)ds=faf(xy-1)((y)dy

for any x in the preimage of Y. Thus (Pgo)(2) equals P([.~f(xy-l)g(y) dy). On the other hand

(P | I)( f | ~)(~) = i~ ( f f f ) (s)~(s- l fc) ds = f~ ( P f ) ( s ) ~ ( s - l x ) as

=fG (Pf)(xY-1)~(y)dy='(fGf(xY-1)((y)dy)

because/~ is G-invariant.

Remark 3.13. Remark 3.10 and Lemma 3.12 imply that P is block diagonal with respect to the decompositions L2(Wi/F) = L~(Wi/F)|

DEFINITION 3.14. We give the name Pa to P's discrete block.


PROPOSITION 3.15. Let T be a G-invariant, properly supported, pseudodifferential operator of negative order on L 2 sections of W~. Then T~, the discrete block of the operator T descended from T, is compact.

Proof. It suffices to show that (T*T)~ is compact. T*2P is G-invariant, properly supported, and pseudodifferential of negative order, as is a positive integer power of it. A high enough power of T*SP will have continuous kernel. By [CM] this kernel K(x, y) can be represented by a compactly supported function k satisfying K(x, y) = k(xy-~). (For simplicity we leave the kernel's values in bounded operators on W,- out of the notation.) By [BG], the action of this continuous, compactly supported function k on L~(G/F) is compact. Thus the same is true of its action on the subspace of H-invariant elements of Wi | LZ(G/F), which equals L~(Wi/F). By the spectral theorem compactness of a power of (T*T)d implies compactness of (T*T)a.

4. From Equivariant K Homology to KK(C, C'G)

Let P be a G-invariant, properly supported, order zero, elliptic pseudodifferential operator on H\G. In this section we recall from [K2] that Inda(P) can be defined via a sequence of steps that leads through KKa(Co(H\G), C).

THEOREM 4.1 [K2]. Let W1 and W 2 be homogeneous vector bundles over H\G. Denote their spaces of L 2 sections by L2(Wi). Let P be a G-invariant, properly supported, order zero, elliptic pseudodifferential operator. Assume that the principal symbol of P satisfies the 'eventually unitary' condition stated in Corollary 3.2. With the natural left actions of G and of Co(H\G) on L2(Wi ),

(LZ(W1) | L2(W2), I ; ~* l ) eg~

Proof. By assumption g E G =~ g(P) - P = 0. Approximating elements of Co(H\G) by smooth, compactly supported functions, one can reduce checking the conditions in the definition of g~ C) to well-known compactness results on compact manifolds.

DEFINITION 4.2. Denote by [P] the element of KKG(Co(H\G), C) represented by

THEOREM 4.3 [K2]. [/~] is determined by the principal symbol of P. Proof. Using G-invariance of P and approximation of elements of Co(H\G) by

compactly supported functions, one can reduce this statement to the K-theoretic version of the Atiyah-Singer index theorem on compact manifolds. This is dis- cussed in more detail in, e.g., [B1].


THEOREM 4.4. [P] Define a right action of C~(G x (H\G)) on C~(H\G) by

(e - a)(x) = f~ e(xg- ~)a(g, xg-1) dg.

Define a C~(G • (H\G))-valued inner product on C~(H\G) by

{ e, , e2 )( g, x) = el (x)e2(xg).

Completing C~(H\G) in the associated C*(G, Co(HkG)) norm, one gets a Hilbert C*(G, Co(H\G))-module, Q(H\G).

DEFINITION 4.5. Denote by [qn] the class in KK(C, C*(G, Co(H\G))) represented by

THEOREM 4.6 [K2]

[qH] | ---- Ind,(P) ~ KK(C, C'G).

Proof. Because this theorem, which plays an important role in our paper, is stated without proof in [K2], we offer some details of the proof. Starting with identification of the tensor product, we check the conditions required by the definition of Kasparov product.

LEMMA 4.7. Let C*(G, L;(W~) | denote the module associated to L2(WI) QL2(W2) by the jc operation. Then

(Q(H\G) | | C*(G, L2(W,) @L2(W2)) = Ewl OEw2.

