A K-theoretic Selberg trace formula

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arXiv:1904.04728v1 [math.NT] 9 Apr 2019

Bram Mesland, Mehmet Haluk S

¸eng¨

un and Hang Wang

Abstract. LetGbe a semisimple Lie group and Γ a uniform lattice in

G. The Selberg trace formula is an equality arising from computing in two different ways the traces of operators on the Hilbert spaceL2_(Γ\G) associated to test functionsf ∈ Cc(G). Barbasch and Moscovici used

the Selberg trace formula to compute multiplicities of unitary represen-tations ofGinL2_{(Γ\G), by introducing an index theory interpretation} of the trace formula.

In this paper we take their index theoretic interpretation further and obtain aK-theoretic interpretation of the trace formula. This relies on a commuting diagram which displays a relationship between K-theory of the maximal groupC∗_{-algebra ofGand that of Γ.}

Mathematics Subject Classification (2010).Primary 58B34; Secondary: 46L80, 19K56.

Keywords.Selberg trace formula, K-theory, groupC∗_{-algebra, uniform}

lattice.

1. Introduction

Let G be a semisimple Lie group and Γ a uniform lattice in G. For this introduction only, assume also that Γ is Gromov hyperbolic (which will be the case ifGis of rank one). We present an identity which we consider as a K-theoretic analogue of the Selberg trace formula, namely, forx∈K0(C∗_(G))

X

π∈Gb

mΓ(π)π∗(x) = X

(γ)∈hΓi

τγ,∗(res(x)). (1)

Let us explain the terms involved. Given π in the unitary dual Gb of G, the multiplicity of πin the quasi-regular representation ofGonL2_{(Γ\G) is} denoted bymΓ(π) andπ∗ :K0(C∗(G))→Zis the homomorphism associated toπ. We lethΓidenote the set of conjugacy classes of Γ and given a conjugacy

B. Mesland and M.H. S¸eng¨un gratefully acknowledge support from the Max Planck

class (γ), the mapτγ,∗:K0(C∗(Γ))→Zis the homomorphism obtained (this is where we use the hypothesis on Γ) from the localized traceτγ:L1(Γ)→C

supported on the conjugacy class (γ). Finally res :K0(C∗_(G))→K0(C∗_(Γ)) is the restriction homomorphism obtained by Kasparov multiplication with a certain (C∗_{(G), C}∗_{(Γ))-bimodule.}

Using higher indices, we can use Equation (1) above to obtain an index theoretic Selberg trace formula. To see this, letKdenote a maximal compact subgroup of G. Let D denote the Dirac operator on the symmetric space G/K, possibly twisted by a finite dimensional representation ofKand denote by IndGD the higher index of D over C∗(G). We show that if one takes

x= [IndGD] in Equation (1), then one obtains the index theoretic Selberg

trace formula X

π∈Gb

mΓ(π)Trsπ(kt) =

X

(γ)∈hΓi

vol(Γγ\Gγ)

Z

Gγ\G

Trskt(x−1γx)dx. (2)

Heret >0 andktis the kernel associated to the heat operatore−tD

2

viewed as a matrix valued smooth function onGwith Trs denoting the supertrace.

Our result allows for the use of techniques from the representation the-ory of the semisimple Lie groupGin the study of theK-theory of the lattice Γ. It is part of a program that explores the use of operator algebras and K-theory in the theory of automorphic forms. See [19, 20] for other such results.

2. Statement of Selberg trace formula

LetGbe a semisimple Lie group,K a maximal compact subgroup and Γ a uniform lattice inG. Letf ∈C∞

c (G) and denote byRΓ :G→ U(L2(Γ\G))

the right regular representation which extends toRΓ_:C∞

c (G)→ L(L2(Γ\G))

via

RΓ(f) = Z

G

f(y)RΓ(y)dy f ∈Cc∞(G).

Because G is liminal and Γ\G is compact, RΓ_{(f) is a Hilbert-Schmidt} op-erator, and in particular it is trace class. The Selberg trace formula is an equality arising from computing the trace ofRΓ_{(f) in two different ways.}

On the geometric side, regardingφ∈L2_{(Γ\G) as a Γ-invariant function} onG, and from

[RΓ_{(f)φ](x) =} Z

G

f(y)[RΓ_{(y)φ](x)dy =} Z

G

f(y)φ(xy)dy

= Z

G

f(x−1y)φ(y)dy= Z

Γ\G

X

γ∈Γ

f(x−1γy)φ(y)dy,

we obtain the Schwartz kernel

K(x, y) =X

γ∈Γ

associated tof. The fact thatf is compactly supported and smooth ensures that the sum is locally finite and thusK is a smooth kernel. We have

TrRΓ(f) = Z

Γ\G

K(x, x)dx= Z

Γ\G

X

γ∈Γ

f(x−1γx)dx

= Z

Γ\G

X

(γ)∈hΓi X

δ∈Γγ\Γ

f(x−1_δ−1_γδx)dx

= X

(γ)∈hΓi

vol(Γγ\Gγ)

Z

Gγ\G

f(x−1γx)dx.

