A KK-theoretic perspective on deformed Dirac operators
aa r X i v : . [ m a t h . K T ] J u l A KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS
YIANNIS LOIZIDES, RUDY RODSPHON AND YANLI SONG
Dedicated to Gennadi Kasparov for his th birthday A BSTRACT . We study the index theory of a class of perturbed Dirac operators on non-compact manifolds of theform D + i c ( X ) , where c ( X ) is a Clifford multiplication operator by an orbital vector field with respect to theaction of a compact Lie group. Our main result is that the index class of such an operator factors as a KK-productof certain KK-theory classes defined by D and X . As a corollary we obtain the excision and cobordism-invarianceproperties first established by Braverman. An index theorem of Braverman relates the index of D + i c ( X ) to theindex of a transversally elliptic operator. We explain how to deduce this theorem using a recent index theorem fortransversally elliptic operators due to Kasparov. I NTRODUCTION
The present work studies, from the perspective of KK-theory, the index theory of a class of Dirac-typeoperators on non-compact manifolds. A Dirac operator D on a non-compact manifold always determinesa K-homology class [ D ] , but extracting a ‘numerical index’ (in a possibly generalized sense) often requiresadditional ingredients, which can be either boundary conditions at infinity, or appropriate devices playingthe role of compactness, relevant choices being dictated by the geometric situation at hand. In some cases,such a device can be a suitable perturbation of D .A standard example on R n is the operator d + d ∗ + ext ( x ) + int ( x ) acting on L ( R n , ∧ T ∗ R n ) , where ext ( · ) ,int ( · ) denote exterior and interior multiplication respectively. Its square is a harmonic oscillator so thatthe operator has index one. On the KK-theoretic side, it is well-known that this operator represents theKK-product of the Bott / dual-Dirac and the Dirac elements, which is then the identity. Extensive general-izations of this calculation (in various forms) laid the foundations of important techniques used to proveKK-theoretic Poincaré duality results in index theory, or more broadly in much of the work done on theBaum-Connes conjecture.Another source of interesting examples is operators of Callias-type, introduced in [ ] . The perturbationhere is a suitable ‘potential’ Φ , and the operator D + Φ is Fredholm. KK-product interpretations of theFredholm index have been provided in [ ] , in [ ] via unbounded KK-theory with recent improvements in [ ] . Loosely speaking, the potential defines a K -theory class [ Φ ] on the manifold, and the index of D + Φ arises as the KK-product [ Φ ] b ⊗ C ( M ) [ D ] .In this article, we shall focus on a class of operators that we will call deformed Dirac operators . Their studyoriginates partly from [ ] and has been systematized by Braverman in [ ] . These operators have foundinteresting applications, notably in the resolution of a conjecture of Vergne on the quantization commuteswith reduction problem [ ] , and subsequent extensions of this work (e.g. [ ] ).Let M be a complete Riemannian manifold equipped with an isometric action ρ of a compact Lie group G ,let E be a G -equivariant Clifford module bundle over M and let D be a Dirac operator acting on sections of E . In our setting the additional geometric data used to obtain a well-defined index is a G -equivariantmap ν : M → g = Lie ( G ) such that the vector field ν : m ∈ M ρ m ( ν ( m )) has a compact vanishing locus;Braverman [ ] referred to such a map as a taming map . Deformed Dirac operators are then operators ofthe form D f ν = D + i f c ( ν ) , where f ∈ C ∞ ( M ) is a function satisfying a growth condition at infinity (seeSection 2), and c ( ν ) is Clifford multiplication. A deformed Dirac operator has a well-defined equivariantindex, similar to transversally elliptic operators (in the sense of Atiyah [ ] ). We will come back to thisanalogy shortly.An important technical consideration in studying D f ν lies in the calculation of the commutator of D with theperturbation. Whereas this commutator is a bounded operator in the two first examples, it is a differentialoperator of order one (in the orbit directions) in the case of deformed Dirac operators, which makes theKK-product factorization of their index much less straightforward.One of the aims of this paper is to provide such a KK-product interpretation. Heuristically, the idea is rathernatural, and involves viewing the perturbation as a dual-Dirac-like element [ ν ] in the orbit directions.This requires an ‘orbital Clifford algebra’ Cl Γ ( M ) introduced recently by Kasparov [ ] . A simple but keyobservation is that the operator D determines a class in the K -homology of the crossed product algebra G ⋉ Cl Γ ( M ) (see the Key Lemma in Section 1); the difficulty with the commutator mentioned above thendisappears.The rest of the paper explores some consequences by revisiting the index theorem, excision and cobordisminvariance properties obtained by Braverman in [ ] . From the perspective developed, the last two pointsbecome almost automatic and follow mostly from functorial arguments. Braverman’s index theorem statesthat the analytic index of a deformed Dirac operator is equal to the topological index of a transversallyelliptic symbol, obtained by deforming the symbol of D by the vector field ν . We show how this result can bededuced from the KK-product factorization and Kasparov’s index theorem for transversally elliptic operators [
20, Theorem 8.18 ] . This makes the relationship between the indices of deformed Dirac operators and oftransversally elliptic operators more transparent. It is also possible, as shown in [ ] , to relate such anindex to an Atiyah-Patodi-Singer-type index, but this will not be discussed here.A final note on quantization commutes with reduction in the case of a Hamiltonian G -space with propermoment map: many obvious similarities between the analytic approach relying on the properties of thedeformed Dirac operator [
29, 23, 17 ] , and the topological one based on K -theory classes of transversallyelliptic symbols [
25, 27, 26 ] may be spotted. In view of our last observation, it seems plausible that bothapproaches become essentially the same (up to Poincaré duality). It would be desirable to develop asynthesis of these methods in the framework of KK-theory, hopefully offering a unifying perspective on theseworks, and optimistically leading to conceptual simplifications. This is partly the motivation of the presentpaper, and will be the topic of future work.The contents of the paper are as follows: • Section 1 reviews some material from [ ] , and in particular the notion of orbital Clifford algebra, whichis used to build a transverse index class from the Dirac operator. • Section 2 contains the main result, explaining how the equivariant index of deformed Dirac operators canbe seen in terms of a KK-product. Preparatory material on deformed Dirac operators is included. • Section 3 revisits the excision and cobordism invariance of the index of deformed Dirac operators obtainedin [ ] , from the point of view developed in Section 2. KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 3 • Section 4 reviews further material from [ ] , and derives Braverman’s index theorem. • Appendices.
Certain arguments in the main body are streamlined if one has the flexibility to work onnon-complete manifolds, and Appendix A explains how to deal with this case. Evident extensions of classicalresults stated in Section 1 have their proofs relegated to Appendices B and C.
Notation.
Throughout the article G denotes a compact Lie group with Lie algebra g . We denote i = p− c ( v ) = −| v | . The notation 〈 v 〉 : = ( + | v | ) / . Given a Hermitian orEuclidean vector bundle V → M on a Riemannian manifold and section s : M → V , we generally write | s | forthe point-wise norm of s , and k s k for the L -norm using the Riemannian volume form. If A is a C ∗ -algebraand E a Hilbert A -module, we generally write K A ( E ) (resp. B A ( E ) ) for the compact (resp. adjointable)operators in the sense of Hilbert modules. Last but not least, we use the notation b ⊗ for graded tensorproducts. Acknowledgements.
We want to address very special thanks to G. Kasparov; how much the present articleowes to his recent work (and the multiple discussions we had about it) will be evident thoughout the reading.Happy 70 th Birthday Genna! We also thank N. Higson and M. Braverman for helpful discussions. Y. Song issupported by NSF grant DMS-1800667.1. T
RANSVERSE K - HOMOLOGY CLASS OF THE D IRAC OPERATOR
Let ( M n , g ) be an even-dimensional Riemannian manifold (not necessarily complete) equipped with an iso-metric action of a compact Lie group G . Let g denote the Lie algebra of G . Let Cliff ( T M ) denote the Cliffordalgebra bundle of M , and Cl τ ( M ) = C ( M , Cliff ( T M )) the C ∗ -algebra of continuous sections vanishing atinfinity.1.1. Orbital Clifford algebra, and a key lemma.
We first review some material from the recent work ofKasparov [ ] . For every m ∈ M , let ρ m : β ∈ g dd t (cid:12)(cid:12)(cid:12)(cid:12) t = e − t β · m ∈ T m M ,denote the infinitesimal action at the point m . We define Γ m = ρ m ( g ) ⊂ T m M to be the tangent space to the orbit G · m at m . We would like to define spaces of ‘smooth’ and ‘continuous’sections of ⊔ m ∈ M Γ m . Since the orbits of a compact Lie group action typically vary in dimension (the map m dim ( Γ m ) is only lower semi-continuous in general), this takes a little care.Let ρ : g M : = M × g → T M denote the smooth bundle map induced by the maps ρ m , i.e. ρ is the anchormap for the action Lie algebroid g M . By post-composition ρ induces a map (also denoted ρ ) on sections. Wedefine the space of smooth and compactly supported sections of Γ to be C ∞ c ( M , Γ ) : = ρ ( C ∞ c ( M , g M )) ⊂ C ∞ c ( M , T M ) .This is a simple instance of a singular foliation in the sense of [ ] : a C ∞ c ( M ) -submodule of the space ofsmooth compactly supported vector fields which is involutive and locally finitely generated. The space ofcontinuous sections of T M vanishing at infinity C ( M , T M ) is the Banach space completion of C ∞ c ( M , T M ) with respect to the supremum norm. We define the space of continuous sections of Γ vanishing at infinity In fact, the results and proofs of this section remain valid even if G is only a locally compact Lie group acting properly andisometrically on M . YIANNIS LOIZIDES, RUDY RODSPHON AND YANLI SONG C ( M , Γ ) to be the closure of C ∞ c ( M , Γ ) in C ( M , T M ) . Dropping the vanishing conditions we obtain similardefinitions of the space of smooth sections and the space of continuous sections. In particular, this endows Γ = ⊔ m ∈ M Γ m with the structure of a continuous field of vector spaces over M , that we call the orbital tangentfield . (Recall that Γ being a continuous field of vector spaces means that it admits a set of sections σ thatgenerate Γ point-wise and such that m
7→ | σ m | is a continuous function on M .) Definition 1.1.
