An equivariant orbifold index for proper actions
aa r X i v : . [ m a t h . K T ] F e b AN EQUIVARIANT ORBIFOLD INDEX FOR PROPERACTIONS
PETER HOCHS AND HANG WANG
Abstract.
For a proper, cocompact action by a locally compact groupof the form H × G , with H compact, we define an H × G -equivariant indexof H -transversally elliptic operators, which takes values in KK ∗ ( C ∗ H, C ∗ G ).This simultaneously generalises the Baum–Connes analytic assemblymap, Atiyah’s index of transversally elliptic operators, and Kawasaki’sorbifold index. This index also generalises the assembly map to ellipticoperators on orbifolds. In the special case where the manifold in ques-tion is a real semisimple Lie group, G is a cocompact lattice and H isa maximal compact subgroup, we realise the Dirac induction map fromthe Connes–Kasparov conjecture as a Kasparov product and obtain anindex theorem for Spin-Dirac operators on compact locally symmetricspaces. Contents
1. Introduction 1Acknowledgements 32. Preliminaries and results 32.1. Preliminaries 32.2. The index 42.3. Free actions by H
63. Proofs of the results 83.1. The analytic assembly map 83.2. Proofs of Theorem 2.8 and Corollary 2.9 104. Symmetric spaces and locally symmetric spaces 124.1. Dirac induction 124.2. An index theorem for Spin-Dirac operators on compact locallysymmetric spaces 14References 161.
Introduction
Two natural ways in which orbifolds occur are as quotients of locally freeactions by compact groups, and of proper actions by discrete groups. In
Date : February 18, 2020. fact, every orbifold can be realised as the quotient of a locally free action bycompact group H on a manifold ˜ M ; see e.g. [9, 26]. A well-known approachto index theory on a compact orbifold M := ˜ M /H is to consider an ellipticoperator D on M as a transversally elliptic operator ˜ D on ˜ M , and definethe index of D as the H -invariant part of the H -equivariant index of ˜ D inthe sense of Atiyah [1].Intuitively, if an orbifold M is realised as the quotient of a proper actionby a discrete group Γ on a manifold X , then the orbifold index of an operatoron M is the Γ-invariant part of its lift to an operator on X . This can bemade precise in terms of the Baum–Connes assembly map, with values in the K -theory of the maximal group C ∗ -algebra of Γ, from which the invariantpart can be obtained by an application of the map given by summing overΓ. Our purpose in this paper is to unify and extend these two approaches toorbifold index theory. Along the way, we construct a generalisation of theBaum–Connes assembly map from manifolds to orbifolds. We also realise theDirac induction map from the Connes–Kasparov conjecture as a Kasparovproduct.For a compact group H and a locally compact group G , and a proper,isometric, cocompact action by H × G on a manifold ˜ M , we define an indexof H × G -equivariant, H -transversally elliptic operators on ˜ M , with values in KK ∗ ( C ∗ H, C ∗ G ). This builds on parts of Kasparov KK -theoretic treatmentof transversally elliptic operators [15]. If H is trivial, then this index isthe Baum–Connes assembly map. If G is trivial, it is Atiyah’s index oftransversally elliptic operators, whose H -invariant part is the realisationof the orbifold index on M = ˜ M /H mentioned above, if H acts locallyfreely. In general, the pairing of this index with the class of the trivialrepresentation of H in K ∗ ( C ∗ H ) generalises the Baum–Connes assemblymap to G -equivariant elliptic operators on the possibly singular space M .Another K -theoretic approach to index theory on orbifolds was developedby Farsi [9]. An index of families of transversally elliptic operators wasconstructed and applied in a KK -theoretic setting by Baldar´e [4, 5].The index in KK ∗ ( C ∗ H, C ∗ G ) is furthest removed form existing indextheory if the action by H on ˜ M is not free, and M is not a smooth manifold.However, even if H acts freely, this index refines existing (orbifold) indices,as the double quotient ˜ M / ( H × G ) may be singular. We investigate the indexin this special case, and find relations with the Baum–Connes assembly map.A natural setting in which this applies is the case of compact locallysymmetric spaces Γ \ G/K , where G is a real semisimple group, Γ < G is acocompact lattice, and K < G is maximal compact. For a class of examplesincluding these spaces, we show that the index of an elliptic operator onΓ \ G/K can be obtained both as the K -invariant part of a transversally el-liptic operator on Γ \ G and as the Γ-invariant part (in a K -theoretic sense) ofthe index of an elliptic operator on G/K . These approaches are unified in the
N EQUIVARIANT ORBIFOLD INDEX FOR PROPER ACTIONS 3 sense that both indices are obtained from the index of a K × Γ-equivariant,transversally elliptic operator on G , which generalises and refines the twoindices on Γ \ G and G/K . In this sense, the index we consider here encodesthe most refined index-theoretic information on Γ \ G/K , and simultaneouslyincorporates the Γ- and K -symmetries. We obtain explicit expressions forthe values of natural traces on the Γ × K -equivariant index of the lift of theSpin-Dirac operator from G/K to G in this context. Acknowledgements.
H. Wang acknowledges support from Thousand YouthTalents Plan and from grants NSFC-11801178 and Shanghai Rising-StarProgram 19QA1403200.2.
Preliminaries and results
Preliminaries.
