Geometric K-homology and the Freed-Hopkins-Teleman theorem
aa r X i v : . [ m a t h . K T ] J u l GEOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMANTHEOREM
YIANNIS LOIZIDES
Abstract.
We construct a map at the level of cycles from the equivariant twisted K-homologyof a compact, connected, simply connected Lie group G to the Verlinde ring, which is inverseto the Freed-Hopkins-Teleman isomorphism. As an application, we prove that two of theproposed definitions of the quantization of a Hamiltonian loop group space—one via twistedK-homology of G and the other via index theory on non-compact manifolds—agree with eachother. Introduction
A remarkable theorem due to Freed, Hopkins and Teleman [19, 21, 20] relates the repre-sentation theory of the loop group LG of a compact Lie group G to the equivariant twistedK-theory of G . In the special case of a connected, simply connected and simple Lie group, thetheorem states that there is an isomorphism of rings R k ( G ) ≃ K G ( G, A ( k +h ∨ ) ). Here R k ( G ) isthe Verlinde ring of level k > LG bas of the loop group, while K G ( G, A ( k +h ∨ ) ) is the equivariant K-homology of G with twist-ing (Dixmier-Douady class) k + h ∨ ∈ Z ≃ H G ( G, Z ). The shift h ∨ is a Lie theoretic constantassociated to G called the dual Coxeter number. Freed-Hopkins-Teleman work with twistedK-theory, which is related by a Poincare duality isomorphism [53].In the proof Freed, Hopkins and Teleman construct a map, at the level of cycles (cid:18) level k positive energyrepresentations of LG bas (cid:19) (cid:18) ( k + h ∨ )-twistedK-theory of G (cid:19) . The construction involves an interesting family of algebraic Dirac operators parametrized bythe space of connections on a G -bundle over S . Computing the equivariant twisted K-theoryof G using techniques from algebraic topology, they are able to show that their map is anisomorphism.It is less clear how to construct a map in the opposite direction (from twisted K-homology to R k ( G )) at the level of cycles . One goal of this article is to describe such a map for analytic cycles or Fredholm modules, which are the cycles for the analytic description of twisted K-homology(cf. [3, 24, 27]). A special class of analytic cycles are those which come from Baum-Douglas-type geometric cycles (cf. [8, 6]), and we also study the specialization of our map to such cyclesand obtain a correspondingly more explicit description.We should remark at the outset that we do not directly build a positive energy representationfrom a cycle, which would be interesting and perhaps preferable. The output will instead bea formal character. Let us give an overview of the construction. The data used to describea twist in the analytic picture is a G -equivariant Dixmier-Douady bundle A over G ; this isa locally trivial bundle of elementary C ∗ algebras over G , equipped with a G -action covering the conjugation action on the base. Such a bundle has an invariant, the Dixmier-Douadyclass DD( A ) ∈ H G ( G, Z ) ≃ Z , and we assume DD( A ) = ℓ >
0. Consider an analytic cyclerepresenting a class x in the twisted K-homology group K G ( G, A ). Restrict x from G to atubular neighborhood U of a maximal torus T inside G . Over U we show that there is aMorita equivalent Dixmier-Douady bundle A U which has an especially simple structure: itsalgebra of continuous sections can be presented as a twisted crossed product algebra Π ⋉ τ C ( U ),where τ is the twist, Π is the integer lattice, and U is a Π-covering space of U . Applying toolsfrom KK-theory (a Green-Julg-type isomorphism and the analytic assembly map), we obtainan element in the K-theory of the group C ∗ algebra for T ⋉ Π τ . There is a map from the latterK-group into the space of formal characters R −∞ ( T ) for T . The image of K G ( G, A ) underthis composition is R −∞ ( T ) W aff − anti , ℓ , the subspace of formal characters that are alternatingunder the action of the affine Weyl group at level ℓ . For ℓ > h ∨ , the space of such charactersis canonically isomorphic to R k ( G ), k = ℓ − h ∨ , via the Weyl-Kac character formula.Our construction provides an elaboration of a remark made by Freed, Hopkins and Telemanin [20, Remark 3.5]. They comment that there ought to be an inverse map from the twistedK-theory of T to a suitable ‘representation ring’ for T ⋉ Π τ perhaps defined using C ∗ algebras,involving an analogue of ‘integration over t ’. Our ‘integration over t ’ map is the analyticassembly map.We study the specialization of our map to ‘D-cycles’ in the sense of Baum-Carey-Wang[6], which are an analogue of Baum-Douglas cycles in geometric K-homology [8]. A D-cyclefor K ( X, A ) is a 4-tuple ( M, E, Φ , S ) consisting of a compact