As usual, in the abovehΓi denotes the conjugacy classes of Γ and Γγ (resp.

Gγ) is the centralizer of γin Γ (resp.G).

On the representation theory side, the Hilbert spaceL2_{(Γ\G) splits as} a direct sum

L2_(Γ\G)≃M

π∈Gb

H⊕mΓ(π)

π (3)

of irreducible unitary representations (π, Hπ)∈Gb ofGwith each appearing

with finite multiplicitymΓ(π). With respect to the Hilbert space decomposi-tion (3), the trace of the operatorRΓ_{(f) onL}2_{(Γ\G) satisfies}

TrRΓ(f) =X

π∈Gb

mΓ(π)Tr(π(f)),

whereπ(f) =R_Gf(y)π(y)dy, which can be regarded as the Fourier transform off atπ. The Selberg trace formula is the equality

X

π∈Gb

mΓ(π)Tr(π(f)) = X (γ)∈hΓi

vol(Γγ\Gγ)

Z

Gγ\G

f(x−1γx)dx. (4)

2.1. Remark. If Γ\Ghas finite volume but is noncompact, the spectral

de-composition ofL2_{(Γ\G) also involves a continuous component andR}Γ_{(f) is} not necessarily a trace class operator. A truncated version of traces need to be introduced in the Selberg trace formula in this setting. See for example [1,18].

3. Index theoretic trace formula

In this section we formulate an analogue of the Selberg trace formula in the context of index theory, which will be proved using the framework ofK-theory in Section4.

Weassumein addition that the rank ofGequals that of K. Note that

as a consequence, the symmetric spaceG/K is of even dimension. Assume that G/K admits a G-equivariant spinc _{structure and let S = S}+ _⊕S− denote the associated spinor bundle onG/K. Let V be a finite dimensional representation ofK andE=G×KV be the associatedG-equivariant vector

G×K (V ⊗S) → G/K with Z2-grading, i.e., D has the form

0 D− D+ ₀

.

Consider the heat operator e−tD2

, which admits a smooth kernel Kt(x, y).

The kernelKt(x, y) satisfies

Kt(gx, gy) =Kt(x, y) x, y ∈G/K, g∈G,

which is equivalent to theG-invariance ofD. RegardingKt(x, y) as a

matrix-valued function onG×G, invariant under theK×K-action

Kt(x, y) =aKt(xa, yb)b−1∈End(V ⊗S) a, b∈K.

We define

kt(x−1y) :=Kt(x, y) x, y∈G.

Thenktis a matrix-valued function on Gthat is invariant under the action

ofK×K:

kt∈[C∞(G)⊗End(V ⊗S)]K×K, kt(x) =akt(a−1xb)b−1, a, b∈K.

Moreover,kt is smooth and of Schwartz type. To be more precise,

kt∈[S(G)⊗End(V ⊗S)]K×K.

where S(G) is Harish-Chandra’s Schwartz algebra of G. See [7, Section 1] and [5, Section 2] for more details.

We now replace the test function f ∈ C∞

c (G) acting on L2(Γ\G) in

the previous section with the heat kernel kt acting on the Z2-graded space

(L2_{(Γ\G)⊗V⊗S)}K _{and formally state the Selberg trace formula associated}

to the heat operator. We writeπ(kt) for the operator obtained from the action

ofkt on (Hπ⊗(V ⊗S))K.

Choose an invariant Haar measure onGso that it is compactible with G-invariant measure on G/K. This in particular means that the volume of the maximal compact subgroupK ofGis assumed to be 1.

3.1. Proposition. LetTrs be the supertrace ofZ2-graded vector spaces. Then

X

π∈Gb

mΓ(π)Trsπ(kt) =

X

(γ)∈hΓi

vol(Γγ\Gγ)

Z

Gγ\G

Trs[kt(x−1γx)γ]dx. (5)

The heat kernel kt is not compactly supported, however it is of Lp

-Schwartz class for all p > 0 (see [5, Prop. 2.4.]). Using this, one can prove that each summand on the right hand side of equality (5) converges, see [29, Thm. 6.2]. In any case, the equality (5) will follow from our main result Theorem4.14below.

3.1. Geometric side

In this subsection we will see that every term on the geometric side can be identified as the L2_{-Lefschetz number associated to a conjugacy class of Γ} using results from [29], and that the sum of allL2_{-Lefschetz numbers over all} conjugacy classeshΓican be identified with the Kawasaki index theorem [14] for the orbifold Γ\G/K.