Let ˜ M be a Riemannian manifold. Let G and H beLie groups, acting isometrically on ˜ M . Suppose that the actions by the twogroups commute. Then G has the induced action on the quotient M :=˜ M /H . Every orbifold occurs in this way, for a compact Lie group H actinglocally freely on ˜ M (see e.g. the introductions to [9, 26], and Remark 2.10).But we will consider more general spaces of the form M = ˜ M /H .Let E → M be a Hermitian, Z -graded, G -equivariant, continuous vectorbundle. Let q : ˜ M → M be the quotient map. Suppose that ˜ E := q ∗ E → ˜ M has the structure of a smooth vector bundle. Definition 2.1. A G -equivariant differential operator on E is an operator D on Γ ∞ ( ˜ E ) H that is the restriction of a G × H -equivariant differential operator˜ D on Γ ∞ ( ˜ E ). Such a differential operator D is elliptic if the operator ˜ D is(or can be chosen to be) transversally elliptic with respect to the action by H .If H acts properly and freely on ˜ M , then this definition reduces to theusual definition of elliptic differential operators. Remark 2.2. If D is a first order differential operator, and H acts properlyand freely, then Definition 2.1 becomes very explicit. In this case D hasa unique pullback along any smooth, G -equivariant map f : N → M to alinear operator f ∗ D on Γ ∞ ( f ∗ E ) satisfying • for all s ∈ Γ ∞ ( E ), ( f ∗ D )( f ∗ s ) = f ∗ ( Ds ); • for all σ ∈ Γ ∞ ( f ∗ E ) and ϕ ∈ C ∞ ( N ),( f ∗ D )( ϕσ ) = σ D ( T f ◦ grad( ϕ )) σ + ϕ ( f ∗ D ) σ. Here σ D is the principal symbol of D . Existence and uniqueness of f ∗ D canbe proved in the same way as one proves that pullbacks of connections arewell-defined. In this case, we may take ˜ D = q ∗ D in Definition 2.1. PETER HOCHS AND HANG WANG
We fix a first order, G -equivariant, self-adjoint, elliptic differential opera-tor D on E , that is odd with respect to the grading, and a lift ˜ D of D to ˜ E as in Definition 2.1. Let ˜ D + and ˜ D − be the restrictions of ˜ D to sections ofthe even and odd parts of ˜ E , respectively.If M , G and H are compact, then a natural definition of the G -equivariantindex of D is(2.1) index G ( D ) = [ker( ˜ D + ) H ] − [ker( ˜ D − ) H ] . This is an element of the representation ring of G . An index formula for suchan index was given by Vergne [26]. It is our goal in this note to generalisethis index to noncompact M and G , assuming the action to be proper, and M/G to be compact.2.2.
The index.
From now on, suppose that H is compact, that G actsproperly on ˜ M (and hence on M ), and that M/G is compact. Then thereis a cutoff function c ∈ C c ( M ), such that for all m ∈ M , Z G c ( gm ) dg = 1for a left Haar measure dg on G . The pullback ˜ c := q ∗ c ∈ C c ( ˜ M ) H then hasthe analogous property.Consider the idempotents p ∈ C ( M ) ⋊ G and ˜ p ∈ C ( ˜ M ) ⋊ G defined by p ( m, g ) = c ( m ) c ( g − m );˜ p ( ˜ m, g ) = ˜ c ( ˜ m )˜ c ( g − ˜ m ) , (2.2)for m ∈ M , ˜ m ∈ ˜ M , and g ∈ G . They define classes[ p ] ∈ KK ( C , C ( M ) ⋊ G );[˜ p ] ∈ KK H ( C , C ( ˜ M ) ⋊ G ) . Let π ˜ M,H : C ( ˜ M ) ⋊ H → B ( L ( ˜ E ))be the ∗ -representation defined by pointwise multiplication by functions in C ( M ), and the unitary representation of H in L ( ˜ E ). Kasparov showed inProposition 6.4 in [15] that the transversally elliptic operator ˜ D defines aclass(2.3) [ ˜ D ] := (cid:2) L ( ˜ E ) , ˜ D p ˜ D + 1 , π ˜ M,H (cid:3) ∈ KK G ( C ( ˜ M ) ⋊ H, C ) . Let j G : KK G ∗ ( C ( ˜ M ) ⋊ H, C ) → KK ∗ ( C ( ˜ M ) ⋊ ( G × H ) , C ∗ G ); j H : KK H ∗ ( C , C ( ˜ M ) ⋊ G ) → KK ∗ ( C ∗ H, C ( ˜ M ) ⋊ ( G × H )) . be descent maps, see 3.11 in [14]. Here we used the fact that the actions by G and H commute. (We use either maximal or reduced crossed-productsand group C ∗ -algebras.) N EQUIVARIANT ORBIFOLD INDEX FOR PROPER ACTIONS 5
Consider the classes j H [˜ p ] ∈ KK ∗ ( C ∗ H, C ( ˜ M ) ⋊ ( G × H )); j G [ ˜ D ] ∈ KK ∗ ( C ( ˜ M ) ⋊ ( G × H ) , C ∗ G ) . Let [1 H ] ∈ KK ( C , C ∗ H ) = R ( H ) be the class of the trivial representationof H . Definition 2.3.