Riemannian manifold M , aHermitian vector bundle E over M , a continuous map Φ : M → X , and a Morita morphism S : Cliff( T M ) Φ ∗ A . If A = C is trivial, S is equivalent to a spin-c structure on M , and werecover an ordinary Baum-Douglas cycle.In the case of a D-cycle ( M, E, Φ , S ) representing a class x ∈ K G ( G, A ), we prove that itsimage under our map is the T -equivariant L -index of a first-order elliptic operator on a Π-covering space of Φ − ( U ), where U ⊃ T is a tubular neighborhood of the chosen maximal torus T in G . Let us give a summary of the proof. Using ( M, E, Φ , S ) we construct an analytic cycle( H, ρ, D ) representing x : the Hilbert space H is the space of L sections of a smooth Hilbertbundle over M and D is a Dirac operator acting on smooth sections. Because the bundle hasinfinite dimensional fibres, D is not necessarily Fredholm, but the action of the C ∗ algebra C ( A ) (continuous sections of A ) along the fibres provides the needed analytic control to makethis a cycle. After passing to the Morita equivalent Dixmier-Douady bundle over U ⊃ T , thefibres are replaced with copies of L (Π) (tensored with a finite dimensional bundle); using acorrespondence that is well-known (for example in the context of Atiyah’s L -index theorem),the operator D then has an alternate interpretation as a Dirac-type operator on a Π-coveringspace Y of Φ − ( U ). Applying the analytic assembly map gives the T -equivariant L -index ofthis operator.The initial motivation for this work was to understand the relationship between twoapproaches—one via D -cycles for twisted K-homology of G [41] and the other via index theoryon non-compact manifolds [35, 36]—to defining a representation-theoretic ‘quantization’ of aHamiltonian LG -space. The construction of a suitable quantization has interesting applica-tions, for example to the Verlinde formula for moduli spaces of flat connections on Riemannsurfaces, cf. [40] for an overview. A corollary of our results is that the two approaches agreewith each other. Indeed for x represented by a D-cycle, the first-order elliptic operator on Y EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 3 mentioned above coincides with the operator studied in [36]. Our construction thus connectsthe index of this operator with the image of x in R k ( G ) under the Freed-Hopkins-Telemanisomorphism.Throughout the paper we have restricted ourselves to the special case that G is connected,simply connected and simple, but the methods likely generalize. We will fairly easily be able tocheck that the map I : K G ( G, A ) → R −∞ ( T ) W aff − anti ,ℓ that we construct is surjective. Withsome additional effort and a little topology, together with a known (and relatively easy) caseof the Baum-Connes conjecture, we could use the construction described here to prove a weakform of the Freed-Hopkins-Teleman theorem (that I is also injective modulo torsion at primesdividing the order of the Weyl group). We hope to return to these questions in the future.There is an overlap of some of our methods with interesting work of Doman Takata onHamiltonian LT -spaces [51, 52]. In particular, Takata also studies an assembly map into theK-theory of a twisted group C ∗ algebra of T × Π. Takata has built infinite dimensional analoguesof several well-known objects from index theory/non-commutative geometry in the setting ofHamiltonian LT -spaces. It would be interesting to explore further connections with his work.Sections 2 and 3 briefly introduce twisted K-homology, loop groups, and the Freed-Hopkins-Teleman theorem. Section 4 contains some results on twisted convolution algebras and gener-alized fixed-point algebras. In Section 5 we prove some basic facts about the C ∗ algebra of thesemi-direct product T ⋉ Π τ that plays a key role. In Section 6 we construct the map, denoted I , from K G ( G, A ) to R −∞ ( T ) W aff − anti , ℓ , and prove that it is inverse to the Freed-Hopkins-Teleman map. Section 7 studies the specialization of I to geometric cycles (D-cycles in thesense of Baum-Carey-Wang), and briefly describes the application to Hamiltonian loop groupspaces. For the reader’s convenience, we have included an appendix with proofs of a couple ofstandard (but not so easy to find) results used in Section 7. Acknowledgements.
I especially thank Eckhard Meinrenken and Yanli Song for interestingdiscussions about quantization of Hamiltonian LG -spaces over the past couple of years. Thework described in this paper is motivated by our joint work on spinor modules for Hamiltonian LG -spaces [35, 36]. I also thank Nigel Higson and Shintaro Nishikawa for helpful suggestionsand for answering several questions about KK-theory. Notation.