The orbital Clifford algebra Cl Γ ( M ) is the C ∗ -subalgebra of Cl τ ( M ) generated by C ( M , Γ ) and C ( M ) . Cl Γ ( M ) is a C ( M ) -algebra, and may equivalently be described as the algebra of continuous sectionsvanishing at infinity of the continuous field of C ∗ -algebras Cliff ( Γ ) = ⊔ m ∈ M Cliff ( Γ m ) , where the continuousfield structure is inherited from that of Γ . Cl Γ ( M ) contains a dense subalgebra generated by C ∞ c ( M , Γ ) and C ∞ c ( M ) , denoted Cl ∞ Γ , c ( M ) .Let E be a G -equivariant Z -graded Hermitian Clifford module bundle over M , and let D be a Dirac operatorassociated to a G -equivariant Clifford connection ∇ on E . Locally, in terms of a local orthonormal frame e , . . . e n , D = n X i = c ( e i ) ∇ e i .where c ( · ) denotes Clifford multiplication. Lemma 1.2.
Let Y be a vector field on M and ∇ LC denote the Levi-Civita connection associated to the metric g.Then, [ c ( Y ) , D ] = − ∇ Y − n X i = c ( e i ) c ( ∇ LC e i Y ) . In particular, if Y is a smooth section of the orbital tangent field Γ , then [ c ( Y ) , D ] is a differential operator oforder in the orbit direction. Let ν : M → g be a smooth map. Introduce the differential operator L ν acting on smooth sections of E by ( L ν ϕ )( m ) = dd t (cid:12)(cid:12)(cid:12)(cid:12) t = e t ν ( m ) · ϕ ( e − t v ( m ) · m ) .Note that if f ∈ C ∞ ( M ) , then L f ν = f L ν . We will use boldface ν : m ∈ M ρ m ( ν ( m )) to denote the vectorfield generated by ν . The difference ∇ ν − L ν is an operator of order 0. Definition 1.3.
The moment map for the pair ( E , ∇ ) (cf. [ ] ) is the smooth section µ E ∈ C ∞ ( M , g ∗ b ⊗ End ( E )) defined by the equation 〈 µ E , ν 〉 = ∇ ν − L ν for all ν ∈ C ∞ ( M , g ) .An element h ∈ C ∞ ( G ) acts on ϕ ∈ C ∞ c ( M , E ) by the convolution operator h ⋆ ϕ ( m ) = Z G h ( g ) g · ϕ ( g − · m ) d g ,where d g denotes a left invariant Haar measure. Key Lemma.
Let ν : M → g be a smooth map. For any h ∈ C ∞ ( G ) and ϕ ∈ C ∞ c ( M , E ) , L ν ( h ⋆ ϕ )( m ) = − ( ν Rm h ) ⋆ ϕ ( m ) , where ν Rm denotes the right-invariant vector field on G generated by ν m : = ν ( m ) ∈ g , which acts by differentiationon h. For any χ ∈ C ∞ c ( M ) , the operator χ [ c ( ν ) , D ] h extends to a bounded operator on L ( M , E ) . KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 5
Proof.
By definition L ν ( h ⋆ ϕ )( m ) = dd t (cid:12)(cid:12)(cid:12)(cid:12) t = Z G h ( g ) e t ν m g · ϕ ( g − e − t ν m · m ) d g = dd t (cid:12)(cid:12)(cid:12)(cid:12) t = Z G h ( e − t ν m g ) g · ϕ ( g − · m ) d g = − ( ν Rm h ) ⋆ ϕ ( m ) where in the second line we used a change of variables.Lemma 1.2 and Definition 1.3 show that the commutator [ c ( ν ) , D ] = −L ν + B where B is a bundle endomor-phism. Let β , ..., β dim ( g ) be a basis of g , and let ν j : M → R be the components of ν relative to the basis. Bythe calculation above L ν ( h ⋆ ϕ ) = X j ν j ( h j ⋆ ϕ ) where h j : = − β Rj h ∈ C ∞ ( G ) . The second statement follows because the bundle endomorphism B and thesmooth functions ν j are bounded on the support of χ . (cid:3) K-homology and the transverse Dirac class. If ( M , g ) is complete then D is essentially self-adjoint,and the standard practice is to attach the following K -homology class to the Dirac operator: [ D M ] = (cid:2) ( L ( M , E ) , F = D ( + D ) − ) (cid:3) ∈ K G ( C ( M )) . (1)Theorem 1.4 below shows that the same pair ( L ( M , E ) , F ) defines a class in a different K-homology group.This observation is due to Kasparov: see Lemma 8.8 of [ ] for the case of the de Rham-Dirac operator.The action of G on L ( M , E ) by convolution and the action of Cl Γ ( M ) on L ( M , E ) by Clifford multiplicationform a covariant pair, hence L ( M , E ) carries a representation of the crossed-product algebra G ⋉ Cl Γ ( M ) . Theorem 1.4. If ( M , g ) is complete, the pair ( L ( M , E ) , F ) determines a class in K ( G ⋉ Cl Γ ( M )) .Proof. It suffices to verify that for every a ∈ G ⋉ Cl Γ ( M ) , [ F , a ] is a compact operator. That the other Fredholmmodule axioms hold is analogous to standard cases. We may assume a = h ⊗ α , with h ∈ C ∞ ( G ) and α ∈ Cl ∞ Γ , c ( M ) , since such elements are dense in G ⋉ Cl Γ ( M ) . Let χ ∈ C ∞ c ( M ) be a bump function equal to1 on the compact set G · supp ( α ) ⊂ M . Hence a = χ a and [ D , a ] = χ [ D , a ] . Following [ ] or [
19, Lemma4.2 ] , we first write F as a Cauchy integral: F = π Z ∞ D ( + λ + D ) − d λ ,so that, [ F , a ] = π Z ∞ ( + λ + D ) − (cid:0) ( + λ ) χ [ D , a ] + D χ [ a , D ] D (cid:1) ( + λ + D ) − d λ . (2)Since D is G -invariant, [ D , a ] = [ D , α ] h , and the latter is a bounded operator by virtue of the Key Lemma.The operator ( + λ )( + λ + D ) − χ [ D , a ] (3)is compact by the Rellich lemma. Since k ( + λ + D ) − k is O ( λ − ) , the norm of (3) is uniformly boundedin λ , and the product ( + λ )( + λ + D ) − χ [ D , a ]( + λ + D ) − is a compact operator with norm O ( λ − ) . For the second integrand ( + λ + D ) − D χ [ a , D ] D ( + λ + D ) − (4)note that ( + λ + D ) − D χ is compact by the Rellich lemma, and has norm O ( λ − ) . Using again the fact that [ a , D ] is bounded, it follows that (4) is a compact operator of norm O ( λ − ) . Thus both integrands in (2) arecompact with norm O ( λ − ) , hence the integral converges in the norm topology to a compact operator. (cid:3) If M is not complete then [
15, Chapter 10 ] explains a slightly more elaborate construction that produces aclass [ D M ] ∈ K G ( C ( M )) from a Dirac operator. (One could also replace the metric on M with one that iscomplete, and this leads to the same K-homology class.) Using similar techniques it is not difficult to do thesame in our setting, and we outline how this is done in Appendix A. Granted this we make the followingdefinition. Definition 1.5.
Let M be a Riemannian manifold with an isometric action of a compact Lie group G . Let D be a G -equivariant Dirac operator acting on sections of a Clifford module bundle E . The Hilbert space L ( M , E ) and the operator D determine a K-homology class [ D M , Γ ] ∈ K ( G ⋉ Cl Γ ( M )) that we refer to as the transverse Dirac class associated to D . If M is complete, this is the class described in Theorem 1.4. For thegeneral case see Appendix A. Remark 1.6.
Two well-known facts about [ D M ] ∈ K G ( C ( M )) are that (i) the class does not depend on themetric, and (ii) in the complete case the operator F can be replaced by χ ( D ) where χ is any ‘normalizingfunction’, cf. [
15, Chapter 10 ] . Similar results hold for [ D M , Γ ] ∈ K ( G ⋉ Cl Γ ( M )) . Given two G -invariantcomplete metrics g , g on M , there is a canonical isometric isomorphism ( Γ , g ) → ( Γ , g ) given fiberwiseby the square-root of the composite map Γ m g ♭ −−→ Γ ∗ m g ♯ −−→ Γ m . (Flat and sharp exponents denote metric contractions.)This induces a canonical isomorphism Cl Γ ( M , g ) → Cl Γ ( M , g ) between the corresponding orbital Cliffordalgebras, and so also between the crossed products by G . These isomorphisms intertwine the correspondingclasses in K ( G ⋉ Cl Γ ( M , g i )) , and in this sense [ D M , Γ ] is independent of the metric.1.3. Significance of the class [ D M , Γ ] . We discuss here briefly why the class [ D M , Γ ] ∈ K ( G ⋉ Cl Γ ( M )) is referred to as a transverse index class, by explaining its relationship with the more familiar class [ D M ] ∈ K G ( C ( M )) . The result stated is included for expository purposes, without proof and at the cost ofsome rigor. However it suggests an interesting geometric interpretation of the class [ D M , Γ ] in the spirit ofnon-commutative geometry and index theory of foliations, and might provide some helpful insights to thereader.Let ( β , . . . , β dim ( g ) ) be a basis of g . We define the orbital Dirac operator by D Γ : G ⋉ Cl ∞ Γ ( M ) → G ⋉ Cl ∞ Γ ( M ) ; D Γ = dim ( g ) X j = c ( ρ ( β j )) L β j where Cl ∞ Γ ( M ) is the smooth version of Cl Γ ( M ) , and L β j denotes Lie differentiation for the diagonal actionof G on G ⋉ Cl ∞ Γ ( M ) given by ( g ⊙ a )( h ) = g · a ( g − h ) . This definition leads (after considerable work ,compare [
20, Definition 8.5 ] ) to the construction of an element [ D Γ ] ∈ KK G ( C ( M ) , G ⋉ Cl Γ ( M )) . The receptacle of the class [ D Γ ] given here is technically not right, and more sophisticated KK-groups have to be used. However,it is sufficient for the purpose of exposition and motivation, especially since it will not be used thereafter. KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 7
If all the orbits of G have the same dimension, this element is the longitudinal index class of a family of Diracoperators over the orbit space M / G . The following theorem extends this observation, and shows that theclass [ D M , Γ ] ∈ KK ( G ⋉ Cl Γ ( M ) , C ) previously constructed should be interpreted as a transverse index class . Theorem 1.7. [ D M ] = [ D Γ ] b ⊗ G ⋉ Cl Γ ( M ) [ D M , Γ ] .Comment on the proof. Kasparov proves this theorem in [
20, Theorem 8.9 ] in the case where the Cliffordmodule is the exterior algebra bundle Λ • T ∗ M b ⊗ C , with D being the de Rham-Dirac operator d + d ∗ . Theresulting class [ d M , Γ ] ∈ K ( G ⋉ Cl τ ⊕ Γ ( M )) lies in a slightly different KK-group, but is in essence the same asthe one from Theorem 1.4 (This class is used later in Section 4). The theorem above can be proved by astraightforward readaptation of Kasparov’s arguments, or by a direct reduction to the special case he dealswith. ƒ Restriction to open sets.