Proof. We define a map between dense subsets. Because the same argument works for each summand, we give it once for a bundle W.

ag = {e e C2(G) : e(hg) = e(g)Vh e H} = Q(H\G),

= { f r Cy(G • G, W):f(g~, hg2) =f (g , , g2)' h - ' 'v'h e H} c

c*(G, L2(W)),

= {4 ~ C?(G, W) : ~(hg) = r h - ' Vh ~ H} ~ E,, .

Define T : ar | ~ ~ cg by

= f e(gu-X)f(u, gu- ' ) du. T(e | Jc


Using approximations to delta functions, one can show that T has dense range in Ew. Thus it suffices to cEeck that T respects inner products.

(r(e, | r(e2 | )~w(g)

= t~ (T(e, | )(s), T(e2 | ds

= f o l f G e l ( su 1)f~(u, su-1) du, foe2(sv l)f2(vg, sv-1) d v ) w d s

= j G ; ; (eI(SU-1)]Cl(bl'SU 1),e2(sv l)f2(vg, sv-l))wdsdvdu.

For each u, make the changes of variables s u - l = z and u v - l = y. The above integral becomes

z,))wdzd, du.

On the other hand,

@1 | f l , e2 | f2 ) o(H\6) | \a))c*(a,c 2(w))(g)

= (-fl, (el , e2}Q(t/\a)",f2}c*(<L2(w))(g)

= fc fG "f2}(ug, z))wdz du

=fafaff~(u,z),faO,(z)ea(zY)f2(y-'ug, zy) dY)w dzdu.

The next lemma provides half of the connection condition in the definition of Kasparov product. The other half of the condition follows by taking adjoints.

LEMMA 4.8. Let e e Q( H\ G). Let ~--e : C*( G, L2(Wi)) ~Ew; be T o T~ where T~ is defined just before Definition 1.4 and T is defined in the proof of Lemma 4.7. Let F be the operator in Ind,(if) and let F2 be the operator in jG([/~]). Then

3--~ o F2 - F o "Y--e e JI(C*(G, L2(WI)), Ew2 ).

Proof. Suppose ~b | ~ e C,Y(G)| Lz(w, ) c C*(G, Lz(w~)).

(J-e o F 2 - - F o ~-e)(~b | ~) = fG O(u)u ~ " K(~) du,

where K is the compact operator from Lz(w1) to L2(W2) arising from the commutator of multiplication by e and the action of the properly supported, order zero pseudodifferential operator P.

On the other hand one can consider the action of a generating element 0~,~ | of Y(C*(G, La(W1)), Ew:) on ~b | r Here

O" e Cc~176 ~ Ew2 and (p | e C~(G) | ~ C*(G, L2(W,)).

56O

As usual we view q and

{0~,~| | ~)}(x) ={~'<~|

JEFFREY FOX AND PETER HASKELL

as functions on G.

a(xg-l)<qo | t/, 0 | ~)C*G| dg

_(a _(a Q p(s)q(y), O(sg)~(y) >wl dy ds dg ~r(xg 1)

_la (o(s)O(sg) ds dg .ta <tl(Y)' ~(y) >Wl dy o-(xg - l )

=faa(xg-l) fa(p(ug-1)O(u)dudg(tl,:)l.2(w,)

=fafaa(xg-l)(o(ug-1)O(u)dgdu<rl,~>L2(wl)

=~faa(xu-17-1)(o(y-l)O(u)dyduQl,~)r2(w,)

To see that 6~" e o F 2 - - F e ~--e is in the closed span generated by such elements, one shows that elements of L2(Wa) can be approximated in L2(W2) by expressions of the form ~G a(x7- ~)(p*(7) dT.

Remark 4.9. By Corollary 3.2 Ind,(/~) e g(C, C'G).

Remark 4.10. Because the operator in [qn] is the zero operator, condition 3 of the definition of Kasparov product is satisfied.