The C ∗ algebras of bounded (resp. compact) operators on a Hilbert space H willbe denoted B ( H ) (resp. K ( H )).If ( V, g ) is a finite dimensional real Euclidean vector space, Cliff( V ) denotes the complexClifford algebra of V , the Z -graded complex algebra generated in degree 1 by the elements v ∈ V subject to the relation v = k v k . For V a real Euclidean vector bundle over M , Cliff( V )denotes the bundle of algebras with fibres Cliff( V ) m = Cliff( V m ). On a Riemannian manifold M , we write Cl( M ) for the algebra of continuous sections of Cliff( T M ) vanishing at infinity.If K is a compact Lie group, Irr( K ) denotes the set of isomorphism classes of irreduciblerepresentations of K , and R ( K ) is the representation ring. The formal completion R −∞ ( K ) = Z Irr( K ) consists of formal infinite linear combinations of irreducibles π ∈ Irr( K ) with coefficientsin Z . When discussing a U (1) central extension Γ τ of a group Γ, we use the notation b γ to denotesome lift to Γ τ of an element γ ∈ Γ.Throughout G denotes a compact, connected, simply connected, simple Lie group with Liealgebra g . Let T ⊂ G be a maximal torus with Lie algebra t . We fix a positive Weyl chamber YIANNIS LOIZIDES t + , and let R + (resp. R − ) denote the positive (resp. negative) roots. The half sum of thepositive roots is denoted ρ , and h ∨ is the dual Coxeter number of G . Since G is simplyconnected, the integer lattice Π = ker(exp : t → T ) coincides with the coroot lattice. The dualΠ ∗ = Hom(Π , Z ) is the (real) weight lattice. There is a unique invariant inner product B on g , the basic inner product, with the property that the squared length of the short co-roots is2. We often use the basic inner product to identify g ≃ g ∗ , and we sometimes write B ♭ , B ♯ forthe musical isomorphisms when we want to emphasize this. The basic inner product has theproperty that B (Π , Π) ⊂ Z , and thus B ♭ (Π) ⊂ Π ∗ .2. Twisted K-homology
Here we give a brief introduction to the analytic description of twisted K-homology. Ourdiscussion is similar to [41, 39] where one can find further details. For further background onanalytic K-homology and KK-theory, see for example [24, 22, 27]. We also recall the definitionof ‘D-cycles’ due to Baum, Carey and Wang [6], which are a version of Baum-Douglas-typegeometric cycles [8] for twisted K-homology.Let X be a locally compact space. A Dixmier-Douady bundle over X is a locally trivialbundle of C ∗ algebras A → X , with typical fibre isomorphic to the compact operators K ( H )for a (separable) Hilbert space H , and structure group the projective unitary group P U ( H )with the strong operator topology. Restricting to a sufficiently small open U ⊂ X , A| U isisomorphic to K ( H ) for a bundle of Hilbert spaces H → U , but this need not be true globally.The notation A op denotes the Dixmier-Douady bundle obtained by taking the opposite algebrastructure on the fibres. The tensor product A ⊗ A of Dixmier-Douady bundles is again aDixmier-Douady bundle.A Morita morphism S : A A between Dixmier-Douady bundles over X is a bundle of A ⊗ A op0 modules S → X , locally modelled on the K ( H ) − K ( H ) bimodule K ( H , H ). Inthe special case A = C , S is called a Morita trivialization of A . Any two Morita morphisms A A are related by tensoring with a line bundle; if the line bundle is trivial, one says theMorita morphisms are .By a theorem of Dixmier and Douady [14], Morita isomorphism classes of Dixmier-Douadybundles are classified by a degree 3 integral cohomology class DD( A ) ∈ H ( X, Z ) known as the Dixmier-Douady class . The Dixmier-Douady class satisfiesDD( A op ) = − DD( A ) , DD( A ⊗ A ) = DD( A ) + DD( A ) . There are modest generalizations to the case where the fibres of A (resp. S ) carry Z -gradings; in this case A (resp. S ) is locally modelled on K ( H ) for a Z -graded Hilbert space H (resp. K ( H , H ) with H , H being Z -graded Hilbert spaces), and the Dixmier-Douadyclass DD( A ) ∈ H ( X, Z ) ⊕ H ( X, Z ). If X carries an action of a compact group G , one candefine G -equivariant Dixmier-Douady bundles, which are classified up to G -equivariant Moritamorphisms by classes in the analogous equivariant cohomology groups.The C ∗ algebraic definition of twisted K-theory goes back to Donovan-Karoubi [15] (in thecase of a torsion Dixmier-Douady class) and Rosenberg [47] (the general case); see also [5, 26].Let A be a G -equivariant Z -graded Dixmier-Douady bundle and C ( A ) the Z -graded G - C ∗ algebra of continuous sections of A vanishing at infinity. One defines the G -equivariant EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 5 A - twisted K-homology of X to be the G -equivariant analytic K-homology of this C ∗ algebra:K Gi ( X, A ) = KK iG ( C ( A ) , C ) , i = 0 , iG ( A, B ) is Kasparov’s KK-theory (cf. [27]). This definition is well known to beequivalent to Atiyah-Segal’s [5] description in terms of homotopy classes of continuous sectionsof bundles with typical fibre the Fredholm operators on a Hilbert space.