Let U be a G -invariant open set of M , let ι U : C ( U ) , → C ( M ) be theextension-by-0 homomorphism, and ι ∗ U : K G ( C ( M )) → K G ( C ( U )) the corresponding restriction map on K-homology. A well-known property of the class [ D M ] ∈ K G ( C ( M )) (cf. [
15, Proposition 10.8.8 ] ) is that ι ∗ U [ D M ] = [ D U ] where [ D U ] ∈ K G ( C ( U )) is the class determined by the restriction D | U .The class [ D M , Γ ] has an analogous property. We will abuse notation slightly and use ι U to also denote theextension-by-0 homomorphism Cl Γ ( U ) , → Cl Γ ( M ) , as well as the induced ∗ -homomorphism between thecrossed products G ⋉ Cl Γ ( U ) , → G ⋉ Cl Γ ( M ) . Thus there is a restriction map ι ∗ U : K ( G ⋉ Cl Γ ( M )) → K ( G ⋉ Cl Γ ( U )) . Proposition 1.8.
The restriction of D to U determines a class [ D U , Γ ] ∈ K ( G ⋉ Cl Γ ( U )) and ι ∗ U [ D M , Γ ] = [ D U , Γ ] . For a proof, see Appendix B.1.5.
Manifolds with boundary.
Let e M be a Riemannian G -manifold with boundary, and let M = ∂ e M be theboundary, equipped with the restriction of the metric and of the G -action. There is a short exact sequence of C ∗ -algebras0 → C ( e M \ M ) → C ( e M ) → C ( M ) → ∂ : K ( C ( e M \ M )) → K ( C ( M )) be the induced boundary homomorphism.Suppose e E → e M is an ungraded Clifford module bundle on the odd-dimensional manifold e M . A Diracoperator e D for e E | e M \ M determines a class [ e D e M \ M ] ∈ K ( C ( e M \ M )) . Let E = e E | ∂ e M , equipped with Z -grading E ± the ± i -eigenbundles of c ( n ) , where n is an inward unit normal vector to the boundary. A well-knownproperty of the class [ e D e M \ M ] (cf. [
15, Proposition 11.2.15 ] ) is that ∂ [ e D e M \ M ] = [ D M ] where [ D M ] ∈ K ( C ( M )) is the class associated to a Dirac operator acting on sections of E .Transverse Dirac classes have an analogous property. The definitions of Γ and of the orbital Clifford algebraCl Γ ( e M ) go through for the manifold with boundary e M . Moreover the definition of Γ is compatible withrestriction to the boundary, in the sense that the restriction of Γ to the boundary (in the sense of continuous YIANNIS LOIZIDES, RUDY RODSPHON AND YANLI SONG fields), coincides with the orbital tangent field of the boundary. We therefore make a slight abuse of notationand write Γ for the orbital tangent fields on each of e M , e M \ M and M .There is a surjective ∗ -homomorphism Cl Γ ( e M ) → Cl Γ ( M ) given by restriction. Since the boundary is G -invariant, there is an extension of C ∗ -algebras0 → G ⋉ Cl Γ ( e M \ M ) → G ⋉ Cl Γ ( e M ) → G ⋉ Cl Γ ( M ) → ∂ : K ( G ⋉ Cl Γ ( e M \ M )) → K ( G ⋉ Cl Γ ( M )) It is straight-forward to adapt the arguments in Theorem 1.4 and Appendix A to show that a Dirac operator e D acting on sections of e E | e M \ M yields a class [ e D e M \ M , Γ ] ∈ K ( G ⋉ Cl Γ ( e M \ M )) . Proposition 1.9.
Let e E be an (ungraded) Clifford module over the odd-dimensional manifold e M, and [ e D e M \ M , Γ ] ∈ K ( G ⋉ Cl Γ ( e M \ M )) the corresponding class. Let E = e E | ∂ e M , equipped with Z -grading E ± the ± i -eigenbundlesof c ( n ) , where n is an inward unit normal vector to the boundary. Then ∂ [ e D e M \ M , Γ ] = [ D M , Γ ] .For a proof, see Appendix C. 2. D EFORMED D IRAC OPERATOR AND
KK-
PRODUCT
In this section we assume ( M , g ) is a complete Riemannian G -manifold (without boundary).2.1. Deformed Dirac operator.
Let us first review some definitions introduced by Braverman [ ] . A tamingmap is a G -equivariant map ν : M → g such that the induced vector field ν : m ∈ M ρ m ( ν ( m )) has a compact vanishing locus. It is convenient toassume that | ν | ≤ ν by a suitable smooth positive function). Following Braverman [ ] , a non-negative G -invariant function f ∈ C ∞ ( M ) G is said to be admissible iflim M ∋ m →∞ f | d f | M + f ( |∇ LC ν | M + | ν | g + |〈 µ E , ν 〉| E ) + = ∞ .(In this expression, | · | M is used to denote the point-wise norms on the vector bundles T M ≃ T ∗ M andEnd ( T M ) induced by the Riemannian metric, | · | E denotes the point-wise norm on the vector bundle End ( E ) induced by the Hermitian structure, and | · | g denotes the norm on the Lie algebra g induced from its innerproduct.) One can show [
8, Lemma 2.7 ] that admissible functions always exist. Definition 2.1.
Let E → M be a Clifford module bundle and let D be a Dirac operator acting on sections of E . Let ν : M → g be a taming map and let f be an admissible function. The deformed Dirac operator is theDirac-type operator D f ν = D + i f c ( ν ) .Intuitively, the assumption that f be admissible ensures that the cross-terms in D f ν can be neglected. This isreminiscent of Kasparov’s technical theorem, which provides operators playing the same role in the generalconstruction of the KK-product. The admissibility property also ensures nice properties of the spectrum of D f ν (cf. proof of Lemma 2.3), which makes it possible to define an equivariant index. KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 9
Theorem 2.2 (Braverman [ ] ) . Let D f ν be a deformed Dirac operator associated to a Z -graded Clifford modulebundle E = E + ⊕ E − . Then the pair ( L ( M , E ) , D f ν ) determines a class [ D f ν ] ∈ KK ( C ∗ ( G ) , C ) , which is independent of the choice of admissible function f . Under the identification KK ( C ∗ ( G ) , C ) ≃ R −∞ ( G ) = Z b G given by the Peter-Weyl theorem, the class [ D f ν ] identifies with its index Ind ( D f ν ) : = X π ∈ b G ( m + π − m − π ) · π ∈ R −∞ ( G ) where m ± π < ∞ is the multiplicity of the irreducible representation π ∈ b G in ker ( D f ν ) ∩ L ( M , E ± ) . To give some idea of what is involved, we outline an argument. Let F f ν = D f ν ( + D f ν ) − . First, for every e ∈ C ∗ ( G ) , [ F f ν , e ] = G -invariance of F f ν . It only remains to see that ( − F f ν ) e is compact, which comesfrom the following lemma: Lemma 2.3.
Let h ∈ C ∗ ( G ) . Then, ( + D f ν ) − h is a compact operator on L ( M , E ) .Proof. (of the lemma) For G compact, the Peter-Weyl theorem states that C ∗ ( G ) is an infinite direct sum overmatrix algebras End ( π ) , π ∈ b G . It suffices to consider the case where h lies in a single summand End ( π ) (inother words, h is a matrix coefficient for π ). Equivalently we must show that the restriction of ( + D f ν ) − to each isotypic component in L ( M , E ) is compact. One has D f ν = D + f | ν | + i (cid:0) f [ D , c ( ν )] + c ( d f ) c ( ν ) (cid:1) In terms of a local orthonormal frame e , ..., e dim ( M ) the commutator writes [ D , c ( ν )] = L ν + X j c ( e j ) c ( ∇ LC e j ν ) + 〈 µ E , ν 〉 .On the π -isotypic component, one has an inequality of semi-bounded operators |L ν | ≤ C π | ν | (the latter is amultiplication operator for the function | ν | on M ) with C π a constant just depending on the representation π . Thus on the π -isotypic component one has an inequality of semi-bounded operators D f ν ≥ D + f € | ν | M − f − (cid:0) f ( C π | ν | g + |∇ LC ν | M + |〈 µ E , ν 〉| E ) + | d f | M | ν | M (cid:1)Š . (5)The definition of admissible function implies that the term in the inner brackets, multiplied by the factor of f − , goes to 0 at infinity. On the other hand | ν | M = M . Consequently on the π -isotypic component, there is an inequality of semi-bounded operators of the form D f ν ≥ D + V where the potential function V is proper and bounded below. It is known that the operator D + V hasdiscrete spectrum (cf. [
21, Appendix B ] for a short proof and further references). This implies D f ν restrictedto the π -isotypical component has discrete spectrum, and hence compact resolvent. (cid:3) KK-product factorization.