The (
H, G ) -equivariant index of ˜ D isindex H,G ( ˜ D ) = j H [˜ p ] ⊗ C ( ˜ M ) ⋊ ( G × H ) j G [ ˜ D ] ∈ KK ∗ ( C ∗ H, C ∗ G ) . The G -equivariant index of D isindex G ( D ) = [1 H ] ⊗ C ∗ H index H,G ( ˜ D ) ∈ KK ∗ ( C , C ∗ G ) . More generally, for any locally compact group G , compact group H , andlocally compact, Hausdorff, proper, cocompact G × H space X , the G -equivariant index map index G : KK G ∗ ( C ( X ) ⋊ H, C ) → K ∗ ( C ∗ G )is defined byindex G ( a ) = [1 H ] ⊗ C ∗ H j H [˜ p ] ⊗ C ( ˜ M ) ⋊ ( G × H ) j G ( a ) , for a ∈ KK G ∗ ( C ( X ) ⋊ H, C ).In the definition of index G ( D ), the Kasparov product with [1 H ] plays therole of taking the H -invariant part of the G × H -equivariant index of ˜ D ,analogously to (2.1).In the rest of this note, we investigate some properties and applications ofthese indices. First of all, the ( H, G )-equivariant index generalises Atiyah’s[1] and Kawasaki’s [17] classical indices.
Lemma 2.4. If M is compact and G = { e } is the trivial group, then index H, { e } ˜ D ∈ KK ( C ∗ H, C ) is the index of ˜ D in the sense of Atiyah. Inparticular, if H acts locally freely on M , then index { e } ( D ) is Kawasaki’sorbifold index of D .Proof. If M is compact, and G = { e } , then we can take c to be the constantfunction 1 on M . Then [˜ p ] is the class of the only ∗ -homomorphism C → C ( ˜ M ). So index H ×{ e } ( ˜ D ) = ψ ∗ [ ˜ D ] ∈ KK ( C ∗ H, C ) , where ψ is the map from ˜ M to a point. The right hand side of the aboveequality is the index of ˜ D in the sense of Atiyah, see Remark 6.7 in [15].Pairing this index with [1 H ] means taking its H -invariant part, which yieldsKawasaki’s index of D if H acts locally freely. (cid:3) Remark 2.5.
Theorem 8.18 in [15] is a topological expression for the class(2.3) in terms of the principal symbol of ˜ D . This directly implies analogous KK -theoretic index theorems for index H,G ( ˜ D ) and index G ( D ). PETER HOCHS AND HANG WANG
The case where H acts trivially on ˜ M is not the most interesting, butwe include it as a consistency check. For any locally compact, Hausdorff,proper, cocompact G -space X , let µ GX : KK G ( C ( X ) , C ) → KK ( C , C ∗ G )be the analytic assembly map [6]. Lemma 2.6. If H acts trivially on a locally compact, Hausdorff, proper,cocompact G -space X , then the map index G : KK G ∗ ( C ( X ) ⋊ H, C ) = KK G ∗ ( C ( X ) , C ) ⊗ ˆ R ( H ) → K ∗ ( C ∗ G ) is given by index G ( a ⊗ [ V ]) = (cid:26) µ GX ( a ) if V = 1 H ;0 otherwise , for a ∈ KK G ∗ ( C ( X ) , C ) and V ∈ ˆ H . Here µ GX is the analytic assemblymap, and ˆ R ( H ) is the completed representation ring of H .Proof. It follows from the definition of the descent map and the Peter–Weyltheorem that the descent map j H : R ( H ) = KK H ( C , C ) → KK ( C ∗ H, C ∗ H ) = End Z ( R ( H ))maps [ V ] ∈ ˆ H to the projection map proj V onto Z [ V ]. Now X/H = X ,˜ c = c and ˜ p = p , so the class[˜ p ] ∈ KK H ( C , C ( X ) ⋊ G ) = KK ( C , C ( X ) ⋊ G ) ⊗ R ( H )equals [ p ] ⊠ [1 H ], where ⊠ is the external Kasparov product. We find that j H [˜ p ] = [ p ] ⊠ proj H ∈ KK ( C , C ( X ) ⋊ G ) ⊗ End Z ( R ( H )) . Now for a ∈ KK G ∗ ( C ( X ) , C ) and V ∈ ˆ H , j G ( a ⊗ [ V ]) = j G ( a ) ⊠ [ V ] ∈ KK G ∗ ( C ( X ) , C ) ⊗ ˆ R ( H ) . So index G ( a ⊗ V ) = [ p ] ⊗ C ( X ) ⋊ G j G ( a ) ⊠ h [ V ] , proj H ([1 H ]) i , which implies the claim. The angular brackets denote the pairing betweenˆ R ( H ) = Hom Z ( R ( H ) , Z ) and R ( H ). (cid:3) Free actions by H . If H acts freely on ˜ M , then M is a smoothmanifold. In that case, the index of Definition 2.3 reduces to the analyticassembly map, see Proposition 2.7. In this sense, the G -equivariant indexgeneralises the analytic assembly map to orbifolds. If, furthermore, G = Γis discrete, then M/ Γ is an orbifold. Even if Γ acts freely on ˜ M , its actionon M is not necessarily free. (Similarly, the action by H on ˜ M /
Γ is not freein general.) This leads to two different ways to realise orbifold indices onspaces of the type that includes compact locally symmetric spaces, Corollary2.9, which is based on Theorem 2.8. We work out the example of locallysymmetric spaces in Subection 4.2.