Remark . A Morita morphism A A defines an invertible element in the groupKK G ( C ( A ) , C ( A )), and hence an isomorphism between the corresponding twisted K-homology groups. Thus, the resulting groups depend only on the Dixmier-Douady class of A . Note however that there may be no canonical isomorphism; different Morita morphismscan lead to genuinely different maps. Any two Morita morphisms are related by tensoring with a Z -graded line bundle, hence the set of Morita morphisms is a torsor for H G ( X, Z ) × H G ( X, Z ). Example . An important example of a Z -graded Dixmier-Douady bundle is the Cliffordalgebra bundle Cliff( T M ) of a Riemannian manifold M . Kasparov’s fundamental class [ D ]is the class in the twisted K-homology group K ( M, Cliff(
T M )) = KK(Cl( M ) , C ) representedby the de Rham-Dirac operator D = d + d ∗ acting on smooth differential forms over M (cf. [27, Definition 4.2]). A Morita trivialization S : Cliff( T M ) C is the same thing as aspinor module for Cliff( T M ). S defines an invertible element [ S ] ∈ KK( C ( M ) , Cl( M )), andthe KK product [ S ] ⊗ Cl( M ) [ D ] ∈ KK( C ( M ) , C ) is the class represented by a spin-c Diracoperator for S . More generally, twisting D by a complex vector bundle E , one obtains a class[ D E ] ∈ KK(Cl( M ) , C ), and the KK product [ S ] ⊗ Cl( M ) [ D E ] is the class represented by theDirac operator coupled to E .2.1. Geometric twisted K-homology.
Baum, Carey and Wang [6] describe a ‘geometric’approach to twisted K-homology, in the spirit of Baum-Douglas geometric K-homology [8] (seealso [9]). Actually in [6] two types of cycles for twisted geometric K-homology are discussed:‘K-cycles’ versus ‘D-cycles’. The geometric K-homology groups defined by both types of cyclesadmit natural maps to the analytic K-homology group described above. In this paper we willonly discuss D-cycles, and only use the even case.
Definition 2.3. [6] Let A be a G -equivariant Z -graded Dixmier-Douady bundle over a locallyfinite G -CW complex X . An (even) D-cycle for ( X, A ) is a 4-tuple ( M, E, Φ , S ) where • M is an even-dimensional smooth closed G -manifold, with a G -invariant Riemannianmetric • Φ : M → X is a G -equivariant continuous map • E is a G -equivariant Hermitian vector bundle over M • S : Cliff( T M ) Φ ∗ A is a G -equivariant Morita morphism. Remark . The terminology ‘D-cycle’ comes from string theory. If M is orientable, theDixmier-Douady class of Cliff( T M ) is the third integral Stieffel-Whitney class W ( M ) (theobstruction to the existence of a spin-c structure on M ). The existence of S impliesΦ ∗ DD( A ) = W ( M ) , which is called the ‘Freed-Witten anomaly cancellation condition’ in string theory. YIANNIS LOIZIDES
The geometric twisted K-homology K G geo ,i ( X, A ) of X is the set of D-cycles modulo an equiv-alence relation analogous to Baum-Douglas geometric K-homology (generated by suitable ver-sions of ‘disjoint union equals direct sum’, ‘bordism’, and ‘bundle modification’), see [6]. Thereis a functorial map K G geo ,i ( X, A ) → K Gi ( X, A ) (1)which is straight-forward to describe at the level of cycles. We will only use the even case i = 0here; the odd case is similar. Let [ D E ] ∈ KK G (Cl( M ) , C ) be the class of the de Rham-Diracoperator on M , coupled to the vector bundle E . The pair (Φ , S ) defines a push-forward map(Φ , S ) ∗ : KK G (Cl( M ) , C ) → KK G ( C ( A ) , C )given as the composition of the Morita morphism Cliff( T M ) Φ ∗ A , with the map inducedby the ∗ -homomorphism Φ ∗ : C ( A ) → C (Φ ∗ A ) . The image of [(
M, E, Φ , S )] in K G ( X, A ) is the push-forward:(Φ , S ) ∗ [ D E ] . (2)The push-forward can alternately be expressed as a KK product[ S ] ⊗ [ D E ] (3)where [ S ] ∈ KK G ( C ( A ) , Cl( M )) is the class defined by the triple ( C ( S ) , Φ ∗ , Remark . A proof that the map (1) is an isomorphism has been announced by Baum,Joachim, Khorami and Schick [10], at least for the non-equivariant case.2.2.