We now come to the main result of the article, which is a KK-productfactorization of the K -homology class [ D f ν ] ∈ KK ( C ∗ ( G ) , C ) .Given two C ∗ -algebras A and B , we denote E ( A , B ) the set of ( A , B ) KK-cycles (or Kasparov A , B -bimodules).Recall the following theorem, which allows to recognize when a KK-cycle arises as a KK-product. Theorem 2.4 (Connes-Skandalis, [ ] ) . Let A , B , C be graded C ∗ -algebras, with A separable. Let ( H , π , F ) ∈ E ( A , B ) , ( H , π , F ) ∈ E ( B , C ) , and let E , E be their respective KK-theory classes. Suppose that F ∈ L C ( H b ⊗ B H ) is a C-linear boundedoperator such that (a) ( H = H b ⊗ B H , π b ⊗ F ) ∈ E ( A , C ) ,(b) F is an F -connection, i.e for every ξ ∈ H , the operators ( ξ b ⊗ . ) F − ( − ) deg ( ξ ) F ( ξ b ⊗ . ) and ( ξ b ⊗ . ) ∗ F − ( − ) deg ( ξ ) F ( ξ b ⊗ . ) ∗ are compact operators. (c) For every a ∈ A, a [ F b ⊗ B F ] a ∗ ≥ modulo compact operators on H.Then, the cycle ( H , π b ⊗ F ) ∈ E ( A , C ) represents the KK-product E b ⊗ B E ∈ KK ( A , C ) . Moreover, the KK -product E b ⊗ B E always admits a representative of this form, which is unique up to (norm-continuous) homotopy. Now, consider a deformed Dirac operator D f ν = D + i f c ( ν ) , where f is an admissible function and ν is thevector field associated to the taming map ν : M → g , with | ν | = ν . The latter condition means that the vector field ν determines a class [ ν ] = (cid:2)(cid:0) Cl Γ ( M ) , i c ( ν ) (cid:1)(cid:3) ∈ KK G ( C , Cl Γ ( M )) .Let j G [ ν ] ∈ KK ( C ∗ ( G ) , G ⋉ Cl Γ ( M )) be its image under the descent map j G : KK G ( C , Cl Γ ( M )) → KK ( C ∗ ( G ) , G ⋉ Cl Γ ( M )) . We can then form theproduct j G [ ν ] b ⊗ G ⋉ Cl Γ ( M ) [ D M , Γ ] ∈ KK ( C ∗ ( G ) , C ) . Theorem 2.5.
The K-homology class [ D f ν ] ∈ KK ( C ∗ ( G ) , C ) of the deformed Dirac operator factors as thefollowing KK-product: [ D f ν ] = j G [ ν ] b ⊗ G ⋉ Cl Γ ( M ) [ D M , Γ ] ∈ KK ( C ∗ ( G ) , C ) . Proof.
The first condition of the Connes-Skandalis criterion (Theorem 2.4) is Theorem 2.2. It suffices to checkthe F -connection condition for ξ = a ∈ G ⋉ Cl Γ ( M ) of the form a = h b ⊗ α with h ∈ C ∞ ( G ) , α ∈ Cl ∞ c ( M ) . Theoperator denoted ( a b ⊗ . ) : L ( M , E ) → ( G ⋉ Cl Γ ( M )) b ⊗ G ⋉ Cl Γ ( M ) L ( M , E ) ≃ L ( M , E ) in the Connes-Skandaliscriterion is given by the action of a ∈ G ⋉ Cl Γ ( M ) on L ( M , E ) , hence we must verify that F f ν a − ( − ) deg ( a ) aF is a compact operator on L ( M , E ) . Let χ ∈ C ∞ c ( M ) be a bump function equal to 1 on the compact set G · supp ( α ) ⊂ M . Let B = D f ν a − ( − ) deg ( a ) a D = [ D , a ] + i f c ( ν ) a .Then B = χ B and it follows from the Key Lemma that B is a bounded operator. Using integral expressionsas in the proof of Theorem 1.4, one has F f ν a − ( − ) deg ( a ) aF = π Z ∞ ( + λ + D f ν ) − (cid:0) ( + λ ) χ B − ( − ) deg ( a ) D f ν χ B D (cid:1) ( + λ + D ) − d λ .As in the proof of Theorem 1.4, the integrand is compact with operator norm O ( λ − ) , hence the integralconverges in norm to a compact operator. The verification for ( a b ⊗ . ) ∗ is similar. KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 11
We now check the positivity condition. Recall that for G compact C ∗ ( G ) is isomorphic to the direct sum over π ∈ b G of matrix algebras End ( π ) . It suffices to consider h ∈ C ∗ ( G ) lying in a single summand End ( π ) . Writethe commutator [ i c ( ν ) , F f ν ] via an integral formula for F f ν as in the proof of Theorem 1.4: h [ i c ( ν ) , F f ν ] h ∗ = π Z ∞ ( + λ + D f ν ) − € ( + λ ) h [ i c ( ν ) , D f ν ] h ∗ + D f ν h [ i c ( ν ) , D f ν ] h ∗ D f ν Š ( + λ + D f ν ) − d λ . (6)The integral formula for F f ν is convergent in the strong operator topology. Here, we have used the G -equivariance of c ( ν ) and D f ν , which implies that they commute with the convolution operator h . Considerthe graded commutator [ i c ( ν ) , D f ν ] = i [ c ( ν ) , D ] + f | ν | .It follows from the admissibility condition on f and our assumption that | ν | = f (cid:0) | ν | − f − ( C π | ν | + |∇ LC ν | + |〈 µ E , ν 〉| ) (cid:1) is bounded below, where C π is the constant appearing in inequality (5); let −∞ < C ≤ P = [ i c ( ν ) , D f ν ] − C is a positive unbounded operator when restricted to the π -isotypical component of L ( M , E ) . Thus h [ i c ( ν ) , D f ν ] h ∗ = hPh ∗ + Chh ∗ ,and hPh ∗ is a positive operator. The contribution of P to the integrand in (6) is a positive operator, and thecorresponding integral converges in the strong operator topology to a positive operator.The contribution to the integral (6) of Chh ∗ is2 C π Z ∞ ( + λ + D f ν ) − € ( + λ ) hh ∗ + D f ν hh ∗ D f ν Š ( + λ + D f ν ) − d λ . (7)The two terms in the integrand are analyzed as in the proof of Theorem 1.4. For example consider C ( + λ + D f ν ) − D f ν hh ∗ D f ν ( + λ + D f ν ) − . (8)By Lemma 2.3 the operator ( + λ + D f ν ) − D f ν h is compact, with norm O ( λ − ) , and the same is true of itsadjoint. Thus (8) is a compact operator with norm O ( λ − ) . It follows that the integral (7) converges in normto a compact operator. (cid:3) Remark 2.6.
In the case when M is compact, the equivariant index of D can be obtained by applying thecollapse map M → pt to the class [ D ] ∈ K G ( C ( M )) . In the present non-compact situation, the result aboveshows that the map ( j G [ ν ] b ⊗ . ) plays a similar role.3. A PPLICATIONS
In this section let M be a complete Riemannian manifold equipped with an isometric action of a compact Liegroup G , and let D f ν = D + i f c ( ν ) be a deformed Dirac operator associated to a ( Z -graded) Clifford modulebundle E → M . Excision for deformed Dirac operators.
A first consequence of the KK-product factorization of D f ν isan excision result for its index, which can be seen as a rough K -theoretic analogue of localization formulasin equivariant cohomology.Recall that we assumed | ν | = U ⊂ M be a G -invariant open set such that | ν | = U , and let ι U : Cl Γ ( U ) , → Cl Γ ( M ) be the extension-by-0 homomorphism. Let ν U = ν | U . The pair ( Cl Γ ( U ) , c ( ν U )) determines a class [ ν U ] ∈ KK G ( C , Cl Γ ( U )) . Proposition 3.1. ( ι U ) ∗ [ ν U ] = [ ν ] ∈ KK G ( C , Cl Γ ( M )) .Proof. Under the obvious identification Cl Γ ( U ) b ⊗ Cl Γ ( U ) Cl Γ ( M ) ≃ Cl Γ ( U ) , the element ( ι U ) ∗ [( Cl Γ ( U ) , c ( ν U ))] =[( Cl Γ ( U ) , c ( ν U ))] b ⊗ Cl Γ ( U ) [ ι U ] is represented by the pair [( Cl Γ ( U ) , c ( ν U ))] ∈ KK G ( C , Cl Γ ( M )) . Then, a homotopybetween this cycle and the cycle ( Cl Γ ( M ) , c ( ν )) is provided by the following ( C , Cl Γ ( M ) b ⊗ C [
0, 1 ]) -cycle ( E , F ) : E = { continuous functions f : [
0, 1 ] → Cl Γ ( M ) : supp ( f ( )) ⊂ U } ; F = i c ( ν ) .That 1 − F = − | ν | is a compact operator on E comes from the fact that | ν | = U , whence theresult. (cid:3) Corollary 3.2. [ D f ν ] = j G [ ν U ] b ⊗ G ⋉ Cl Γ ( U ) [ D U , Γ ] .Proof. This follows from the KK-product factorization of Theorem 2.5, Proposition 3.1, plus associativity ofthe Kasparov product: [ D f ν ] = ( j G [ ν U ] b ⊗ [ ι U ]) b ⊗ [ D M , Γ ] = j G [ ν U ] b ⊗ ([ ι U ] b ⊗ [ D M , Γ ]) together with the fact that [ ι U ] b ⊗ G ⋉ Cl Γ ( M ) [ D M , Γ ] = [ D U , Γ ] (Proposition 1.8). (cid:3) The corollary together with another application of Theorem 2.5 on the manifold U , imply that the indexof D f ν can be computed from the index of a deformed Dirac operator on U . This operator is determinedup to suitable homotopy by the condition that it represents the KK-product j G [ ν U ] b ⊗ G ⋉ Cl Γ ( U ) [ D U , Γ ] . Notehowever that one cannot simply restrict D f ν to U ; one should for example complete the metric on U andalso replace f | U with a function that is admissible for U . This result was proved by Braverman [ ] usingthe cobordism invariance of the index (see the next section). Here we obtain it as a consequence of theKK-product factorization.3.2. Cobordism invariance of the index.
We will reprove the following result of Braverman [ ] , whichleads directly to the cobordism invariance of the index of the deformed Dirac operator. Theorem 3.3.
Let M be a Riemannian G-manifold which is the boundary of a Riemannian G-manifold e M. Let e E be a G-equivariant (ungraded) Clifford module bundle over e M, and let E = e E | M be the induced Clifford modulebundle over the boundary M with Z -graded subbundles E ± given by the ± i -eigenbundles of c ( n ) , where n is theinward unit normal vector to the boundary. Let D be a Dirac operator associated to E, let e ν : e M → g be a tamingmap and let ν be its restriction to M. Thenj G [ ν ] b ⊗ G ⋉ Cl Γ ( M ) [ D M , Γ ] = ∈ K ( C ∗ ( G )) .3.2.1. Review of cobordism invariance in the standard case.