Because Γ acts properly and cocompactly onG/K, there exists a cutoff functionconG/Kwith respect to the action of Γ, i.e.,c:G/K→[0,1] such that

X

γ∈Γ

c(γx) = 1 ∀x∈G/K.

Define theL2_{-Lefschetz number associated to the conjugacy class (γ) ofγ∈Γ} by

indγD:= tr(sγ)e−tD

2

= tr(γ)e−tD−D+−tr(γ)e−tD+D− where

tr(γ)S:= X

h∈(γ) Z

G/K

c(x)Tr[h−1KS(hx, x)]dx,

for any (γ)-trace class operatorSwith smooth Schwartz kernelKS, see (3.22)

of [29]. By Theorems 3.23 and 6.1 of [29], indγDadmits a fixed point formula:

for h ∈ (γ) the kernel h−1_K

t(hx, h) localizes to the submanifold (G/K)γ

consisting of points inG/Kfixed byγ. By [29, Theorem 6.2], theL2-Lefschetz number for G/K admits the following expression analogous to the orbital integral on the geometric side of (4):

indγD= vol(Γγ\Gγ)

Z

Gγ\G

Trs[kt(x−1γx)γ]dx. (6)

3.2. Example. If Γ acts freely onG/K, then (G/K)γ is empty and all orbital

integrals vanish except for the term whereγ is the group identity. The geo-metric side of (5) in this special case is equal to vol(Γ\G)·indeD. Here indeD

is a topological index calledL2_{-index (denoted ind}

L2D) for the symmetric

spaceG/K introduced in [7].

Denote by DΓ the operator on Γ\G/K descending from the one on G/K. It is a Dirac operator on the compact orbifold Γ\G/K. Recall that by the Kawasaki orbifold index formula, the Fredholm index indDΓ can be decomposed as a sum over conjugacy classes of Γ and the summands are exactly theL2_{-Lefschetz numbers:}

indDΓ= X (γ)∈hΓi

indγD. (7)

3.3. Remark. In the special case of a free action, indDΓis the Fredholm index ofDΓ on the closed manifold Γ\G/K. By Atiyah’sL2_{-index theorem [2], we} obtain

vol(Γ\G/K) indL2D= indDΓ,

which is a special case of (7). Noting that K is assumed to have volume 1, this formula is compatible with Example3.2.

3.2. Spectral side

Recall the Plancherel decomposition ofL2_(G):

L2(G) = Z ⊕

b

G

(Hπ∗⊗Hπ)dµ(π),

Accordingly, the Dirac operatorDon [L2_(G)⊗V⊗S]K _{has a decomposition}

into a family of Dirac type operatorsDπ on [Hπ⊗V ⊗S]K, parameterized

by irreducible unitary representations (π, Hπ)∈G.b

We recall the definition and properties of Dπ introduced in [21] and

Section 7 of [7]. See also [13] for a comprehensive treatment. LetH∞ π be the

space ofC∞_{-vectors for theG-representationπ. TheG-invariant operatorD} can be written as a finite sumD=P_iRΓ_(X

i)⊗AiwhereXibelongs to the

universal enveloping algebra of gC and A±_i ∈ Hom(V ⊗S±, V ⊗S∓). Here

RΓ _{stands for the right regular representation. ThenD}

π is given by

Dπ: [Hπ∞⊗V ⊗S]K→[Hπ∞⊗V ⊗S]K, Dπ=

X

i

π(Xi)⊗Ai.

It is proved in [7,21] that Dπ is essentially self-adjoint on the dense domain

[H∞

π ⊗V ⊗S]K. Its closure (also denotedDπ) is a Fredholm operator whose

Fredholm index

indDπ = dim(kerD+π)−dim(kerD−π)

is equal to Trsπ(kt) on the spectral side of (5). Indeed, by Proposition 2.1

of [5], one has

Tr(π(k±t )) = Tr(e−tD

∓

πD

±

π).

Then by the McKean-Singer formula, we obtain

indDπ= Tr(e−tD

−

πD

+

π)−Tr(e−tD

+

πD− p π )

= Tr(π(kt+))−Tr(π(kt−)) = Trs(π(kt)). (8)

3.4. Remark. For an irreducible representation V of K, the dimensions of

[Hπ⊗V ⊗S+]K and [Hπ⊗V ⊗S−]K are finite (see [3,7, 5]). The index of

the Dirac operatorDπcorresponding toπcan be calculated as the difference

indDπ= Trsπ(kt) = dim[Hπ⊗V ⊗S+]K−dim[Hπ⊗V ⊗S−]K.

3.5. Proposition. The equality (5) is equivalent to the equality of indices: X

π∈Gb

mΓ(π) indDπ=

X

(γ)∈hΓi indγD.