N EQUIVARIANT ORBIFOLD INDEX FOR PROPER ACTIONS 7 If H acts freely on ˜ M , then D is an elliptic differential operator on thesmooth vector bundle E → M in the usual sense. Then it defines a K -homology class[ D ] := (cid:2) L ( E ) , D √ D + 1 , π M (cid:3) ∈ KK G ( C ( M ) , C ) , where π M is defined by pointwise multiplication. Proposition 2.7. If H acts freely on ˜ M , then index G ( D ) = µ GM [ D ] . This proposition will be proved in Section 3.1.Consider, for the moment, the case where G = Γ is discrete, and H = { e } is trivial. Then M = ˜ M , ˜ D = D is elliptic, and ˜ M /
Γ = M/ Γ is an orbifold.Consider the ∗ -homomorphism P Γ : C ∗ max Γ → C given on l (Γ) ⊂ C ∗ max Γ by summing over Γ. Let [ P Γ ] ∈ KK ( C ∗ max Γ , C )be the corresponding class. If we use maximal crossed products and group C ∗ -algebras, then Proposition 2.7, and Theorem 2.7 and Proposition D.3 in[21], imply that(2.4)index Γ ( D ) ⊗ C ∗ max Γ [ P Γ ] = (cid:0)P Γ (cid:1) ∗ index Γ ( D ) = dim(ker( D + ) Γ ) − dim(ker( D − ) Γ ) . The right hand side is the index of the operator D Γ on the compact orbifold M/ Γ induced by D . This gives another realisation of the orbifold index interms of the index of Definition 2.3.This construction applies more generally. Let I G : C ∗ max ( G ) → C bethe continuous extension of the integration map on L ( G ), and [ I G ] ∈ KK ( C ∗ max G, C ) the corresponding KK -class. Theorem 2.8.
Suppose that G is unimodular and that H acts freely on ˜ M . The multiplicity of every irreducible representation of H in ker( ˜ D ) G isfinite, and (2.5)index H,G ( ˜ D ) ⊗ C ∗ max G [ I G ] = [ker( ˜ D + ) G ] − [ker( ˜ D − ) G ] ∈ KK ( C ∗ H, C ) = ˆ R ( H ) . In the setting of Theorem 2.8, we denote the right hand side of (2.5) byindex H ( ˜ D G ). If the action by G on ˜ M is free, then this is the index of thetransversally elliptic operator ˜ D G on Γ ∞ ( ˜ E/G ) = Γ ∞ ( ˜ E ) G induced by ˜ D . Corollary 2.9. If G is unimodular and H acts freely on ˜ M , then ker( ˜ D ) H × G is finite-dimensional, and dim(ker( ˜ D + ) H × G ) − dim(ker( ˜ D − ) H × G ) = [1 H ] ⊗ C ∗ H index H,G ( ˜ D ) ⊗ C ∗ max G [ I G ]= ( I G ) ∗ (index G ( D ))= [1 H ] ⊗ index H ( ˜ D G ) . (2.6) PETER HOCHS AND HANG WANG If G = Γ is discrete, and H acts locally freely, then ˜ M / ( H × Γ) is anorbifold, and the left hand side of (2.6) is the orbifold index of the operatoron ˜
M / ( H × Γ) induced by ˜ D . Then Corollary 2.9 shows that index H,G ( ˜ D ) isa common refinement of the two indices index G ( D ) and index H ( ˜ D G ), whichrefine the orbifold index of the operator ˜ D H × Γ on ˜ M / ( H × Γ) induced by ˜ D in two different ways. In this sense, index H,G ( ˜ D ) contains the most refinedindex-theoretic information about ˜ D H × Γ . This applies for example in thecase of compact locally symmetric spaces, see Section 4.2.Theorem 2.8 and Corollary 2.9 are deduced from Proposition 2.7 in Sec-tion 3.2. Remark 2.10.
Let M be a complete Riemannian manifold of dimension n and let Γ be a discrete group acting properly, cocompactly and isometricallyon M . Then X := M/ Γ is a compact orbifold. Let P be the O( n )-framebundle of X . Denote H = O( n ). Then P is a compact manifold acted onfreely by H . One can lift the H -frame bundle P from X to M to obtain aprincipal H -bundle ˜ M over M . Then ˜ M has free, commuting actions by H and Γ, and X = P/H = ˜
M / (Γ × H ) . Let D X be an elliptic differential operator on X . It can be realised aseither a Γ-equivariant elliptic differential operator D Γ M on M , or an H -transversally elliptic operator D HP on P . These two operators have a com-mon lift to an H × Γ-equivariant, H -transversally elliptic operator ˜ D on ˜ M .Corollary 2.9 implies that the orbifold index of D X can be obtained fromthe ( H, Γ)-index of ˜ D asindex( D X ) = [1 H ] ⊗ C ∗ H index H, Γ ( ˜ D ) ⊗ C ∗ max Γ [ P Γ ]= ( P Γ ) ∗ (index Γ ( D Γ M )) = [1 H ] ⊗ C ∗ H index H ( D HP ) . It is an interesting question in what way the contributions to index( D X )from singularities in the quotient of M by Γ or the quotient of P by H arerelated. 3. Proofs of the results
The analytic assembly map.