Let us first recall the Baum-Douglas Taylor proofof cobordism invariance in the standard case (cf. [
3, p.765 ] ), i.e we assume e M (and then M ) is compact,and ignore the G -action. The key C ∗ -algebra extension is0 → C ( e M \ M ) → C ( e M ) r −→ C ( M ) → KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 13 where r denotes restriction to the boundary. The proof of cobordism invariance is based on the analogue ofProposition 1.9: ∂ [ e D ] = [ D ] where [ e D ] ∈ K ( C ( e M \ M )) is the K-homology class defined by the Dirac operator e D on the odd-dimensional(open) manifold e M \ M , and ∂ is the boundary homomorphism in the six term exact sequence (in K-homology) associated to (9).Let e p (resp. p ) denote the homomorphism C → C ( e M ) (resp. C → C ( M ) ) obtained from the collapsingmap e M → pt (resp. M → pt). Hence r ◦ e p = p ⇒ e p ∗ ◦ r ∗ = p ∗ . (10)We have p ∗ [ D ] = e p ∗ ◦ r ∗ ◦ ∂ [ e D ] but the middle composition r ∗ ◦ ∂ = ƒ Proof of Theorem 3.3.
The relevant C ∗ -algebra extension to consider in this case is0 → G ⋉ Cl Γ ( e M \ M ) → G ⋉ Cl Γ ( e M ) r −→ G ⋉ Cl Γ ( M ) →
0. (11)where r is also the restriction map. The replacements for the collapsing maps e p , p are the taming maps e ν , ν which define elements j G [ e ν ] ∈ KK ( C ∗ ( G ) , G ⋉ Cl Γ ( e M )) , j G [ ν ] ∈ KK ( C ∗ ( G ) , G ⋉ Cl Γ ( M )) respectively. Then,we have j G [ ν ] = j G [ e ν ] b ⊗ [ r ∗ ] which is the analogue of equation (10) (we regard the ∗ -homomorphism r as an element [ r ∗ ] ∈ KK ( G ⋉ Cl Γ ( e M ) , G ⋉ Cl Γ ( M )) here). Thus j G [ ν ] b ⊗ [ D M , Γ ] = j G [ e ν ] b ⊗ [ r ∗ ] b ⊗ [ e D M , Γ ] = j G [ e ν ] b ⊗ [ r ∗ ] b ⊗ ∂ [ e D e M , Γ ] ,where the second equality uses Proposition 1.9. But [ r ∗ ] b ⊗ ∂ [ e D e M , Γ ] = r ∗ ◦ ∂ [ e D e M , Γ ] and r ∗ ◦ ∂ = ƒ
4. D
EFORMED D IRAC OPERATORS AND TRANSVERSALLY ELLIPTIC OPERATORS
In this section, we provide a KK-theoretic proof of the following theorem due to Braverman [
8, Theorem 5.5 ] (see also [
24, 22 ] ). Theorem 4.1.
Let M be a complete Riemannian G-manifold equipped with an isometric action of a compactLie group G, and let D f ν be a deformed Dirac operator. Then, the equivariant index of D f ν in R −∞ ( G ) is equalto the index (in Atiyah’s sense) of the transversally elliptic symbol σ ν ( ξ ) = i c ( ξ + ν ) obtained by deforming thesymbol of the Dirac operator using the vector field ν . Such transversally elliptic deformations have interesting applications; we mention for example the work ofParadan [ ] on the quantization-commutes-with-reduction theorem in symplectic geometry.The idea of the proof is relatively simple: we observe that with the appropriate KK-groups, the product of [ ν ] and of an appropriate symbol class [ σ M , Γ ] of D is the K-theory class of the transversally elliptic symbol σ ν defined above. The result then follows from our KK-product factorization and a KK-theoretic Poincaréduality theorem for transversally elliptic operators obtained by Kasparov [ ] .The first four subsections of this section might be viewed as a brief further ‘invitation’ to Kasparov’s work [ ] ; we do not attempt to be exhaustive, but rather describe a small sample of the many new constructionsand results contained in loc. cit. , in view of deriving Theorem 4.1.4.1. The transverse de Rham and Dolbeault classes.
Recall that there is a canonical K-homology class [ d M ] ∈ KK G ( Cl τ ( M ) , C ) associated to the de Rham-Dirac operator acting on differential forms. DenotingCl τ ⊕ Γ ( M ) = Cl τ ( M ) b ⊗ C ( M ) Cl Γ ( M ) , a similar construction to 1.4 applied to the de Rham-Dirac operatorproduces a class [ d M , Γ ] ∈ KK ( G ⋉ Cl τ ⊕ Γ ( M ) , C ) that we refer to as the transverse de Rham class (cf. [ ] ).If E is a G -equivariant Clifford module bundle on M , then the class [ D M ] ∈ KK G ( C ( M ) , C ) associated to theDirac operator factors as a KK-product [ D M ] = [ E ] b ⊗ Cl τ ( M ) [ d M ] where [ E ] ∈ R KK G ( M ; C ( M ) , Cl τ ( M )) (cf. [ ] ) is the class represented by the cycle having Hilbert module C ( M , E ) and the zero operator. One has a similar result for the classes [ D M , Γ ] , [ d M , Γ ] . To state it, recall thatthere is a product in R KK: b ⊗ M : R KK G ( M ; A , B ) × R KK G ( M ; C , D ) → R KK G ( M ; A b ⊗ C ( M ) C , B b ⊗ C ( M ) D ) .The following statement can be checked without difficulty, using for instance Theorem 2.4. Proposition 4.2.
There is a factorization [ D M , Γ ] = j G ([ E ] b ⊗ M Cl Γ ( M ) ) b ⊗ G ⋉ Cl τ ⊕ Γ ( M ) [ d M , Γ ] ∈ KK ( G ⋉ Cl Γ ( M ) , C ) .Let [ d ξ ] ∈ R KK G ( M ; C ( T M ) , Cl τ ( M )) be the class implementing the KK-equivalence between the algebras C ( T M ) and Cl τ ( M ) , which can be described explicitely via a family of Dirac operators acting on the fibresof the bundle π T M : T M → M (cf. [
20, Definition 2.5 ] ). Definition 4.3 ( [ ] , Definition 8.17) . Let Cl Γ ( T M ) : = C ( T M ) ⊗ C ( M ) Cl Γ ( M ) (beware this is not exactlythe orbital Clifford algebra of the G -manifold T M ). The transverse Dolbeault class is the product [ ∂ cl T M , Γ ] = j G ([ d ξ ] b ⊗ M Cl Γ ( M ) ) b ⊗ G ⋉ Cl τ ⊕ Γ ( M ) [ d M , Γ ] ∈ KK ( G ⋉ Cl Γ ( T M ) , C ) . (12)The symbol σ ( ξ ) = i c ( 〈 ξ 〉 − ξ ) of the bounded transform F = D ( + D ) − / of the Dirac operator determinesa class [ σ M ] ∈ R KK G ( M ; C ( M ) , C ( T M )) . By [
20, Proposition 3.10 ] , [ σ M ] b ⊗ C ( T M ) [ d ξ ] = [ E ] ∈ R KK G ( M ; C ( M ) , Cl τ ( M )) . (13) If A and B be C ( M ) - C ∗ -algebras, recall that the bivariant K-group R KK G ( M ; A , B ) is defined the same way as KK ( A , B ) , with thefollowing additional requirement: if ( H , F ) is a KK-cycle, then for every f ∈ C ( M ) , a ∈ A , b ∈ B , ξ ∈ H , one has ( f a ) ξ b = a ξ ( f b ) KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 15
In Kasparov’s terminology, the element in (13) is referred to as the
Clifford symbol of D . Proposition 4.2 andequations (12), (13) give us the formula [ D M , Γ ] = j G ([ σ M ] b ⊗ M Cl Γ ( M ) ) b ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] ∈ KK ( G ⋉ Cl Γ ( M ) , C ) . (14)4.2. Transversally elliptic symbols and the symbol algebra S Γ ( M ) . For the purpose of motivation, sup-pose M is a compact Riemannian manifold (we will drop the compactness assumption shortly). Let A be a G -equivariant pseudo-differential operator with symbol σ A acting on sections of a G -equivariant Hermitianvector bundle E . The support supp ( σ A ) of σ A is the subset of T ∗ M ≃ T M where σ A fails to be invertible.The operator A is said to be transversally elliptic if supp ( σ A ) ∩ T G M is compact, where T G M ≃ T ∗ G M = ann ( Γ ) is the conormal space to the G -orbits. In this case Atiyah proved [ ] that the restriction A π ( π ∈ b G ) of A toeach isotypical component is Fredholm, hence A has a well-defined ‘index’, index ( A ) = X π ∈ b G index ( A π ) π ∈ R −∞ ( G ) = Z b G .Moreover, the index depends only on the class in K G ( T G M ) = K G ( C ( T G M )) defined by the symbol.However the K-theory group of the algebra C ( T G M ) turns out to not be ideal for the purpose of stating anindex theorem. Kasparov’s replacement for C ( T G M ) in this context is the following. Definition 4.4 ( [ ] , Definition-Lemma 6.2) . Let M be a Riemannian manifold (not necessarily compact)with an isometric action of a compact Lie group G . The symbol algebra S Γ ( M ) is the norm-closure in C b ( T M ) (the algebra of continuous bounded functions on T M ) of the set of all smooth, bounded functions b ( m , ξ ) on T M , which are compactly supported in the m variable, and satisfy the following two conditions:(a) The exterior derivative d m b ( m , ξ ) in m is norm-bounded uniformly in ξ , and there is an estimate | d ξ b ( m , ξ ) | ≤ C ( + | ξ | ) − for a constant C which depends only on b and not on ( m , ξ ) .(b) The restriction of b to T G M belongs to C ( T G M ) .Another useful description of the symbol algebra S Γ ( M ) , which interprets its elements as symbols havingnegative order in the transverse directions, is the following: Lemma 4.5 ( [ ] , Definition-Lemma 6.2) . Under item (a) in the definition of S Γ ( M ) above, item (b) isequivalent to the following estimate: for any ǫ > there exists a constant c ǫ > such that | b ( m , ξ ) | ≤ c ǫ 〈 ϕ m ( ξ ) 〉 〈 ξ 〉 + ǫ , ∀ m ∈ M , ξ ∈ T m M .Given a G -equivariant Z -graded Hermitian vector bundle E , we can similarly define a Hilbert S Γ ( M ) -module, denoted S Γ ( E ) , as the norm-closure in the space of bounded sections of the pull-back bundle π ∗ T M E satisfying similar conditions to those in Definition 4.4 (using the norm on the fibres of π ∗ T M E induced by theHermitian structure).We now return to our usual setting, with M a complete Riemannian G -manifold. From now on, we refer totransversally elliptic operators (or symbols) according to the following definition. Definition 4.6.