5. Ind.(P) | IFI

As always H is a compact subgroup of G and F is a discrete, torsion-free subgroup of G having finite covolume, ff is a pseudodifferential operator on H\G having all the properties listed in Corollary 3.2, including an 'eventually unitary' principal symbol. In this section we show that the discrete block of the descended operator, Pa, defines an element [Pd] of KK(C, C). We define an element [17] of KK(C*G, C). We show that Inda(/~) | [F] = [Pd]. LEMMA 5.1. Recall that G acts on L2(G/F) by (g �9 ~)(x) = ~(g-~x). This representation has a discrete subrepresentation L ~(G/F). Let a~ be the representation of C*G on L~(G/F) arising from the above representation of G as in Remark 1.13. Then

Proof. [BG] shows that this triple satisfies the condition of Lemma 1.15.


DEFINITION 5.2. We denote by [F] the element of KK(C*G, C) represented by the cycle of the preceding lemma.

LEMMA 5.3. Let P and fi' be as in-Corollary 3.2. Let Pe and P'a denote the discrete blocks of the associated descended operators. Then

Proof. See Proposition 3.15.

Remark 5.4. We denote by [Pal] the class in KK(C, C) represented by the cycle of Lemma 5.3. Under KK(C, C) _--- 2, [Pd] is identified with index (P~:Lad(W1/F) LZ(Wz/F)). By [CM] and [M] this number equals the L2-index of P.

THEOREM 5.5. Let P and [Pa] have the meanings assigned them in Lemma 5.3 and Remark 5.4. Then Inda(/~) | [F] = [Pal] ~ KK(C, C).

Proof. Proposition 3.8 and Remark 3.9 identify the modules involved. Lemma 5.3 gives condition 2 of the Kasparov product. Lemma 3.12 gives condition 3. To prove the connection condition of the definition of the product, we observe that for f ~ C y(G, W)tt, a dense subset of Ew, T s = Qf of (2.14). By Remark 2.15 and [BG] Qs is compact.

6. Reduction to Locally Symmetric Spaces and Langlands' Conjecture

In this section we discuss a general application and an interesting special case of the chain of reasoning leading from equivariant K homology of homogeneous spaces to L2-indices of operators on locally homogeneous spaces. We have already pointed out in Proposition 3.15 that for operators of the kind we are studying, the principal symbol determines the L2-index. Here we point out conditions under which one can prove equality of L2-indices for operators on locally homogeneous spaces determined by different compact groups.

We also show that these conditions are satisfied in the following setting. Let K be a maximal compact subgroup of a connected G, and assume G has a compact Cartan subgroup T ~ K. Each discrete series representation of G can be realized using either a Dolbeault operator on T\G or a Dirac operator in K\G. [Sch], [ASch], [Pa] Working in equivariant K homology, we show that for a given discrete series Ind a of the associated Dolbeault operator and Ind~ of the associated Dirac operator are equal in KK(C, C'G). It follows that for any torsion-free discrete U of finite covolume the descended Dolbeault and Dirac operators have equal L2-indices. For a broad class of discrete series F. Williams has shown that the La-index of the descended Dirac operator equals the multiplicity of the discrete series representation in L2(G/U) (see [W]). Our reasoning shows that Williams' theorem implies that the same statement is true for the descended Dolbeault operator. We thank the referee for informing us that Williams [W1], [W2] has


proven this index-theoretic version of a conjecture of Langlands [L], as well as other generalizations of the conjecture.

Henceforth let H and K be compact subgroups of G with H c K.

NOTATION 6.1. The identity map on G induces a map f : H\G--,K\G. There is an associated map f : Co(K\G)~Co(H\G) defined by f (a )=a of There is an associated map

(p" C*(G, Co(K\G)) ~ C*(G, Co(H\G))

defined for

b ~ C2(G, Co(K\G)) by q~(b)(g) =f(b(g)).

Remark 6.2 (see, e.g., [B1]). Let p :A--+B be a G-equivariant map of C* algebras. Give the algebra B its natural structure as Hilbert B-module. Let a c A act on the left of B by multiplication by p(a). With this structure

NOTATION 6.3. Denote by [p] the class in KKG(A, B) defined by the cycle of Remark 6.2. In particular for the f and q~ of notation 6.1, we have

and

If] ~ KKG(Co(K\G), Co(H\G))

[(p] ~ KK(C*(G, Co(K\G)) , C*(G, Co(H\G))).