The left hand side is finite sum (see Corollary4.3below) and the convergence

of the right hand side is determined by (7).

Let (η, Hη) be a discrete series representation, and µ∈t∗C the

Harish-Chandra parameter ofη. There is a standard way, due to Atiyah-Schmid [3], to realize Hη as the L2-kernel of a Dirac type operator on G/K. For an

irreducible representationVν ∈R(K) of K with highest weightµ ∈t∗C, let

Dν _{denote the associated Dirac operator defined on the homogeneous vector}

bundle

G×K(Vν⊗S)→G/K.

Denoting by ρc the half sum of compact positive roots, then Hη can be

identified with theL2_{-kernel ofD}µ+ρc_{.It is also proved in [3, Section 4] that}

dim(Hη∗⊗Vν⊗S±)K = 0,

unlessµ=ν+ρc. From this, one can derive Proposition3.5 whenD=Dη.

See also [4] for some concrete examples.

3.6. Example. Let (η, Hη) ∈ Gb be a discrete series with Harish-Chandra

paramaterµ. Letπbe an arbitrary discrete series. It follows from [3] that

ind(Dµ+ρc₎

π= dim[Hπ∗⊗Vµ+ρc⊗S

+_]K_−dim[H∗

π⊗Vµ+ρc⊗S

−_]K₌

(

1 η=π 0 η6=π.

Assume further that Γ is a torsion-free. Then the index theoretic Selberg trace formula (5) reduces to

mΓ(η) = vol(Γ\G/K)indL2Dµ+ρc,

recovering a main result of Pierrot [22], which is a special case of [17].

4. K-theoretic Selberg Trace Formula

4.1. Decomposition of right regular representation ofG

Let us consider again the right regular representationRΓ _ofGonL2_(Γ\G). As mentioned earlier, C∞

c (G) acts on L2(Γ\G) by compact operators. We

use the standard notation K(X) for the algebra of compact operators on a HilbertC∗_{-moduleX. AsC}∞

c (G) is a dense subalgebra of the maximal group

C∗_-algebraC∗_{(G) ofG, we obtain a ∗-homomorphism}

RΓ_:C∗_(G)→K(L2_(Γ\G)).

This induces a homomorphism onK-theory:

As G is liminal, the above observation applies to any irreducible unitary representation as well. Indeed, for any (π, Hπ)∈G, we have homomorphismsb

π:C∗(G)→K(Hπ), π∗:K0(C∗(G))→Z.

The morphism π∗ determines an element in KK(C∗(G),C) which we will denote [π].Observe that we have

π∗(x) =x⊗C∗(G)[π],

for everyx∈K0(C∗(G)), where⊗C∗_(G)denotes the Kasparov product.

4.1. Proposition. For everyx∈K0(C∗_{(G))there exists a finite subsetS}

x⊂Gb

such that

1. π∗(x) = 0 for allπ∈Gb\Sx,

2. RΓ ∗(x) =

P

π∈Sx

π∗(x).

Proof. Letx∈K0(C∗_{(G)) be given by the even (C, C}∗_{(G)) Kasparov module}

(X, F). Since 1∈C_{acts onX by a projection, after replacingX bypX and}

F bypF p, we may without loss of generality assume thatC_{acts unitally on}

X andF2−1∈K(X). SinceC∗(G) acts onL2(Γ\G) by compact operators, we have thatK(X)⊗1 ⊂ K(X⊗RΓ L2(Γ\G)) (see [16, Proposition 4.7]).

The operator

F⊗1 :X⊗RΓL2(Γ\G)→X⊗_RΓL2(Γ\G),

satisfies (F⊗1)2_{−1 = (F}2_{−1)⊗1∈K(X)⊗1⊂K(X⊗L}2_{(Γ\G)). Thus} F⊗1 is a self-adjoint unitary modulo compact operators, and therefore it is Fredholm. The integerRΓ

∗(x)∈Zis thus gives by the index ofF⊗1. Since there is a direct sum splitting

X⊗RΓL2(Γ\G)≃

M

π∈Gb

(X⊗πHπ)⊕mΓ(π),

it follows thatF⊗1 =L_π∈GbFπ where the operators are defined to be

Fπ:=F⊗π1 : (X⊗πHπ)⊕mΓ(π)→(X⊗πHπ)⊕mΓ(π).

SinceF ⊗1 is Fredholm, its kernel and cokernel are finite dimensional and thus it follows thatFπ is unitary for all but finitely manyπ∈G. Thereforeb

the set

Sx:={π∈Gb:π∗(x) = Ind (Fπ)6= 0},

is finite and the restriction

FGb\Sx :=F⊗1 :

M

π∈Gb\Sx

(X⊗πHπ)⊕mΓ(π)→

M

π∈Gb\Sx

(X⊗πHπ)⊕mΓ(π),

is Fredholm of index 0. Moreover

FSx:

M

π∈Sx

(X⊗πHπ)⊕mΓ(π)→

M

π∈Sx

obviously satisfies Ind (FSx) =

P

π∈Sx

π∗(x). Since

RΓ∗(x) = Ind (F⊗1) = Ind (FSx) =

X

π∈Sx

π∗(x),

the result follows.