Let A be a properly supported, odd, self-adjoint G -invariant pseudodifferential operator oforder 0 acting on sections of a G -equivariant Z -graded Hermitian vector bundle E . We will say that A (orits symbol σ A ) is transversally elliptic if for every a ∈ C ( M ) , a · ( − σ A ) ∈ S Γ ( M ) . By the ‘symbol’ σ A of A , we will mean a section of π ∗ TM End ( E ) in the usual Hörmander ( ρ = δ = ) class, defined everywhereand not required to be homogeneous, whose equivalence class modulo symbols of lower order is the class of the principal symbol of A . Atiyah proved a stronger result, that the index determines a distribution on G . Since S Γ ( M ) ⊂ K S Γ ( M ) ( S Γ ( E )) (the compact operators on S Γ ( E ) in the Hilbert module sense), a transver-sally elliptic symbol determines a class [ σ A ] = [( S Γ ( E ) , σ A )] ∈ R KK G ( M ; C ( M ) , S Γ ( M )) .By construction there is a ∗ -homomorphism ι ∗ T G M : S Γ ( M ) → C ( T G M ) , hence a map R KK G ( M ; C ( M ) , S Γ ( M )) → R KK G ( M ; C ( M ) , C ( T G M )) .In this sense the element [ σ A ] ∈ R KK G ( M ; C ( M ) , S Γ ( M )) can be viewed as a ‘refinement’ of the ‘naive’ classin R KK G ( M ; C ( M ) , C ( T G M )) defined by the symbol.4.3. The class f M , Γ . Recall the trivial bundle g M = M × g and the anchor map ρ : g M → T M describing thevector fields generated by the G -action. We now fix a G -invariant metric ( − , − ) g M on the bundle g M suchthat g ( ρ ( β ) , ρ ( β )) ≤ ( β , β ) g M . Using the metrics on g M , T M the anchor ρ has a transpose ρ ⊤ : T M → g M . Definition 4.7.
We define a smooth bundle map ϕ : T M → T M to be the composition ϕ = ρ ◦ ρ ⊤ . Remark 4.8.
The range of ϕ is contained in Γ ⊂ T M , and ϕ is, roughly speaking, a smooth version of fibre-wise orthogonal projection T m M → Γ m . For simplicity suppose the metric on g M is constant. Let β , ..., β dim ( g ) be an orthonormal basis of g , and β M = ρ ( β ) , ..., β dim ( g ) M = ρ ( β dim ( g ) ) the corresponding vector fields on M .Let X be a vector field. Then ϕ ( X ) = dim ( g ) X j = g ( X , β jM ) β jM .If the action of G is free, then ϕ ( X ) is, to a first approximation, the projection of X to the orbit directions(with some re-scaling of its components). At the other extreme, in a neighborhood of an isolated fixed point,the length | β jM | is O ( r ) where r is the distance to the fixed-point, and consequently the length | ϕ ( X ) | is O ( r ) (the typical example would be the vector field r ∂ θ in R ).The following definition is one of the main reasons to introduce the symbol algebra S Γ ( M ) . Definition 4.9. [
20, pp.1344–1345 ] The element [ f M , Γ ] ∈ R KK G ( M ; S Γ ( M ) , Cl Γ ( T M )) is the class repre-sented by the pair ( Cl Γ ( T M ) , f M , Γ ) where at a point ( m , ξ ) ∈ T m M , the operator f M , Γ ( m , ξ ) is left Cliffordmultiplication by − i ϕ m ( ξ ) 〈 ϕ m ( ξ ) 〉 − .Note that for b ∈ S Γ ( M ) , the estimate in Lemma 4.5 shows that the product b ( m , ξ )( − f M , Γ ( m , ξ ) ) = b ( m , ξ ) 〈 ϕ m ( ξ ) 〉 − belongs to C ( T M ) ⊂ Cl Γ ( T M ) = K Cl Γ ( T M ) ( Cl Γ ( T M )) . Hence the pair ( Cl Γ ( T M ) , f M , Γ ) does define a KK-cycle. Remark 4.10.
The class [ f M , Γ ] should be viewed as the symbol class of the orbital Dirac element sketchedin Section 1.3. On the other hand, it implements a KK-equivalence between S Γ ( M ) and Cl Γ ( T M ) .4.4. Kasparov’s index theorem for transversally elliptic operators.
Let X be a compact Riemannian man-ifold equipped with an isometric action of the compact Lie group G . Let A be a G -equivariant, odd, self-adjointorder-0 pseudodifferential operator acting on sections of a Z -graded Hermitian vector bundle E . Supposethe symbol σ A is transversally elliptic in the sense of Definition 4.9. Then • The symbol determines a class [ σ A ] ∈ R KK ( X ; C ( X ) , S Γ ( X )) . • By [ ] , [
20, Proposition 6.4 ] , the pair ( L ( X , E ) , A ) determines a class [ A ] ∈ KK ( G ⋉ C ( X ) , C ) , andmoreover index ( A ) ∈ R −∞ ( G ) ≃ K ( C ∗ ( G )) is the push-forward of [ A ] under the map X → pt. KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 17
Kasparov’s index theorem relates these two KK-theory classes. To state it, it is convenient to introduce avariant of the symbol class.
Definition 4.11 ( [ ] , Definition 8.13) . The tangent Clifford symbol class [ σ tcl A ] is the KK-product [ σ tcl A ] = [ σ A ] b ⊗ S Γ ( X ) [ f X , Γ ] ∈ R KK G ( X ; C ( X ) , Cl Γ ( T X )) .Kasparov provides (see the paragraph following [
20, Definition 8.13 ] ) the following explicit cycle ( E Γ , S Γ ) representing the class [ σ tcl A ] : the Hilbert module is the tensor product E Γ = C ( T X , π ∗ T X E ) b ⊗ C ( T X ) Cl Γ ( T X ) and the operator S Γ is N / ( σ A b ⊗ ) + N / ( b ⊗ f X , Γ ) (15)where the weights N , N = − N ∈ C ∞ b ( T X ) take the form N ( x , ξ ) = 〈 ξ 〉 〈 ξ 〉 + 〈 ϕ x ( ξ ) 〉 , N ( x , ξ ) = 〈 ϕ x ( ξ ) 〉 〈 ξ 〉 + 〈 ϕ x ( ξ ) 〉 . (16)For later use, we note that the weights N , N are chosen according to Kasparov’s technical theorem, andhave the following important properties: Lemma 4.12. N / · S Γ ( X ) ⊂ C ( T X ) and N ( − f X , Γ ) ∈ C ( T X ) . In fact, the first inclusion holds even when X is non-compact (this will be used later). The lemma shows that N (cid:0) K S Γ ( X ) ( S Γ ( E )) b ⊗ (cid:1) ⊂ K Cl Γ ( T X ) (cid:0) S Γ ( E ) b ⊗ S Γ ( X ) Cl Γ ( T X ) (cid:1) and N ( − f X , Γ ) ∈ K Cl Γ ( T X ) (cid:0) S Γ ( E ) b ⊗ S Γ ( X ) Cl Γ ( T X ) (cid:1) ,as in the general construction of the KK-product (cf. the proof of [
7, Theorem 18.4.3 ] ). It follows quicklythat ( E Γ , S Γ ) is indeed a cycle representing [ σ tcl A ] . It is not hard and rather instructive to check this facttogether with the previous lemma by hand.With these preparations, we can finally state Kasparov’s index theorem for transversally elliptic operators. Theorem 4.13 ( [ ] , Theorem 8.18) . Let A be a transversally elliptic operator on a compact RiemannianG-manifold X . Then, [ A ] = j G ([ σ tcl A ]) b ⊗ G ⋉ Cl Γ ( T X ) [ ∂ cl T X , Γ ] ∈ KK ( G ⋉ C ( X ) , C ) . Remark 4.14.
Kasparov gives several other variants of the index theorem, but this version is best suited toour purposes. Moreover, his theorem still applies if X and G are non-compact, as long as G acts properly andisometrically on X . We will only need the compact case.4.5. Transversally elliptic symbols on open manifolds.