Remark 6.4 (see, e.g., [BI]). For p A ~ B there are maps

p," KKG(D, A) --+ KKG(D, B) and p* : KKO(B, D) --+ KKO(A, D).

For [c~] e KK~(D, A), p,([cr =[cr | [P]. For [fl] e KKa(B, D), p*([fl]) =[p] | [fl].

Remark 6.5. For the f and f of notation 6.1, the notations f , and _f* have the

same meaning.

LEMMA 6.6. [~p] =jc( [ f ] ) . Proof. Compare definitions.

LEMMA 6.7. For the [qHJ and [qx] of Definition 4.5,

[qK] @C*(G, Co(K\G)) [(,0] ~ [qH]"

Proof. Because the operator in each cycle in zero, we focus on proving equality of modules. We work with dense submodules, and we represent functions on


homogeneous spaces by invariant functions on G.

J " = {e E C~(G) " e(kg) = e(g) Vk ~ K},

~ ' = { f ~ C~(G x G) : f ( g , , hg2) = f ( g , , g2) Vh ~ H},

cg, = {~ ~ Cy(G) : ~(hg) : ~(g) Vh E H}.

Define T' : ~ " | ~ ' --+ cg, by

| = fa e(gs -1)f(s, gs -l) ds. T'(e

Using approximations to delta functions, one can show that T' has dense range in Q(H\G). Thus it suffices to check that T' respects inner products.

( T'(e, | T'(e2 | ) e~.~G~(g,, g2)

= T'(el | )(g2)T'(e2 | )

= fGY,(gas ')~(s, gzs ') ds f6ez(g2g, u-~)f2(u, gzg, u ' )du .

{el @fl , e2 | )0(K',C)| (gl, g2)

= }.~(S, g s - ' ) { ( e , , e2)o(K,,G)f2}(sg,, g2 s - ' ) ds

t ~

= Jafl(s, g2s l ) ;G el(g2s 1)eE(g2s-lr)J2(r l sg l 'g2s- lr )dr ds.

Now for each s and gl, make the change of variable u = r-~sg~.

Remark 6.8. Recall that I H : K K G ( C o ( H \ G ) , C ) ~ K K ( C , C * G ) is the map defined by IH([Cq) = [qz~] | For a pseudodifferential operator /~ of the type we consider on H\G, I~([P]) = Ind,(/~). Recall also the map

f , = f * : KKG(Co(H\G), C) --* KKa(Co(K\G), C)

of Remark 6.5.

T H E O R E M 6.9. For [c~] ~ KKG(Co(H\G), C)

I x ( f . ([a])) = I.([~1).

Proof.

I n ( f . (M)) = [qK ] | C*<a,Co(K\a))jG(~] | Co(.\a) [~])

= [qK] | Co(,V\C~ ([~P] |

= ([q,v] | [@]) |

= [ q . ] | c*(a,c o(U~G)~ JG([~]) = IH ([~1).

COROLLARY 6.10. Let fi, respectively Q, be a pseudodifferential operator of the type we consider on K\G, resp. H\G. I f [fi] =f , ( [Q]) , then .for a given F, index(Pal) = index(Qa), i.e. L ~ - index(P) = L 2 - index(Q).

Proof. See Remark 5.4 and Theorem 5.5.


We now consider a special case of the above, where H is a compact Cartan subgroup, which we denote T, and K is a maximal compact subgroup containing T. We show how Williams' theorem implies Langlands' conjecture. For the rest of the paper we assume that G is connected and that K\G has a G-invariant spin structure. The last condition can always be achieved by passing to a double cover. We fix a K-invariant complex structure on T\K, which gives it the structure of a rational algebraic variety. G-invariant complex structures on T\G are assumed to be compatible with the complex structure.