Twisted K-homology of G . Let G be a compact, connected, simply connected, simpleLie group acting on itself by conjugation. Then H G ( G, Z ) ≃ Z , while H G ( G, Z ) = H G ( G, Z ) = H G ( G, Z ) = 0. There is a canonical generator of H G ( G, Z ); in de Rham cohomology, it isrepresented by the equivariant Cartan 3-form η G ( ξ ) = η − B ( θ L + θ R , ξ ) , η = B ( θ L , [ θ L , θ L ]) , where ξ ∈ g and θ L (resp. θ R ) denotes the left (resp. right) invariant Maurer-Cartan form.Thus G -equivariant Dixmier-Douady bundles A over G are classified up to Morita equivalenceby an integer ℓ ∈ Z , and moreover any two Morita morphisms are 2-isomorphic, see Remark2.1. Although we will not use it, it is known that the twisted K-homology group K G ( G, A )carries a ring structure; in this picture the ring structure originates from a canonical Moritamorphism A ⊠ A Mult ∗ A , where Mult : G × G → G is the group multiplication, cf. [19, 39].3. Loop groups and the Freed-Hopkins-Teleman Theorem
In this section we briefly introduce the loop group LG and its important class of projectivepositive energy representations, cf. [44]. We then recall the Freed-Hopkins-Teleman theorem,which relates loop groups to twisted K-homology.To obtain a Banach-Lie group, we take LG to consist of maps S = R / Z → G of some fixedSobolev level s > . The basic inner product defines a central extension of the Lie algebra L g by R , with cocycle c ( ξ , ξ ) = Z B ( ξ ( t ) , ξ ′ ( t )) dt. (4) EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 7
This extension integrates to a U (1) central extension LG bas of LG , that we will call the basiccentral extension . For G connected, simple, and simply connected, U (1) central extensions of LG are uniquely determined by their Lie algebra cocycle, which must be an integer multiple ofthe generator (4); thus U (1) central extensions are classified by Z , with LG bas correspondingto the generator 1 ∈ Z .For later reference, note that the loop group can be written as a semi-direct product LG = G ⋉ Ω G , where Ω G = { γ ∈ LG | γ (0) = γ (1) } is the based loop group, and G ֒ → LG identifies G with the constant loops in LG . Our assumptions on G imply that any U (1) central extensionof G is trivial, hence in particular the restriction of LG bas to the constant loops is trivial.Let T ⊂ G be a fixed maximal torus and let Π = ker(exp : t → T ) be the integral lattice. Theproduct group T × Π may be viewed as a subgroup of LG , where T is embedded as constantloops and Π as exponential loops: η ∈ Π corresponds to the loop s ∈ R / Z exp( sη ) ∈ T . Therestriction of the central extension LG bas to T × Π is a central extension1 → U (1) → T ⋉ Π bas → T × Π → . We discuss the subgroup T ⋉ Π bas ⊂ LG bas in detail in Section 5.3.1. Positive energy representations.
The loop group has a much-studied class of projec-tive representations known as positive energy representations, which have a detailed theoryparallel to the theory for compact groups cf. [25, 44]. Let S ⋉ LG denote the semi-directproduct constructed from the action of S on LG by rigid rotations. This action lifts to anaction on the basic central extension. A positive energy representation is a representation of LG bas which extends to a representation of the semi-direct product S ⋉ LG bas such that theweights of S are bounded below. One can always tensor a positive energy representation by a1-dimensional representation of S , hence one often normalizes positive energy representationsby requiring that the minimal S weight is 0, and we always assume this.For an irreducible positive energy representation, the central circle of LG bas acts by a fixedweight k ≥ level . There are finitely many irreducible positive energy representationsat any fixed level, parametrized by the ‘level k dominant weights’: weights λ ∈ Π ∗ ∩ t ∗ + satisfying B ( λ, θ ) ≤ k , where θ ∈ R + is the highest root of g . Equivalently the level k weights Π ∗ k =Π ∗ ∩ k a , where a ⊂ t + is the fundamental alcove, which we identify with a subset of t ∗ usingthe basic inner product.Let R k ( G ) denote the free abelian group of rank ∗ k ) generated by Z -linear combinationsof the level k irreducible positive energy representations. There is a canonical isomorphism(‘holomorphic induction’, cf. [19]) R k ( G ) ≃ R ( G ) /I k ( G )where R ( G ) is the representation ring of G and I k ( G ) is the Verlinde ideal consisting of char-acters vanishing on the conjugacy classes of the elementsexp (cid:16) ξ + ρk + h ∨ (cid:17) , ξ ∈ Π ∗ k . In particular R k ( G ) is a ring, known