Atiyah [ ] (see also [
27, Section 3 ] ) defined adistributional index more generally for any element α M ∈ K G ( T G M ) where M is a not-necessarily compactRiemannian G -manifold. The construction proceeds as follows. Atiyah proves [
1, Lemma 3.6 ] that onecan find a Z -graded Hermitian vector bundle E = E ⊕ E on M and σ M ∈ C b ( T M , π ∗ T M
End ( E )) an odd,self-adjoint bundle endomorphism whose restriction to T G M represents the class α , and such that one has σ M = π − T M ( K ) for a G -invariant compact subset K of M . Choose a Hermitian vector bundle F → M such that e E = E ⊕ F is trivial, and fix a trivialization. Let e E = E ⊕ F and e σ M = σ M ⊕ id F . Via e σ M we obtain a trivialization of ( E ⊕ F ) | M \ K . Choose a relatively compact G -invariant open neighborhood U of K , and let ι U , M : U , → M be the inclusion; we will use the same symbol for the induced open inclusion T G U , → T G M . The pair ( e E | U , e σ M | U ) represents a class α U ∈ K G ( T G U ) and α M = ( ι U , M ) ∗ α U by construction.Choose a G -equivariant open embedding ι U , X of U into a compact G -manifold X ; again we use the samesymbol for the induced open inclusion T G U , → T G X . Using the trivializations over U \ K , the bundle e E | U and endomorphism e σ M | U can be extended trivially to X (denoted e E X , e σ X respectively) and represent the class α X = ( ι U , X ) ∗ α U ∈ K G ( T G X ) . Atiyah definesindex ( α M ) = index ( A X ) ∈ R −∞ ( G ) where A X is any transversally elliptic operator on X such that the (naive) K-theory class of its symbol is α X .Atiyah proves an excision property [
1, Theorem 3.7 ] showing that the index can be determined just fromdata on U , and hence the construction is independent of the various choices.We can reformulate this construction and Atiyah’s excision result in the language of Theorem 4.13: supposethat one manages to choose σ M such that, in addition to the conditions above, one has ( − σ M ) ∈ S Γ ( M ) .Then σ M determines a class [ σ M ,c ] = [( S Γ ( E ) , σ M )] ∈ KK G ( C , S Γ ( M )) refining the class α M . The subscript‘c’ is to emphasize that this is a K-theory class whose support is compact over M , in contrast with the symbolsdefining elements of the group R KK G ( M ; C ( M ) , S Γ ( M )) that were considered in Section 4.2. One thenobtains similar classes [ e σ U ,c ] = [( S Γ ( e E | U ) , e σ | U )] ∈ KK G ( C , S Γ ( U )) refining α U , [ e σ X ,c ] ∈ [( S Γ ( e E X ) , e σ X )] ∈ KK G ( C , S Γ ( X )) refining α X , and moreover [ e σ X ,c ] = ( ι U , X ) ∗ [ e σ U ,c ] , [ σ M ,c ] = ( ι U , M ) ∗ [ e σ U ,c ] . (17)Let [ e σ tcl X ,c ] , [ e σ tcl U ,c ] , [ σ tcl M ,c ] be the corresponding tangential Clifford symbols obtained by KK-product with f X , Γ , f U , Γ , f M , Γ respectively. Functoriality of the classes f − , Γ under open embeddings implies the tangentialClifford symbols satisfy analogous formulae to (17).Let p : X → pt be the collapse map, and [ σ A , X ] ∈ R KK G ( X ; C ( X ) , S Γ ( X )) the class defined by the symbol of A X , so that p ∗ [ σ A , X ] = [ e σ X ,c ] . By Theorem 4.13,index ( A X ) = p ∗ [ A X ] = j G ([ e σ tcl X ,c ]) b ⊗ G ⋉ Cl Γ ( T X ) [ ∂ cl T X , Γ ] .Equations (17), as well as the functoriality of the KK-product and of the transverse Dolbeault class, give theequivalent formulaeindex ( A X ) = j G ([ e σ tcl U ,c ]) b ⊗ G ⋉ Cl Γ ( T U ) [ ∂ T U , Γ ] = j G ([ σ tcl M ,c ]) b ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] .We thus obtain the following formula for the index in Atiyah’s sense of α M = ι ∗ T G M [ σ M ,c ] ∈ K G ( T G M ) :index ( ι ∗ T G M [ σ M ,c ]) = j G ([ σ tcl M ,c ]) b ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] . (18)4.6. Proof of Theorem 4.1 (beginning).
Using the vector field ν , define σ ν ( ξ ) = i c ( 〈 ξ 〉 − ξ + ν ) .Since ν is a section of Γ , the support supp ( σ ν ) ∩ T G M = { ( x , 0 ) ∈ T M : ν ( x ) = } . By assumption the vanish-ing locus of ν is compact, hence the pair ( C ( T G M , π ∗ T G M E ) , σ ν ) represents an element α ν = [ σ ν ] ∈ K G ( T G M ) ,and so has a distributional index. We can now prove that index ( D f ν ) = index ( α ν ) as a consequence of theKK-product factorization (Theorem 2.5) and Kasparov’s index theorem 4.13.As a first step, let us re-write the right-hand-side of Theorem 4.1 in the language of Section 4.5. Recall thatwe assumed | ν | ≤
1, with equality outside a G -invariant relatively compact open set U ⊂ M . Define σ ν ( ξ ) = i c (cid:0) ( − | ν | ) / 〈 ξ 〉 − ξ + ν (cid:1) . KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 19
The symbols σ ν , σ ν define the same class α ν ∈ K G ( T G M ) . Since ( − | ν | ) has compact support, onehas ( − σ ν ) ∈ S Γ ( M ) . By the discussion in Section 4.5, the pair ( S Γ ( E ) , σ ν ) represents a class [ σ ν ,c ] ∈ KK G ( C , S Γ ( M )) that refines α ν ∈ K G ( T G M ) . By equation (18),index ( α ν ) = j G ([ σ tcl ν ,c ]) b ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] . (19)The next subsection explains how to factor out [ ν ] ∈ KK G ( C , Cl Γ ( M )) in the equation above.4.7. The symbol class of the transverse Dirac element.
Recall that the same operator D defines K-homology classes in two different groups, [ D M ] ∈ K G ( C ( M )) , [ D M , Γ ] ∈ K ( G ⋉ Cl Γ ( M )) . The order-0 symbol σ ( ξ ) = i c ( 〈 ξ 〉 − ξ ) (the symbol of F = D ( + D ) − / ) determines an element [ σ M ] = [( C ( T M , π ∗ T M E ) , σ )] ∈R KK G ( M ; C ( M ) , C ( T M )) . The analogue of [ D M , Γ ] at the level of symbols is the following. Lemma 4.15.
The pair ( S Γ ( E ) , σ ) represents a class [ σ M , Γ ] ∈ R KK G ( M ; Cl Γ ( M ) , S Γ ( M )) . With this in hand, a classical KK-product formula of Kasparov (cf. [
7, Proposition 18.10.1 ] ) provides a symbolanalogue of the factorization [ D f ν ] = j G ([ ν ]) b ⊗ G ⋉ Cl Γ ( M ) [ D M , Γ ] . Lemma 4.16. [ σ ν ,c ] = [ ν ] b ⊗ Cl Γ ( M ) [ σ M , Γ ] ∈ KK G ( C , S Γ ( M )) . Applying the lemma to equation (19) we obtainindex ( α ν ) = j G ([ ν ] b ⊗ Cl Γ ( M ) [ σ tcl M , Γ ]) b ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] . (20)4.8. End of the proof of Theorem 4.1.
By Theorem 2.5 and equation (14), one has [ D f ν ] = j G (cid:0) [ ν ] b ⊗ Cl Γ ( M ) ([ σ M ] b ⊗ M Cl Γ ( M ) ) (cid:1)b ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] .Comparing this to equation (20), we see that the proof of Theorem 4.1 is completed by the following result,which is the symbol analogue of Theorem 1.7. Proposition 4.17. [ σ M ] b ⊗ C ( M ) Cl Γ ( M ) = [ σ tcl M , Γ ] ∈ KK G ( Cl Γ ( M ) , Cl Γ ( T M )) .Proof. Note that H : = S Γ ( E ) b ⊗ S Γ ( M ) Cl Γ ( T M ) ≃ C ( T M , π ∗ T M E ) b ⊗ C ( T M ) Cl Γ ( T M ) ≃ C ( T M , π ∗ T M E ) b ⊗ C ( M ) Cl Γ ( M ) as Hilbert Cl Γ ( T M ) = C ( T M ) b ⊗ C ( M ) Cl Γ ( M ) -modules; thus the KK-elements on the left and right hand sidesare naturally represented on the same Cl Γ ( T M ) -module H . The representations of Cl Γ ( M ) differ however;we denote the representation for [ σ tcl M , Γ ] (resp. [ σ M ] b ⊗ C ( M ) Cl Γ ( M ) ) by π (resp. π ), where for a ∈ Cl Γ ( M ) , π ( a ) = c ( a ) b ⊗ π ( a ) = b ⊗ a (here 1 b ⊗ a denotes the operator e b ⊗ f ( − ) deg ( e ) deg ( a ) e b ⊗ a f ).The operator representing [ σ M ] b ⊗ M Cl Γ ( M ) is σ ( m , ξ ) b ⊗ = i c ( 〈 ξ 〉 − ξ ) b ⊗
1. The operator representing theproduct [ σ tcl M , Γ ] = [ σ M , Γ ] b ⊗ S Γ ( M ) [ f M , Γ ] ∈ KK G ( Cl Γ ( M ) , Cl Γ ( T M )) ,can be taken to be the same as that in (15), namely S = N / ( σ b ⊗ ) + N / ( b ⊗ f M , Γ ) where the weights N = − N ∈ C b ( T M ) are as in equation (16); indeed the only additional conditionthat needs to be checked is the compactness of the commutators [ π ( a ) , S ] , and this follows from theobservation that for a ∈ C ( M , Γ ) one has g ( a , 〈 ξ 〉 − ξ ) ∈ S Γ ( M ) , together with Lemma 4.12. We perform a ‘rotation’ homotopy simultaneously on the operator S and representation π . For t ∈ [
0, 1 ] let π t ( a ) = cos ( π t ) c ( a ) b ⊗ + sin ( π t ) b ⊗ a , S t = N / ( σ b ⊗ ) + N / f M , Γ , t ,where f M , Γ , t ( m , ξ ) = sin ( π t ) i c ( 〈 ϕ m ( ξ ) 〉 − ϕ m ( ξ )) b ⊗ − cos ( π t ) b ⊗ i 〈 ϕ m ( ξ ) 〉 − ϕ m ( ξ ) .It is clear that π , π , S coincide with the previous definitions. Let us check that this is a homotopy ofKasparov cycles. The commutator condition for the representation follows because [ π t ( a ) , f M , Γ , t ] = a ∈ Cl Γ ( M ) and t ∈ [