Let us prove Proposition 2.7. Supposethat H acts freely on ˜ M , so that M is smooth. Then we have a Moritaequivalence bimodule M between C ( ˜ M ) ⋊ H and C ( M ), see Situation 2in [24]. This is a left C ( ˜ M ) ⋊ H and right Hilbert C ( M )-bimodule, definedas the completion of C c ( ˜ M ) in the inner product( ϕ , ϕ ) C ( M ) ( m ) = Z H ¯ ϕ ( hm ) ϕ ( hm ) dh, for ϕ , ϕ ∈ C c ( ˜ M ) and m ∈ M . The right action by C ( M ) on M isdefined by pointwise multiplication after pullback along q . The left action N EQUIVARIANT ORBIFOLD INDEX FOR PROPER ACTIONS 9 by C ( ˜ M ) ⋊ H , denoted by π M , is defined by the standard actions by C ( ˜ M )and H on C c ( ˜ M ). This yields an invertible class[ M ] := [ M , , π M ] ∈ KK G ( C ( ˜ M ) ⋊ H, C ( M )) . Lemma 3.1.
We have [ ˜ D ] = [ M ] ⊗ C ( M ) [ D ] ∈ KK G ( C ( ˜ M ) ⋊ H, C ) . Proof.
We use the unbounded picture of KK -theory [3, 18, 22]. Denotingsets of unbounded KK -cycles by the letter Ψ, we have( ˜ D ) = ( L ( ˜ E ) , ˜ D, π M ) ∈ Ψ G ( C ( ˜ M ) ⋊ H, C );( D ) = ( L ( E ) , D, π M ) ∈ Ψ G ( C ( M ) , C );( M ) = ( M , , π M,H ) ∈ Ψ G ( C ( ˜ M ) ⋊ H, C ( M )) . We will show that(3.1) ( ˜ D ) = ( M ) ⊗ C ( M ) ( D ) . First note that we have an isomorphism of C ( ˜ M ) ⋊ H -modules M ⊗ C ( M ) L ( E ) ∼ = −→ L ( ˜ E ) , mapping ϕ ⊗ s to ϕq ∗ s , for ϕ ∈ C c ( ˜ M ) and s ∈ L ( E ). Now Theorem 13 in[18] states that the equality (3.1) holds if(1) for all ϕ ∈ C ∞ c ( ˜ M ), the operators˜ D ◦ T ϕ − T ϕ ◦ D : Γ ∞ c ( E ) → L ( ˜ E ) and D ◦ T ∗ ϕ − T ∗ ϕ ◦ ˜ D : Γ ∞ c ( ˜ E ) → L ( E )are bounded, where T ϕ denotes tensoring with ϕ ;(2) the resolvent of ˜ D is compatible with the zero operator in the senseof Lemma 10 in [18], which is a vacuous condition; and(3) a positivity condition that trivially holds because the operator in thecycle ( M ) is zero.To verify the first condition, we note that, since ˜ D is a first order operator,˜ D ◦ T ϕ − T ϕ ◦ D = σ ˜ D ( dϕ ) ⊗ , which is a bounded operator. And D ◦ T ∗ ϕ − T ∗ ϕ ◦ ˜ D is minus the adjoint ofthe above operator, hence also bounded. (cid:3) Next, consider the maps(3.2) [˜ p ] ∈ KK H ( C , C ( ˜ M ) ⋊ G ) j H −−→ KK ( C ∗ H, C ( ˜ M ) ⋊ ( G × H )) −⊗ C
0( ˜ M ) ⋊ ( G × H ) j G [ M ] −−−−−−−−−−−−−−−→ KK ( C ∗ H, C ( M ) ⋊ G ) [1 H ] ⊗ C ∗ H − −−−−−−−→ KK ( C , C ( M ) ⋊ G ) ∋ [ p ] . Lemma 3.2.
The composition of the maps (3.2) maps the class [˜ p ] to [ p ] . Proof.
We have [˜ p ] = [˜ p ( C ( ˜ M ) ⋊ G ) , , π C ] , where π C is the representation of C by scalar multiplication. Hence j H [˜ p ] = [˜ p H ( C ( ˜ M ) ⋊ ( G × H )) , , π H ] , where π H is the representation of C ∗ H defined by convolution on H , and˜ p H is the idempotent in C ( ˜ M ) ⋊ ( G × H ) defined by extending ˜ p constantin the H -direction. (Recall that H is compact.)Now let M G be the ( C ( ˜ M ) ⋊ ( G × H ) , C ( M ) ⋊ G )-bimodule constructedfrom M as in the definition of j G . Then M G implements the Morita equiv-alence between C ( ˜ M ) ⋊ ( G × H ) and C ( M ) ⋊ G , and˜ p H ( C ( ˜ M ) ⋊ ( G × H )) ⊗ C ( ˜ M ) ⋊ ( G × H ) M G = p ( C ( M ) ⋊ G ) . So j H [˜ p ] ⊗ C ( ˜ M ) ⋊ ( G × H ) j G [ M ] = [ p ( C ( M ) ⋊ G ) , , π H ] . Finally, pairing with [1 H ] means replacing π H by π C , so the claim follows. (cid:3) Proof of Proposition 2.7.