as the level k Verlinde ring .There is an alternate description of R k ( G ) that will be crucial for us later on; this descriptionplays a significant role in the proof of the Freed-Hopkins-Teleman Theorem as well. An element YIANNIS LOIZIDES of R k ( G ) is uniquely determined by its multiplicity function , a map m : Π ∗ k → Z . It is known that Π ∗ k is precisely the set of weights contained in the interior of the shifted, scaledalcove ( k + h ∨ ) a − ρ . The latter is a fundamental domain for the ρ -shifted level ( k + h ∨ ) actionof the affine Weyl group W aff = W ⋉ Π, given by w • k +h ∨ ξ = ( w, η ) • k +h ∨ ξ = w ( ξ + ρ ) − ρ + ( k + h ∨ ) η, ξ ∈ t ∗ , w ∈ W, η ∈ Π . (5)Thus, m has a unique extension to a map m : Π ∗ → Z which is alternating under (5), i.e. m ( w • k +h ∨ ξ ) = ( − l ( w ) m ( ξ ) , where l ( w ) is the length of the affine Weyl group element w . The extension of m vanishes onthe boundary of the fundamental domain ( k + h ∨ ) a − ρ . This defines an isomorphism of abeliangroups R k ( G ) ∼ −→ R −∞ ( T ) W aff − anti , ( k +h ∨ ) (6)where the right hand side denotes the formal characters of T which are alternating under theaction (5).That (6) is an isomorphism can be deduced more or less immediately from the Weyl-Kaccharacter formula (cf. [25, 44]). A positive energy representation has a formal character χ ∈ R −∞ ( S × T ) given by a formula analogous to the Weyl character formula for compactLie groups, but with the numerator and denominator both being formal infinite expressions.As in the Weyl character formula, the denominator ∆ is a universal expression (the same forany χ ∈ R k ( G )); multiplying χ by ∆ and then restricting to q = 1 ∈ S , one obtains anelement ( χ · ∆) | q =1 ∈ R −∞ ( T ) W aff − anti , ( k +h ∨ ) , and this correspondence is one-one.3.2. Dixmier-Douady bundles from positive energy representations.
Loop groups areclosely related to twisted K-theory of G . One manifestation of this is that positive energyrepresentations can be used to construct Dixmier-Douady bundles over G .Let ℓ > LG τ denote the central extension of LG corresponding to ℓ times the basicinner product ( ℓ = 1 corresponds to LG bas ). Let V be a level ℓ positive energy representation,or in other words, a positive energy representation of LG τ such that the central circle acts withweight 1. The dual space V ∗ carries a negative energy representation such that the centralcircle acts with weight −
1. Let
P G denote the space of quasi-periodic paths in G of Sobolevlevel s > , that is, P G is the space of paths γ : R → G such that γ ( t ) γ ( t + 1) − is a fixedelement of G , independent of t ∈ R . The group LG × G acts on P G , with LG acting by rightmultiplication, and G by left multiplication (cf. [35] for further discussion). The map q : γ ∈ P G γ ( t ) γ ( t + 1) − ∈ G makes P G into a G -equivariant principal LG -bundle over G . The adjoint action of LG τ on thealgebra of compact operators K ( V ∗ ) descends to an action of LG , and the associated bundle A = P G × LG K ( V ∗ ) (7)is a Dixmier-Douady bundle over G such that DD( A ) = ℓ ∈ Z ≃ H G ( G, Z ), cf. [39]. EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 9
The Freed-Hopkins-Teleman theorem.
The following is a special case (for G con-nected, simply connected, simple) of the Freed-Hopkins-Teleman theorem. Theorem 3.1 (Freed-Hopkins-Teleman [19, 21, 20]) . Let k > , and let h ∨ be the dual Coxeternumber of G . Let A be a G -equivariant Dixmier-Douady bundle over G with DD( A ) = k +h ∨ ∈ Z ≃ H G ( G, Z ) . The group K G ( G, A ) = 0 , and there is an isomorphism of rings R k ( G ) ≃ K G ( G, A ) . Let ι : { e } ֒ → G be the inclusion of the identity element in G . Consider the model (7)for A . The Hilbert space V ∗ gives a (canonical) G -equivariant Morita trivialization of ι ∗ A .Freed-Hopkins-Teleman prove that their isomorphism R k ( G ) → K G ( G, A ) moreover fits into acommutative diagram R ( G ) −−−−→ R k ( G ) ≃ y ≃ y K G (pt) ( ι,V ∗ ) −−−−→ K G ( G, A ) (8)where the top horizontal arrow is the quotient map and the bottom horizontal arrow is inducedby the evaluation map ι ∗ : C ( A ) → A| e composed with the Morita trivialization V ∗ : A| e C .4. Crossed products and twisted K-homology
In this section we describe some general facts involving crossed product algebras, centralextensions, and generalized fixed-point algebras. Throughout this section Γ, S , N are locallycompact, second countable topological groups equipped with left Haar measure, and A is aseparable C ∗ algebra.4.1. Twisted crossed-products.