For the quasi-regular representation, the above proposition gives a de-composition ofRΓ

∗.

4.2. Corollary. For x∈K0(C∗_{(G)), we have}

RΓ ∗(x) =

X

π∈Gb

mΓ(π)π∗(x),

understood as a sum with only finitely many nonzero terms.

As is well-known, K0(C∗_{(G)) is the recipient of the so-called higher} indices which we recall now. A Dirac type operatorDonG/K has aG-index

IndG(D) := [p]⊗C0(G/K)⋊GjΓ([D])∈K0(C∗(G))

defined by the descent mapjΓ :K∗

Γ(C0(X))→ KK(C0(G/K)⋊G, C∗(G)) followed by a compression on the left by the projection p∈ C0(G/K)⋊G given by

p(x, g) :=c(gx)c(x), x∈G/K, g∈G

wherec:G/K→[0,∞) is a continuous compactly supported function satis-fyingR_Gc(s−1_x)2_{ds= 1 for allx∈G/K. See, for example, [10, Section 4.2]} for details.

4.3. Corollary. Let D be a G-invariant Dirac type operator on G/K with

index IndGD∈K0(C∗(G)). Then

(i) Givenπ∈G, we haveb π∗(IndGD) = indDπ,

(ii) indDπ 6= 0for at most finitely many π∈G.b

It is possible to obtain refined information about the localized indices using representation theory. We have already remarked in Example3.6that the index ofDπ is nonzero whenπis in the discrete series. The next

propo-sition is Prop. 3 of [15] (see also [7, Prop. 7.3] and [8, Lem. 1.1] for related results).

4.4. Proposition. Let π∈Gbbe tempered. Ifπis not in the discrete series nor

a limit of discrete series, then the index ofDπ is zero.

4.5. Remark. Recall that there is a surjective morphismλ:C∗_(G)→C∗

r(G)

and the Connes-Kasparov isomorphismR(K)≃K0(C∗

r(G)) assuming

exis-tence of G-equivariant spinc_{-structure of G/K. The Connes-Kasparov}

iso-morphism is defined through Dirac induction

R(K)≃K0(Cr∗(G)), [Vµ]7→IndG,r(DVµ).

Here Vµ is an irreducible unitary representation of K with highest weight

G×KVµ →G/K.The twisted Dirac operatorDVµ has an equivariant index

IndG(DVµ) ∈ K0(C∗(G)) so that λ∗(IndG(DVµ)) = IndG,r(DVµ). That is,

there is a commutative diagram

R(K) IndG

/

/

IndG,r

&

&

▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

▲ K0(C

∗_(G))

λ∗

K0(C∗ r(G)).

If G is not K-amenable, i.e., λ is not an isomorphism, then not all x ∈ K0(C∗_{(G)) are represented by the equivariant index of some elliptic operator.} For example, ifGhas property T, the Kazhdan projection defines a nontrivial element [p]∈K0(C∗_{(G)) butλ}

∗[p] = 0∈K0(Cr∗(G)).

We will discuss a special case of Corollary4.2where

x= IndG(DVµ)∈K∗(C∗(G)).

Recall that

R∗(IndG(DVµ)) =1∗(IndΓ(DVµ)) = indDΓVµ

whereDVµ

Γ is the operator DVµ descended to Γ\G/K. Then by [21, Section 3] and [5, (1.2.4)]

indDVµ

Γ = X

π∈Gb

mΓ(π)ind(DVµ₎

π.

Furthermore, the summand on the right hand side is nonvanishing only when the infinitesimal characterχπ of π∈Gb coincides with that ofµ+ρc, hence

it is a finite sum. See [3] and [5, (1.3.7)-(1.3.8)]. In summary, we obtain

R∗(x) =R∗(IndG(DVµ)) =

X

π∈G,χb π=χµ+ρc

mΓ(π)ind(DVµ₎

π.

A typical example of such x is represented by the integrable discrete series. The integrable discrete series representations ofGappear as isolated points inGb(see [28]). As a result, they arise as generators [pπ] forK0(C∗(G)).

Recall that every discrete series representation ofGgives rise to a generator ofK0(C∗

r(G)) (see [25, Thm. 4.6]). Soλ∗[pπ] = [pπ]∈K0(Cr∗(G)).Ifπis an

integrable representation, it follows from Example3.6and Corollary4.2that

RΓ∗([pπ]) =mΓ(π).