DEFINITION 6.11. Let L . . . . denote the Hilbert space of L 2 differential forms of type (0, q) on T \K for q even. Let L ~ have the analogous meaning for q odd. Let L denote the graded Hilbert space L . . . . �9 L ~

Remark 6.12. The action of K on T\K determines a unitary action of K on L.

D EFINITI ON 6.13. Let Co(G, L) I~ denote the set of continuous L-valued functions on G that vanish at infinity and that satisfyf(kg) = f ( g ) . k -1 for all k E K. The grading on L determines a grading on this space of functions.

Remark 6.14. There is a right action of Co(K\G ) = Co(G) I~ on the above space of functions defined via pointwise multiplication. This action is compatible with the C0(G)K-valued inner product on Co(G, L) x defined by

/-f, ,f2 >Co(C,L)K(g) = (,fl (g), f2 (g))L-

D EFINITI ON 6.15. Let ~ = (a + #*)(1 + (•+ {-,)2)-~/2, where ffis the Dolbeault

operator on T\K. Define an operator D on C0(G, L) K by (Df)(g) = @(f(g)).

Remark 6.16. One can identify Co(G, L) K with a space of sections of a bundle

over T\G. Pointwise multiplication defines a left action of Co(T\G) on this space of sections. Each element of Co(T\G) acts by an operator adjointable with respect to the Co (G) K-valued inner product on Co(G, L) K.

LEMMA 6.17. With the structure described above

(Co(G, L) K, D) ~ gG(Co(T\G ), Co(K\G)).

Proof. There are natural G-actions on the algebras and module defined by (g . f)(g') =f(g'g). D is G-invariant. All structure is compatible with the G-action.

The algebra of compact operators on Co(G, L) K equals the algebra of continuous K-invariant oVf(L)-valued functions on G that vanish at infinity. Using that observation, one can compute with principal symbols to verify the conditions in the definition of Kasparov bimodule.

DEFINITION 6.18. Denote by [D] the class in KK~(Co(T\G), Co(K\G)) represented by the cycle of the preceding lemma.

DEFINITION 6.19. For a chosen complex structure on T\G, let ~ be the line bundle equal to the square root of the canonical bundle associated with directions orthogonal to the fibers o f f : T\G ~ K \ G .


Remark 6.20. Each complex structure on T\G determines in a natural way a spin C structure on T\G. By the discussion in Section 3, we can associate to the spin c Dirac operator on T\G, twisted by s a class IDle] ~ KKa(Co(T\G), C). By the same discussion, we can associate to the spin Dirac operator on K\G a class [D] ~ KKC(Co(K\G), C).

THEOREM 6.21. [D] | = [D>]. Proof. The main idea of the proof is that the sharp product of the principal

symbol of the first operator with the pullback of the principal symbol of the second operator equals the principal symbol of the third operator. [Gi] The modules can be identified by defining an identification of dense subsets and checking that it respects inner products. The connection condition is checked by calculation with principal symbols. Similar calculations establish the positivity condition by reducing it to qffa)D2qffa) * plus a compact operator.

Remark 6.22. There is a distinghished class 1A e KK~(A, A) that is represented by

:1) For fil ~ KKC(B, A) and f12 ~ KKa(A, B),

fil @A 1,4 = fil and 1 a | f12 = fia-

LEMMA 6.23. If__] | [D] = lco(K\a ~. Proof. Because T\K is a rational algebraic variety, L . . . . splits into a direct sum

of the one-dimensional K-invariant kernel of ~ and its orthogonal complement, which ~ maps isomorphically to L ~ This partial isomorphism is homotopic to a partial isometry. The Kasparov product splits as a direct sum of lc0(K\~ } and a degenerate Kasparov bimodule.

COROLLARY 6.24. [q0] -c @C*(G, Co(T\G))J ([D]) = lc.(~,c0(K\a) ). Proof. This is a consequence of Lemma 6.23 and identities in [K1].

PROPOSITION 6.25. f , ( [ D > ] ) = [D].

Proof. This proposition is a consequence of Theorem 6.21, Lemma 6.23, and the associativity of the Kasparov product.