By one of the equivalent definitions of the analyticassembly map, and by Lemma 3.2, we have µ GM [ D ] = [ p ] ⊗ C ( M ) ⋊ G j G [ D ]= [1 H ] ⊗ C ∗ H j H [˜ p ] ⊗ C ( ˜ M ) ⋊ ( G × H ) j G [ M ] ⊗ C ( M ) ⋊ G j G [ D ]= [1 H ] ⊗ C ∗ H j H [˜ p ] ⊗ C ( ˜ M ) ⋊ ( G × H ) j G ([ M ] ⊗ C ( M ) [ D ]) . So the claim follows from Lemma 3.1. (cid:3)
Proofs of Theorem 2.8 and Corollary 2.9.
We now deduce The-orem 2.8 from Proposition 2.7 and the results in the appendix to [21], andthen deduce Corollary 2.9 from Theorem 2.8. We still assume that H actsfreely on ˜ M .Let V ∈ ˆ H . We will write[ V ] ∈ KK ( C ∗ H, C );[ V ] ∈ KK ( C , C ∗ H )(3.3)for the classes defined by V . Consider the elliptic operator ˜ D ⊗ V onΓ ∞ ( ˜ E ) ⊗ V . Let E V → M be the quotient of ˜ E ⊗ V by H . (Note that M is smooth and E V is a well-defined vector bundle because H acts freely on˜ M .) Let D V be the restriction of ˜ D ⊗ V toΓ ∞ ( M, E V ) = (Γ ∞ ( ˜ E ) ⊗ V ) H . This is a G -equivariant, elliptic operator. The following result generalisesProposition 2.7, which we use in its proof. Proposition 3.3.
For all V ∈ ˆ H , µ GM [ D V ] = [ V ] ⊗ C ∗ H (index H,G ( ˜ D )) . N EQUIVARIANT ORBIFOLD INDEX FOR PROPER ACTIONS 11
Proof.
For any G - C ∗ -algebra B and ( G × H × H )- C ∗ -algebra A , letRes H × H ∆( H ) : KK G ( A ⋊ ( H × H ) , B ) → K G ( A ⋊ H, B )be defined by restriction to the diagonal in H × H (and similarly for thecase where G = { e } ). Consider the action by G × H × H on ˜ M , where thesecond factor H acts trivially. Consider the diagram(3.4) KK G ( C ( ˜ M ) ⋊ ( H × H ) , C ) index H × H,G / / Res H × H ∆( H ) (cid:15) (cid:15) KK ( C ∗ ( H × H ) , C ∗ G ) Res H × H ∆( H ) (cid:15) (cid:15) KK G ( C ( ˜ M ) ⋊ H, C ) index H,G / / KK ( C ∗ H, C ∗ G ) [1 H ] ⊗ C ∗ H − (cid:15) (cid:15) KK G ( C ( M ) , C ) µ GM / / [ M ] ⊗ C M ) − ∼ = O O KK ( C , C ∗ G )The bottom part of this diagram commutes by Proposition 2.7. Because thesecond factor H acts trivially on ˜ M , the projection ˜ p also defines a class in KK H × H ( C , C ( ˜ M ) ⋊ G ), which we denote by [˜ p ] H × H . Let1 C ∗ H ∈ KK ( C ∗ H, C ∗ H )be the identity element. Again, because the second factor H acts triviallyon ˜ M , we have(3.5) j H × H [˜ p ] H × H = j H [˜ p ] ⊠ C ∗ H ∈ KK ( C ∗ ( H × H ) , C ( ˜ M ) ⋊ ( G × H × H )) = KK ( C ∗ H ⊗ C ∗ H, C ( ˜ M ) ⋊ ( G × H ) ⊗ C ∗ H ) , where ⊠ denotes the exterior Kasparov product. This equality implies thatthe top part of (3.4) commutes. By Lemma 3.1, we have[ M ] ⊗ C ( M ) [ D V ] = Res H × H ∆( H ) [ ˜ D ⊗ V ] . Here we view ˜ D ⊗ V as a G × H × H -equivariant operator, where thesecond factor H acts trivially on ˜ M , and on E ⊗ V via its action on V . Bycommutativity of (3.4), we therefore have(3.6) µ GM [ D V ] = [1 H ] ⊗ C ∗ H (cid:0) Res H × H ∆( H ) (index H × H,G ( ˜ D ⊗ V )) (cid:1) . Next, again using the fact that the second factor H acts trivially on ˜ M ,we have[ ˜ D ⊗ V ] = [ ˜ D ] ⊠ [ V ] ∈ KK G ( C ( ˜ M ) ⋊ ( H × H ) , C ) = KK G (( C ( ˜ M ) ⋊ H ) ⊗ C ∗ H, C ) . Here ⊠ again denotes the exterior Kasparov product. By this equality and(3.5),index H × H,G ( ˜ D ⊗ V ) = ( j H [˜ p ] ⊠ C ∗ H ) ⊗ C ( ˜ M ) ⋊ ( G × H ) ⊗ C ∗ H ( j G [ ˜ D ] ⊠ [ V ] )= index H,G ( ˜ D ) ⊠ [ V ] . (3.7) Finally, we have for all C ∗ -algebras A and all x ∈ KK ( C ∗ H, A ),[1 H ] ⊗ C ∗ H Res H × H ∆( H ) ( x ⊠ [ V ] ) = [ V ] ⊗ C ∗ H x. Combining this equality with (3.6) and (3.7), we conclude that the desiredequality holds. (cid:3)
Proof of Theorem 2.8.