Letx∈ K0(C∗_{(G)) now be arbitrary. Then there exists finitely many} λµ ∈Kb with multiplicity mλµ such that

λ∗(x) = X

λµ∈Kb

mλµIndG,rD

Vµ_.

Letx0=P_λµ∈KbmλµIndGD

Vµ_{.By definition}

where R∗(x−x0) does not come from index theory and R∗(x0) can be cal-culated by

R∗(x0) = X

λµ∈Kb

R∗(IndGDVµ) =

X

π∈Gb

mΓ(π) X

λµ∈K,b

χπ=χµ+ρc

mλµind(D

Vµ₎

π.

Thus if we were to consider x0 in K0(C∗

r(G)), then x0 can be decomposed

into a finite sum involving elliptic operators on G/K. Note however that we cannot work withC∗

r(G) in this paper, as one knows that non-tempered

representations ofGmay enterRΓ _{[27, p.177].}

4.2. Decomposition of the trivial representation ofΓ

Consider a uniform lattice Γ inGand denote by1_{the trivial representation}

of Γ. The corresponding representation of C∗_{(Γ) is a ∗-homomorphism and} hence a trace

1_:C∗_(Γ)→C X

γ∈Γ aγγ7→

X

γ∈Γ aγ.

Thus there is a morphism ofK-theory:

1_∗:K0(C∗_(Γ))→Z_.

Let γ ∈ Γ with conjugacy class (γ). Then we have a well-defined localized trace on the Banach subalgebraL1_(Γ)⊂C∗_(Γ),

τγ :L1(Γ)→C,

X

g∈Γ agg7→

X

g∈(γ) ag.

Fora, b∈L1_{(Γ) we have the tracial propertyτ}

γ(a∗b) =τγ(b∗a) with respect

to the convolution product∗. This implies the existence of the morphism

τγ,∗:K0(L1(Γ))→C.

Letι:L1_(Γ)→C∗_{(Γ) be the inclusion map. It is straightforward to observe}

that _X

γ∈hΓi

τγ,∗=1∗ι∗:K0(L1(Γ))→Z. (9)

Given a Dirac type operator D on G/K, we can form its Γ-index IndΓ(D) which lands inK0(C∗_{(Γ)) following the recipe we outlined right} be-fore Corollary4.3. If we choose a properly supported parametrix, the index class can be presented in a smaller algebraC_{Γ⊗ Rwhere Ris the algebra}

of operators with smooth kernels and compact support. In particular, the Γ-index homomorphism IndΓ factors through the L1_{-index homomorphism}

KΓ 0(G/K)

Ind_Γ,L1

−−−−−→ K∗(L1(Γ)). That is, there is IndΓ,L1(D) ∈ K0(L1(Γ))

such that

ι∗(IndΓ,L1(D)) = IndΓ(D).

The following lemma is proved in Definition 5.7 and Proposition 5.9 in [29].

4.6. Lemma.

Then together with (9) we have

1_{∗(IndΓ(D)) =} X

(γ)∈hΓi

indγ(D).

LetDΓbe the operator on the quotient orbifold Γ\G/Kinduced byD. Note that1_{∗(IndΓ(D)) is equal to the Fredholm index ofDΓ (see [6]). Therefore,}

we have an equality of indices obtained from an identity onK-theory:

ind(DΓ) = X (γ)∈hΓi

indγ(D).

The reader should compare this to (7).

4.3. The restriction map

Let Γ⊂Gbe a discrete subgroup and consider the restriction map

ρ:Cc(G)→Cc(Γ)⊂C∗(Γ),

defined by ρ(f)(γ) := f(γ). The map ρ defines a positive definite C∗_(Γ) valued inner product onCc(G) by

hf, hi(γ) :=ρ(f∗∗h)(γ) = Z

G

f∗(ξ)g(ξ−1γ)dµG(ξ).

Upon completion, a (C∗_{(G), C}∗_(Γ))-C∗_{-bimodule, denoted Res, where the} action ofC∗_{(G) is induced by convolution onC}

c(G). Denote by [g] the class

ofg inG/Γ. The commutativeC∗_{-algebraC0(G/Γ) acts onC}

c(G) from the

left via

(f·φ)(g) :=f([g])φ(g),

and this extends to an action by adjointable endomorphisms ofRes_{. Elements}

ofC0(G/Γ) do not act compactly unlessGis discrete.

By Rieffel’s imprimitivity theorem (see [24]) the algebra of compact endomorphisms ofRes_is

K(Res_{) =C}∗_(G⋉G/Γ).

The dense subalgebraCc(G⋉G/Γ) acts on the dense submoduleCc(G)⊂Res

via

(K·φ)(g) := Z

G

K(ξ,[ξ−1g])φ(ξ−1g)dµGξ.