Let Γ be a locally compact group with left invariant Haarmeasure, and let Γ τ be a U (1)-central extension:1 → U (1) → Γ τ → Γ → . Normalize Haar measure on Γ τ such that the integral of a function over Γ τ is given by firstaveraging over U (1) (using normalized Haar measure) followed by integration over Γ. A choiceof section Γ → Γ τ is not needed. In detail, for f ∈ C c (Γ τ ) let¯ f ( b γ ) = Z U (1) f ( z b γ ) dz. (9)Then ¯ f is a U (1)-invariant function on b Γ so descends to a function on Γ, and Z Γ τ f ( b γ ) d b γ = Z Γ ¯ f ( γ ) dγ. (10)Let A be a Γ- C ∗ algebra. Note that A can be regarded as a Γ τ - C ∗ algebra such that thecentral circle in Γ τ acts trivially. The (maximal) crossed product algebra Γ τ ⋉ A = C ∗ (Γ τ , A )(we use both notations interchangeably) decomposes into a direct sum of its homogeneousideals Γ τ ⋉ A = M n ∈ Z (Γ τ ⋉ A ) ( n ) (11) where (Γ τ ⋉ A ) ( n ) denotes the norm closure (in the maximal crossed product algebra Γ τ ⋉ A )of the set of compactly supported functions a ∈ C c (Γ τ , A ) satisfying a ( z − b γ ) = z n a ( b γ ) , z ∈ U (1) , b γ ∈ Γ τ . There is a ∗ -homomorphism from C ∗ ( U (1)) into the multiplier algebra M (Γ τ ⋉ A ) (cf. [13,II.10.3.10-12]) making Γ τ ⋉ A into a C ∗ ( U (1)) = C ( Z )-algebra, and the ideals (Γ τ ⋉ A ) ( n ) are thefibres. The decomposition (11) is also not difficult to prove directly. A short calculation using(9) shows that the (Γ τ ⋉ A ) ( n ) are 2-sided ideals, and hence one has a ∗ -homomorphism fromthe right hand side of (11) to Γ τ ⋉ A . One also has a ∗ -homomorphism in the opposite direction,given by ‘taking Fourier coefficients’. For further details see for example [31, Proposition 3.2]or [51]. Definition 4.1.
We define the τ - twisted crossed product algebra Γ ⋉ τ A to be the idealΓ ⋉ τ A = (Γ τ ⋉ A ) (1) . The special case A = C gives the twisted group C ∗ algebra C ∗ τ (Γ) = C ∗ (Γ τ ) (1) . Remark . One often sees the twisted crossed-product algebra defined in terms of a cocyclefor the central extension, cf. [38]. One can translate to this definition by choosing a sectionΓ → Γ τ . One reason we take the approach above is that later on we will consider the actionof a second group S (cid:8) Γ ⋉ τ A , and it seems slightly awkward to describe this in terms of asection Γ → Γ τ ; for example, it is not clear to us that one can find an S -invariant section.The twisted crossed product algebra Γ ⋉ τ A has the important universal property that non-degenerate ∗ -representations of Γ ⋉ τ A are in 1-1 correspondence with covariant pairs ( π A , π τ Γ ),where π A is a ∗ -representation of A , π τ Γ is representation of Γ τ such that the central circle actswith weight 1 (a τ -projective representation of Γ), and π τ Γ ( b γ ) π A ( a ) π τ Γ ( b γ ) − = π A ( b γ · a ) (12)for all b γ ∈ Γ τ , a ∈ A .The space L (Γ τ ) splits into an ℓ -direct sum of its homogeneous subspaces L (Γ τ ) = M n ∈ Z L (Γ τ ) ( n ) , (13)where L (Γ τ ) ( n ) denotes the subspace of L (Γ τ ) consisting of functions f ∈ L (Γ τ ) satisfying f ( z − b γ ) = z n f ( b γ ) , z ∈ U (1) , b γ ∈ Γ τ . Recall the left and right regular representations of Γ τ on L (Γ τ ) are given, respectively, by λ ( b γ ) f ( b γ ) = f ( b γ − b γ ) , ρ ( b γ ) f ( b γ ) = f ( b γ b γ ) . Both actions preserve the decomposition (13).