Proposition 3.3 implies that( I G ) ∗ index H,G ( ˜ D ) = M V ∈ ˆ H ([ V ] ⊗ C ∗ H ( I G ) ∗ index H,G ( ˜ D )) ⊗ [ V ] = M V ∈ ˆ H (( I G ) ∗ µ GM ( D V )) ⊗ [ V ] By unimodularity of G , Theorem 2.7 and Proposition D.3 in [21] imply thatthe latter expression equals M V ∈ ˆ H (cid:0) dim(ker( D + V ) G ) − dim(ker( D − V ) G ) (cid:1) ⊗ [ V ] = [ker( ˜ D + ) G ] − [ker( ˜ D − ) G ] . (cid:3) Proof of Corollary 2.9.
Associativity of the Kasparov product and Theorem2.8 imply that [1 H ] ⊗ C ∗ H index H ( D G ) = ( I G ) ∗ index G ( D ) . Because G is unimodular, Theorem 2.7 and Proposition D.3 in [21] implythat the right hand side equalsdim(ker( ˜ D + ) H × G ) − dim(ker( ˜ D − ) H × G ) . (cid:3) Symmetric spaces and locally symmetric spaces
In this section, we consider a Lie group G , a maximal compact subgroup K < G , the manifold ˜ M = G , and the operator ˜ D which is the pullbackof the Spin-Dirac operator on G/K . Then we obtain a realisation of Diracinduction as a Kasparov product, and an index formula for compact locallysymmetric spaces.4.1.
Dirac induction.
Using the (
H, G )-equivariant index from Definition2.3, we can realise the Dirac induction map from the Connes–Kasparovconjecture [6, 7, 20, 29] as a Kasparov product.We will sometimes consider a particular transversally elliptic Dirac-typeoperator. Let G be an almost connected Lie group, and let K < G bemaximal compact. Let p ⊂ g be the orthogonal complement of k withrespect to an Ad( K )-invariant inner product on g . Suppose that the adjointrepresentation K → SO( p ) of K in p lifts to a homomorphism(4.1) K → Spin( p ) . N EQUIVARIANT ORBIFOLD INDEX FOR PROPER ACTIONS 13 (This is true of we replace G by a double cover if necessary.) Let ∆ p be thestandard representation of Spin( p ). We view it as a representation of K viathe map (4.1).Let { X , . . . , X l } be an orthonormal basis of p . We denote the left regularrepresentation of G in C ∞ ( G ) by L and the Clifford action by p on ∆ p by c . For V ∈ ˆ K , consider the operator(4.2) D VG,K := l X j =1 L ( X j ) ⊗ c ( X j ) ⊗ V on the space(4.3) C ∞ ( G ) ⊗ ∆ p ⊗ V of sections of the trivial G × K -equivariant vector bundle G × (∆ p ⊗ V ) → G. The operator D VG,K is G × K -equivariant, and K -transversally elliptic. Sowe in particular have the element(4.4) D-Ind := index K,G ( D C G,K ) ∈ KK ∗ ( C ∗ K, C ∗ G ) . Here C is the trivial representation of K . If G/K is even-dimensional,then ∆ p is Z -graded, and this element lies in even KK -theory. For odd-dimensional G/K , ∆ p is ungraded, and this index lies in odd KK -theory. Proposition 4.1.
For all V ∈ ˆ K , [ V ] ⊗ C ∗ K D-Ind = D-Ind GK [ V ] ∈ K ∗ ( C ∗ G ) , where [ V ] is as in (3.3) , and on the right hand side, D-Ind GK is the Diracinduction map.Proof. Let V ∈ ˆ K . Let D VG/K be the restriction of D VG,K to the space of K -invariant elements of (4.3), which is the space of sections of the vectorbundle G × K (∆ p ⊗ V ) → G/K.
Proposition 3.3 implies that for all V ∈ ˆ K ,[ V ] ⊗ C ∗ K D-Ind = µ GG/K [ D VG/K ] = D-Ind GK [ V ] . (cid:3) Let η ∈ KK ∗ ( C ∗ G, C ∗ K ) be the dual-Dirac element; see for exampleSection 2.2 of [19]. By Proposition 4.1, a sufficient condition for injectivityof Dirac induction isD-Ind ⊗ C ∗ G η = 1 C ∗ K ∈ KK ( C ∗ K, C ∗ K ) , whereas a sufficient condition for surjectivity is η ⊗ C ∗ K D-Ind = 1 C ∗ G ∈ KK ( C ∗ G, C ∗ G ) . Bijectivity of Dirac induction was proved in [7, 20]. See also forthcomingwork by Higson, Song and Tang.4.2.
An index theorem for
Spin -Dirac operators on compact locallysymmetric spaces.