4.7. Lemma. Letf ∈Cc(G)andh∈C0(G/Γ). Then(f·h)(g, x) :=f(g)h(x)

is an element ofCc(G⋉G/Γ)whose action on Cc(G)is given by

(f·h)φ(g) = Z

G

f(ξ)h([ξ−1g])φ(ξ−1g)dµGξ=

Z

G

f(ξ)(h·φ)(ξ−1g)dµGξ.

(10)

Consequently we have thatC∗_{(G)·C0(G/Γ)⊂K(Res).}

Proof. The functionf·hsatisfies supp (f·h)⊂suppf×supphand therefore

has compact support, that is it is an element ofCc(G⋉G/Γ). The remaining

4.8. Proposition. Let Γ ⊂ Gbe a discrete cocompact group and Res _the

as-sociated(C∗_{(G), C}∗_{(Γ))-bimodule defined above. ThenC}∗_{(G)acts onResby}

compact module endomorphisms. Hence the bimodule Res_{defines an element}

[Res_]∈KK0(C∗_{(G), C}∗_(Γ)).

Proof. AsG/Γ is compact we have thatC0(G/Γ) =C(G/Γ) and the constant

function 1∈C0(G/Γ). By Lemma4.7it follows thatf·1∈K(Res_{). It follows}

from Equation (4.7) that the action of f on Cc(G) coincides with that of

f ·1 onCc(G) and thusCc(G) acts by compact endomorphisms. Hence by

continuity all of C∗_{(G) acts compactly. The remaining claims now follow}

directly.

4.9. Definition. The restriction map

res :K0(C∗(G))→K0(C∗(Γ))

is given by the KK-product with Res _{∈ KK0(C}∗_{(G), C}∗_{(Γ)) in}

Proposi-tion4.8:

res([x]) := [x]⊗C∗_(G)[Res].

Denote by 1_:C∗_(Γ)→C_{the∗-homomorphism induced by the trivial}

representation of Γ onC_:

1_{:f 7→}X

γ∈Γ

f(γ), f ∈Cc(Γ),

and by [1_]∈KK0(C∗_(Γ),C_{) itsK-homology class.}

4.10. Proposition. We have a commutative diagram

K∗(C∗(G))

−⊗[RΓ_]

/

/

res

Z

K∗(C∗(Γ))

−⊗[1]

;

;

① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ①

(11)

Here[RΓ] is the element inK0(C∗(G))determined by the morphism RΓ∗.

Proof. Let (X, F) be a (C_{, C}∗_{(G)) Kasparov module with}C_{acting unitally}

(as in the proof of Lemma4.1). Then by definition

[(X, F)]⊗C∗(G)[RΓ] =RΓ∗([(X, F)])∈Z,

is given by the index of the Fredholm operator

F⊗1 :X⊗RΓL2(Γ\G)→X⊗_RΓL2(Γ\G).

On the other hand,

By Proposition4.8we haveC∗_{(G)→K(Res) and the trivial representation}

1_{is finite dimensional. Thus a successive application of [16, Proposition 4.7]}

shows that

(F⊗1⊗1)2−1 = (F2−1)⊗1⊗1∈K(X)⊗1⊂K(X⊗C∗_(G)Res⊗_C∗_(Γ)C),

proving that F⊗1⊗1 is Fredholm. Therefore res([(X, F)])⊗C∗_(Γ)[1]∈ Z equals the index ofF⊗1⊗1 and to show the diagram commutes, it is sufficient to show that there is a unitary isomorphism

Φ :Res_⊗C∗_(Γ)C−∼→L2(Γ\G), (12)

of Hilbert spaces intertwining theC∗_{(G)-representations. Define a map}

Φ :Cc(G)⊗Cc(Γ)C→C(Γ\G), Φ(f)(Γg) =

X

γ∈Γ

f(g−1γ).

Note that the above map is well defined, i.e., the right hand sides coincide for Γg= Γg′_{.Then on one hand}

hΦ(f),Φ(h)i= Z

G/Γ

Φ(f)(Γg)Φ(h)(Γg)d(Γg)

= Z

G/Γ X

γ∈Γ

f(g−1_γ)X

δ∈Γ

h(g−1δ)d(Γg)

=X

δ∈Γ Z

G

f(g−1_)h(g−1_δ)dg,

whereas on the other hand

hf ⊗1, g⊗1i=X

δ∈Γ

hf, hiC∗(Γ)(δ) = X

δ∈Γ Z

G

f(g−1_)h(g−1_{δ)dg=hΦ(f),Φ(h)i.}

Hence the map Φ is an isometry and extends to an injection. Since the action of Γ onGis properly discontinuous, functions of small support on Γ\Gare in the image of Φ, and a partition of unity argument shows that Φ is surjective, so it extends to a unitary isomorphism. Forξ∈Gwe have

ξ(Φ(f))(Γg) = Φ(f)(Γgξ) =X

γ∈Γ

f(ξ−1g−1γ) =X

γ∈Γ

(ξf)(g−1γ) = Φ(ξf)(Γg),

hence Φ intertwines the G-representations and therefore the C∗_(G)- repre-sentations. Thus we proved the isomorphism (12).