LEMMA 6.26 [P]. Let H be an arbitrary compact subgroup of G, Let V be a finite-dimensional unitary representation of H. Replacing C~(H\G) by

{~ ~ c ~ ( a , v ) ~(hg) = ~ ( g ) . h ~ Vh e H } ,

one can proceed as in Theorem 4.4 to define an element [q~] of -K(C, C*(G, Co(H\G)) ) that is represented by a module of sections of the vector bundle V XHG-+H\G.


Remark 6.27. Recall that a pseudodifferential operator (of the kind we consider) P mapping sections of W1 to sections of W2 can be 'twisted' by a bundle V to define a pseudodifferential operator Pv mapping sections of V| to sections of V | W2. The principal symbol of Pv equals I v @ (the principal symbol of/%).

THEOREM 6.28. [qV] | Proof. This is a slight adaptation of the proof of Theorem 4.6.

PROPOSITION 6.29. Let ~ be the irreducible representation of T associated to a root 2, and let V be the irreducible representation of K having highest weight 2. Then

[q~] .G D | Co(.\o))J ([ ]) = [q~]-

Proof. The product breaks into a direct sum of a degenerate Kasparov bimodule plus another Kasparov bimodule. The Borel-Weil theorem identifies this second bimodule.

THEOREM 6.30. For ~ and V related as in the preceding proposition, Inda(spin Dirac operator on K \ G twisted by V x f~ G) equals Inda(spin c Dirac operator on T\G twisted by 5P | "t r x r G).

Proof. By Theorem 6.28 it suffices to show

[q~] .a D | ([_]) = [ q ~ ] .o .o

| ([D])| Co(K\o))J ([D])

[qV] .G n = | ([ ] |

[qV] .G D ~ = | ([ ~]).

Proposition 6.29 gives the first equation. The second is an identity involving jo and Kasparov products. The third equation follows from Theorem 6.21.

COROLLARY 6.31. For a given F, the L2-indices of the operators descended from the operators of Theorem 6.30 are equal.

Proof. Section 5 shows that Ind, determines the L2-index.

Remark 6.32. [Gi] The principal symbol of the spin c Dirac operator on T\G can be identified with the principal symbol of the Dolbeault operator on T\G. Thus Ind, (twisted spin c Dirac) = Inda (twisted Dolbeault), and the descended operators have equal L2-indices. Here we are twisting by 5~ | ~ x r G.

Remark 6.33 [ASch], [Sch]. The kernels of the Dirac operator of Theorem 6.30 and the Dolbeault operator of Remark 6.32 realize the discrete series representation with Harish-Chandra parameter 2 + Pc.

THEOREM 6,34 [W]. Choose a F as in the introduction. For a broad class of discrete series representations, the L 2-index of the descended Dirac operator equals the multiplicity in L 2(G /F) of the discrete series realized by the Dirac operator.


COROLLARY 6.35. The same statement is true with Dolbeault operator replacing Dirac operator, i.e. the index-theoretic version of Langlands' conjecture is true for the discrete series considered in [W].

Proof. See Remark 6.32 and Corollary 6.31.

Remark 6.36. We thank the referee for informing us that Williams has proven Corollary 6.35. [Wl], [W2].

References

[ASchl

[BaM]

[BauDT]

[B1]

[BG]

[Br]

[BrS]

[CM]

[CSk]

[FH]

[ai]

IN1]

[K21

[K31

[K4]

ILl

[M]

[M/i]

[Pal [V]

[Schl

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[Sk] Skandalis, G.: Some remarks on Kasparov theory, J. Funct. Anal. 56 (1984), 337 347. [W] Williams, F.: Note on a theorem of H. Moscovici, J. Funet. Anal. 72 (1987), 28-32.

[W1] Williams, F.: An L2-Riemann Roch problem, in Darstellungstheorie Reduktiver Lie-Gruppen und Automorphe Darstellungen, Mathematisches Forschungsinstitut Oberwolfach (1987), abstract and preprint.

[W2] Williams, F.: Lectures on the Spectrum of L2(F\G), Pitman Res. Notes in Math., Longman Scientific and Technical, Harlow, Essex (1991).

index theory on locally homogeneous spaces

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