Consider a compact locally symmetric space: an orb-ifold of the form X = Γ \ G/K where G is a connected, semisimple Lie group, K < G is a maximal compact subgroup and Γ is a cocompact, discrete sub-group in G .In several contexts [10, 16, 27], it was shown that traces defined by orbitalintegrals on discrete groups are useful tools to extract information fromclasses in K -theory of group C ∗ -algebras. (This is also true for orbitalintegrals on semisimple Lie groups [13] and higher analogues [25].) For adiscrete group Γ, the orbital integral of a function f ∈ l (Γ) over a conjugacyclass ( γ ) of an element γ ∈ Γ, is the sum of f over ( γ ): τ γ ( f ) = X h ∈ ( γ ) f ( h ) . We assume that this trace τ γ extends to a continuous functional on a densesubalgebra A (Γ) ⊂ C ∗ Γ, closed under holomorphic functional calculus (asmooth subalgebra for short). For example, this is true for every group if γ = e , and for every γ if G has real rank one. Lemma 4.2. If G has real rank one, then τ γ defines a continuous linearfunctional on a smooth subalgebra of C ∗ red Γ .Proof. If G has real rank one, then G/K has negative sectional curvature,and hence it is a hyperbolic space in the sense of Definition 3.1 in [23].Since Γ acts cocompactly on
G/K , the Svarˇc–Milnor lemma implies that itis quasi-isometric to
G/K , with respect to any word-length metric. Hyper-bolicity of metric spaces is preserved by quasi-isometries (see 7.2 in [11]),so that Γ is hyperbolic. See also Section 2.7 in [11]. Proposition 5.5 in [23]implies that an algebra as in the lemma exists if Γ is hyperbolic. (cid:3)
Examples of groups with real rank one are O( n,
1) and U( n, G is at least two, then Γ is not hyperbolic in general. (Then thesectional curvature of G/K does not have a negative upper bound, as
G/K admits an embedding of a Euclidean space of dimension at least two.)By Corollary 2.9, in a sense the most refined index-theoretic informationabout an eliptic operator D X on the compact symmetric space X = Γ \ G/K is the ( K, Γ)-index of its lift ˜ D to G . A natural way to obtain potentiallycomputable numbers from the index of ˜ D in KK ( C ∗ K, C ∗ Γ) is to evaluatethe component in ˆ R ( K ) at a group element (where this makes sense), andto apply suitable traces to the component in C ∗ Γ, such as traces defined byorbital integrals.Because of our assumption that τ γ extends to a smooth subalgebra A ( G )of C ∗ G , it induces a map τ γ : K ( C ∗ Γ) = K ( A ( G )) → C . N EQUIVARIANT ORBIFOLD INDEX FOR PROPER ACTIONS 15
Therefore, it is a natural problem to compute the element(4.5) τ γ (index K, Γ ( ˜ D )) ∈ ˆ R ( K ) ⊗ C . We will do this for the operator D C G,K as in (4.2).We denote the character of a finite-dimensional representation space V of K , by χ V . If V is Z -graded, as V = ∆ p is, then χ V is the difference of thecharacters of the even and odd parts of V . Proposition 4.3. (a) if γ is not a torsion element and nontrivial, then τ γ (index K, Γ ( ˜ D )) = 0 .Suppose that G has discrete series representations, and let D C G,K be as in (4.2) . (b) For γ = e , τ e (index K, Γ ( D C G,K )) = vol(Γ \ G ) X π d π [ V π ] , where d π is the formal degree of the discrete series representation π of G , and V π ∈ ˆ K is the irreducible representation corresponding to π via Dirac induction. (c) If γ is a regular element of a compact Cartan subgroup of G containedin K , then τ γ (index K, Γ ( D C G,K )) = ( − dim( G/K ) / vol( Z G ( γ ) /Z Γ ( γ )) χ ∆ p ( γ ) X [ V ] ∈ ˆ K χ V ( γ )[ V ] . Proof.
For V ∈ ˆ K , let D V be as in Subsection 3.2. By Proposition 3.3,(4.6) index K, Γ ( D C G,K ) = X V ∈ ˆ K [ V ] ⊗ µ Γ G/K ( D V ) ∈ ˆ R ( K ) ⊗ K ∗ ( C ∗ Γ) = KK ( C ∗ K, C ∗ Γ) . If γ is not a torsion element, then it has no fixed points in G/K . As aspecial case of Theorem 5.10 in [27], this implies that τ γ (index Γ ( D V )) = 0,so part (a) follows.For a semisimple element g ∈ G , let τ Gg be the corresponding orbitalintegral: τ Gg ( f ) = Z G/Z G ( g ) f ( xgx − ) d ( xZ G ( g )) , for f such that this converges.If ˜ D = D C G,K as in (4.2), then for all discrete series representations π of G , the index formula Theorem 6.12 in [28] implies in particular that τ e ( µ Γ G/K ( D V π )) = vol(Γ \ G ) τ Ge (index G ( D V π ))By Lemma 5.4 in [12], τ Ge (index G ( D V π )) = d π . And if V ∈ ˆ K does notcorrespond to a discrete series representation, then index e D V π = 0. SeeCorollary 7.3.B in [8]. So part (b) follows. For part (c), we use the fact that τ γ ( µ Γ G/K ( D V )) = vol( Z G ( γ ) /Z Γ ( γ )) τ Gγ ( µ GG/K ( D V )) . This follows from the topological expression for these indices in Theorem5.10 in [27]. So τ γ (index K, Γ ( ˜ D )) = vol( Z G ( γ ) /Z Γ ( γ )) X [ V ] ∈ ˆ K τ Gγ ( µ GG/K ( D V ))[ V ] . Part (c) now follows from Theorem 3.2 in [13]. (cid:3)
A version of part (b) of Proposition 4.3 was used by Atiyah and Schmidto obtain formal degrees of discrete series of G [2]. Remark 4.4.
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University of Adelaide
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