4.4. Selberg trace formula in K-theory

We are ready for the K-theoretic Selberg trace formula. Let ι : L1_{(Γ) →} C∗_{(Γ) be the natural inclusion and ι}

4.11. Theorem. Letx∈K0(C∗_{(G)). Assume that there existsy∈K0(L}1_(Γ))

such thatι∗(y) = res(x)∈K0(C∗(Γ)). Then

X

π∈Gb

mΓ(π)π∗(x) = X

(γ)∈hΓi τγ,∗(y).

4.12. Remark. We can drop the hypothesis on xby putting conditions on

Γ. For example, if Γ admits the polynomial growth condition of each of its conjugacy classes (see [26]), or if Γ is a word hyperbolic group (see [23]), then the trace τγ : L1(Γ) → C can be extended to a subalgebra of C∗(Γ)

that is stable under holomorphic functional calculus and thus gives rise to a well-defined morphism

τγ,∗:K0(C∗(Γ))→C.

It then follows that X

π∈Gb

mΓ(π)π∗(x) = X

(γ)∈hΓi

τγ,∗(res)

forallx∈K0(C∗_(Γ)).

With a little more work, we can derive a general index theoretic trace formula from the above. For this, we need to bring in the equivariant K-homology groups into the picture.

4.13. Lemma. The following diagram commutes:

KG

0(G/K)

IndG

/

/

r

K∗(C∗(G))

res

KΓ 0(G/K)

IndΓ

/

/K∗(C∗(Γ))

where the left vertical arrow r is the restriction map in equivariant

KK-theory.

Proof. Every cycle representing an element ofKG

0(G/K) is given by ([L2(G)⊗ V]K_{, F) whereV is aZ2-graded finite dimensional representation ofK, and}

Fis a bounded properly supportedG-invariant operator on the Hilbert space H = [L2_(G)⊗V]K_{.Its images under Ind}

Gand under IndΓ◦rare represented

by cycles of the form (EG, F), (EΓ, F). Here the Hilbert C∗(G)-module EG

(resp.C∗_{(Γ)-moduleEΓ) is given by completion of [C}

c(G)⊗V]K with respect

to theCc(G)-valued (Cc(Γ)-valued) inner product determined by

hf1, f2i(t) = Z

G

hf1(s), tf2(t−1s)iHds f1, f2∈H

fort∈G(t∈Γ). Recall thatRes_{is the closure ofCc(G).To show that}

we only need to observe thatEΓ =EG⊗C∗_(G)Res. This follows directly from the isomorphism ofCc(Γ)-modules

[Cc(G)⊗V]K⊗Cc(G)Cc(G)Cc(Γ)≃[Cc(G)Cc(Γ)⊗V]

K_,

which is compatible with the inner products.

By Theorem4.11and Lemmas4.3,4.6and4.13we obtain the following theorem, giving the Selberg trace formula in its index theory form.

4.14. Theorem. The following diagram commutes

KG 0(G/K) IndG / / r

K∗(C∗(G))

RΓ

∗= P

π∈Gb

mΓ(π)π∗

" " ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ res KΓ

0(G/K) _Ind

Γ

/

/

IndΓ,L1

% % ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑

❑ K∗(C

∗_(Γ))

1_∗ //Z

K0(L1_(Γ))

i∗

O

O

P

(γ)∈hΓi

τγ,∗

< < ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① .

In particular, if D is a G-invariant Dirac type operator on G/K, the above

commutative diagram gives rise to the equality X

π∈Gb

mΓ(π)π∗(IndGD) =

X

(γ)∈hΓi

τγ,∗(IndΓ,L1D),

which reduces to the index theoretic Selberg trace formula: X

π∈Gb

mΓ(π)indDπ=

X

(γ)∈hΓi

indγD. (13)

4.15. Remark. Denote byDΓ the Dirac type operator on Γ\G/K whose lift

toG/K isD. Then

indDΓ= X (γ)∈hΓi

indγD.

Also [IndGD]⊗C∗_(G)[RΓ] =RΓ_∗(Ind_GD) =P

π∈GbmΓ(π)indDπ.Then (13) is reduced to

[IndGD]⊗C∗(G)[RΓ] = [DΓ]∈KK(C,C) =Z.

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Mehmet Haluk S¸eng¨un

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Hang Wang

Research Center for Operator Algebras, School of Mathematical Sciences

East China Normal University, Science building A505, 3663 North Zhongshan Rd, Shanghai 200062, P.R.China