0, 1 ] . For f ∈ C ( M ) , the function f ( − S t ) is the same as f ( − S ) except for anadditional cross-term − ( π t ) N / ( m , ξ ) · N / ( m , ξ ) · f ( m ) · g (cid:0) ξ 〈 ξ 〉 , ϕ m ( ξ ) 〈 ϕ m ( ξ ) 〉 (cid:1) . (21)The product f ( m ) · g ( 〈 ξ 〉 − ξ , 〈 ϕ m ( ξ ) 〉 − ϕ m ( ξ )) ∈ S Γ ( M ) . Since N ≤
1, Lemma 4.12 implies that (21) liesin C ( T M ) . From this it follows that π t ( a )( − S t ) ∈ K Cl Γ ( T M ) ( H ) for all a ∈ Cl Γ ( M ) .After the homotopy, the representations of Cl Γ ( M ) on H for the two cycles agree, and we are left with theoperator S = N / ( σ b ⊗ ) + N / f M , Γ ,1 , f M , Γ ,1 ( m , ξ ) = i c ( 〈 ϕ m ( ξ ) 〉 − ϕ m ( ξ )) b ⊗ [ σ b ⊗ f M , Γ ,1 ] is the function 〈 ξ 〉〈 ϕ m ( ξ ) 〉 g ( ξ , ϕ m ( ξ )) and g ( ξ , ϕ m ( ξ )) = g ( ξ , ρ m ρ ⊤ m ( ξ )) ≥
0. It follows that the operator [ σ b ⊗ S ] is positive (and a fortiori positive modulo compacts). We can therefore apply a well-known criterion ofConnes-Skandalis (cf. [
7, Proposition 17.2.7 ] ) to conclude that the cycles ( H , π , S ) , ( H , π , σ b ⊗ ) areoperator homotopic. (cid:3) A PPENDIX
A. T
HE CASE OF NON - COMPLETE MANIFOLDS
This appendix follows up Section 1, and uses the same notation. Recall that on non-complete manifolds, themain issue comes from the possible non-self-adjointness of the Dirac operator, so that K -homology classeshave to be constructed with slightly more care. Adapting the techniques given in [ ] or [
15, Chapter 10 ] ,we generalize the construction of the class [ D M , Γ ] ∈ K ( G ⋉ Cl Γ ( M )) to the case where M is not complete.Throughout this section ∼ stands for equality modulo compact operators.Let χ : R → [ −
1, 1 ] be a ‘normalizing function’, i.e a continuous odd function which is positive on ( ∞ ) and tends to 1 at ∞ , and let H = L ( M , E ) . Cover M with relatively compact G -invariant open sets U j and let f j be a G -invariant partition of unity subordinate to the cover. Let D j be a G -equivariant essentiallyself-adjoint operator agreeing with D on U j (for example, compress D between suitable G -invariant bumpfunctions with support contained in a compact neighborhood of U j ). Let F = X j f j χ ( D j ) f j which converges in the strong operator topology to a bounded self-adjoint operator. KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 21
Lemma A.1 ( [ ] Lemma 10.8.3) . Let D , D be essentially self-adjoint first order differential operators on Mwhich restrict to the same elliptic operator on some open subset U ⊂ M. Let g ∈ C ( U ) . Then χ ( D ) g ∼ χ ( D ) g. Now, let a = h b ⊗ α ∈ G ⋉ Cl Γ ( M ) where h ∈ C ∞ ( G ) and α ∈ Cl ∞ Γ , c ( M ) . Choose a G -invariant compactlysupported cut-off function f equal to 1 on the support of α , and let D f be an essentially self-adjoint operatorthat agrees with D in a neighborhood of the support of f . Then, the lemma above (combined with the G -invariance of f ) shows that [ F , a ] ∼ [ χ ( D f ) , a ] ; a ( F − ) ∼ a ( χ ( D f ) − ) .Following the proof of Theorem 1.4 in the complete case, the operators on the right hand sides are compact,so that ( H , F ) is a Fredholm module.Moreover, if F ′ is an operator constructed the same way as F but from a different partition of unity, LemmaA.1 shows for every a ∈ G ⋉ Cl Γ ( U ) , a ( F F ′ + F ′ F ) a ∗ ∼ a χ ( D f ) a ∗ ≥ f being a function depending on a as above), which is a well-known sufficient condition for F ′ to be norm-continuously homotopic to F (see [ ] ). Therefore, the K -homology class [( H , F )] ∈ K ( G ⋉ Cl Γ ( M )) does notdepend on the choice of the partition of unity (and the cover). Finally, if M is complete, a similar calculationshows that [( H , F )] = [( H , χ ( D ))] in K ( G ⋉ Cl Γ ( M )) .A PPENDIX
B. P
ROOF OF P ROPOSITION [
15, Proposition 10.8.8 ] . We include it for the convenience of thereader. Let U be a G -invariant open set of M , and ι ∗ U : K ( G ⋉ Cl Γ ( M )) → K ( G ⋉ Cl Γ ( U )) be the associatedextension-by-0 homomorphism. Recall we want to prove that ι ∗ U [ D M , Γ ] = [ D U , Γ ] . Proof.
Let ( H , F ) : = ( L ( M , E ) , F = D ( + D ) − ) be the Fredholm module of Theorem 1.4, and let P denotethe orthogonal projection H → H U = L ( U , S ) (given by multiplication by the characteristic function of thesubset U ). Then P F P : H U → H U is a bounded operator, and ( H U , P F P ) is a Fredholm module over G ⋉ Cl Γ ( U ) (to see this, note that P commutes with G ⋉ Cl Γ ( U ) , and P | H U = Q be the orthogonal projection to L ( M \ U , S | M \ U ) , i.e Q = − P . In terms of the decomposition H = PH ⊕ QH , the operator F becomes the 2 × F = (cid:18) P F P P FQQF P QFQ (cid:19) .Notice that for a ∈ G ⋉ Cl Γ ( U ) , aQ =
0. Moreover (recall ∼ stands for equality up to compact operators) aP FQ = PaFQ ∼ P FaQ = aF ∼ (cid:18) aP F P
00 0 (cid:19) . This shows that the restriction of the G ⋉ Cl Γ ( M ) -Fredholm module ( H , F ) to a G ⋉ Cl Γ ( U ) -Fredholm moduleequals ( H U , P F P ) up to a locally compact perturbation (the entries QF P , P FQ and
QFQ in the matrix for F )and a degenerate module (namely ( QH , 0 ) ). Thus ι ∗ U [( H , F )] = [( H U , P F P )] .It remains to check that the cycle ( H U , P F P ) for K ( G ⋉ Cl Γ ( U )) is operator homotopic to ( H U , F U ) where F U = Σ j f j χ ( D j ) f j is the operator constructed in Appendix A. Let a = h b ⊗ α , h ∈ C ∞ ( G ) , α ∈ Cl ∞ Γ , c ( U ) . Fix j and consider a ∗ ( P F P f j χ ( D j ) f j + f j χ ( D j ) f j P F P ) a (22)as an operator on H U . Note that Pa = a since α has support contained in the G -invariant set U . Thus P F P f j χ ( D j ) f j a ∼ P F Pa f j χ ( D j ) f j = P Fa f j χ ( D j ) f j ∼ P F f j χ ( D j ) f j a .Applying similar arguments to the other factors of P in (22), it follows that, modulo compact operators, theoperator in (22) is a ∗ ( F f j χ ( D j ) f j + f j χ ( D j ) f j F ) a and the latter is positive modulo compact operators, by the results of Appendix A applied to the operator F on M . We obtain that the operator in (22) is positive modulo compact operators. Since a f j vanishes for allbut finitely many j , we conclude that a ∗ ( P F P F U + F U P F P ) a ≥ K ( H U ) .This proves ( H U , P F P ) is homotopic to the cycle ( H U , F U ) from Appendix A. (cid:3) A PPENDIX
C. P
ROOF OF P ROPOSITION [
15, Proposition 11.2.15 ] or [ ] . It is included for theconvenience of the reader.Recall the context: M = ∂ e M is the boundary of a Riemannian G -manifold e M , and let W = e M r M . Considerthe C ∗ -algebra extension:0 → G ⋉ Cl Γ ( W ) → G ⋉ Cl Γ ( e M ) → G ⋉ Cl Γ ( M ) → ∂ in K -homology. Let e E → e M be an ungraded Clifford module bundle, e D a Dirac operator acting on sections of e E , and [ e D W , Γ ] ∈ K ( G ⋉ Cl Γ ( W )) the corresponding K-homologyclass. The restriction to the boundary E = e E | ∂ e M becomes a Z -graded Cliff ( T M ) -module bundle with thegraded subbundles E ± being the ± i -eigenbundles of c ( n ) , where n is the inward unit normal vector to theboundary. Let D be a Dirac operator acting on sections of E and [ D M , Γ ] ∈ K ( G ⋉ Cl Γ ( M )) the correspondingK-homology class. We want to show that ∂ [ e D W , Γ ] = [ D M , Γ ] . KK-THEORETIC PERSPECTIVE ON DEFORMED DIRAC OPERATORS 23
Proof.
Let ǫ > ( ǫ ) × M ⊂ W be a collar neighborhood of M for which G acts trivially on the ( ǫ ) part. We then have the following morphisms of extensions0 / / G ⋉ Cl Γ ( W ) / / G ⋉ Cl Γ ( e M ) / / G ⋉ Cl Γ ( M ) / / / / G ⋉ Cl Γ (( ǫ ) × M ) / / extension-by-0 O O G ⋉ Cl Γ ([ ǫ ) × M ) / / extension-by-0 O O G ⋉ Cl Γ ( M ) / / / / C ( ǫ ) b ⊗ (cid:0) G ⋉ Cl Γ ( M ) (cid:1) ≃ O O / / C [ ǫ ) b ⊗ (cid:0) G ⋉ Cl Γ ( M ) (cid:1) ≃ O O / / G ⋉ Cl Γ ( M ) / / δ : K • + (cid:0) C ( ǫ ) b ⊗ ( G ⋉ Cl Γ ( M )) (cid:1) → K • ( G ⋉ Cl Γ ( M )) .Since the class [ e D W , Γ ] does not depend on the choice of the metric, we can equip ( ǫ ) × M with the productmetric. This way, the class [ e D W , Γ ] ∈ K ( G ⋉ Cl Γ ( W )) identifies over the collar neighborhood with the exteriorKK-product [ D ( ǫ ) ] b ⊗ [ D M , Γ ] , where D ( ǫ ) is the Dirac operator on ( ǫ ) and [ D ( ǫ ) ] ∈ K ( C ( ǫ )) . But themap [ D ( ǫ ) ] b ⊗ . is inverse to the suspension isomorphism, so that δ ([ D ( ǫ ) ] b ⊗ [ D M , Γ ]) = [ D M , Γ ] The conclusion then follows from the naturality of the boundary map. (cid:3) R EFERENCES [ ] M. F. Atiyah,
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