Definition 4.3.
The left τ - twisted regular representation of Γ is the restriction of the leftregular representation of Γ τ on L (Γ τ ) to the subspace L τ (Γ) := L (Γ τ ) (1) . EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 11
The restriction of the right regular representation to L τ (Γ) is the right ( − τ )- twisted regularrepresentation of Γ. (Note that under the right regular representation, the central circle of Γ τ acts on L τ (Γ) with weight − Dixmier-Douady bundles from crossed-products.
Let X be a locally compact Haus-dorff space with a continuous proper action of a locally compact group Γ. The quotient X/ Γequipped with the quotient topology is then also a locally compact Hausdorff space. Let Γ acton L (Γ) by right translation, and on K := K ( L (Γ)) by the adjoint action. Define the algebraof sections of a field of C ∗ -algebras over X/ Γ, suggestively denoted C ( X × Γ K ), consisting ofΓ-equivariant continuous maps X → K vanishing at infinity in X/ Γ. The algebra C ( X × Γ K )is an example of a generalized fixed-point algebra .The following result is attributed to Rieffel (for example [45, Proposition 4.3], [46]); seeespecially [18, Corollary 2.11] for a statement formulated in the same terms used here. Anotherreference is [31, Proposition 4.3] where a quite general statement appears for twisted convolutionalgebras of locally compact proper groupoids. Proposition 4.4.
Let X be a locally compact Hausdorff space with a continuous proper actionof a locally compact group Γ , and let K = K ( L (Γ)) . Then Γ ⋉ C ( X ) ≃ C ( X × Γ K ) . Remark . We mention briefly how a map Γ ⋉ C ( X ) → C ( X × Γ K ) is constructed. Usingconventions as in [27, Section 3.7], a function a ∈ C c (Γ , C c ( X )) is sent to the Γ-equivariantfamily K a ( x ) ∈ K , x ∈ X of compact operators defined by the family of integral kernels k a ( γ , γ ; x ) = µ ( γ ) − a ( γ γ − ; γ x ) (14)where µ : Γ → R > is the modular homomorphism of Γ. Remark . Proposition 4.4 can be viewed as a generalization of the Stone-von Neumanntheorem (obtained from the special case Γ = R acting on X = R by translations). Moregenerally for X = Γ, Proposition 4.4 specializes to a well-known isomorphismΓ ⋉ C (Γ) ∼ −→ K ( L (Γ)) . (15)Using equation (14) one verifies that the induced map on multiplier algebras sends C (Γ) tomultiplication operators and Γ to the left regular representation.If the action of Γ on X is free, then X → X/ Γ is a principal Γ-bundle, and the generalizedfixed-point algebra is the algebra of continuous sections vanishing at infinity of the associatedbundle A = X × Γ K . (16)This is a Dixmier-Douady bundle, with typical fibre K ( L (Γ)). In fact A is Morita trivial withMorita trivialization X × Γ L (Γ).To obtain something more interesting from the construction (16), we adjust it slightly intwo ways. First we consider the equivariant situation, where a second group S acts on X and L (Γ). It may happen that the Morita trivialization X × Γ L (Γ) is not S -equivariant. Second,we replace Γ ⋉ C ( X ) with a twisted crossed product algebra, as in Definition 4.1. This willbe important later on, when central extensions of the loop group come into the picture. Consider a semi-direct product S ⋉ Γ, where S , Γ are locally compact groups, and assumethe S action preserves Haar measure on Γ. Let Γ τ be a U (1)-central extension, and assumethe action of S on Γ lifts to an action on Γ τ , so that we have a U (1)-central extension1 → U (1) → S ⋉ Γ τ → S ⋉ Γ → . The right ( − τ )-twisted regular representation ( L τ (Γ) , ρ ) (Definition 4.3) extends to a repre-sentation of S ⋉ Γ τ (such that the central circle acts with weight −
1) according to ρ ( s, b γ ) f ( b γ ) = f ( s − b γ s b γ ) . (17)The adjoint action Ad( ρ ) on K = K ( L τ (Γ)) descends to an action of S ⋉ Γ. Let X be a locallycompact S ⋉ Γ-space, such that the action of Γ on X is proper. The generalized fixed pointalgebra C ( X × Γ K ) is an S - C ∗ algebra. Remark . For later reference note that the left τ -twisted regular representation ( L τ (Γ) , λ )(Definition 4.3) also extends to a representation of S ⋉ Γ τ (such that the central circle actswith weight 1) according to λ ( s, b γ ) f ( b γ ) = f ( b γ − s − b γ s ) . If A is a ( S ⋉ Γ)- C ∗ algebra, the twisted crossed product Γ ⋉ τ A is an S - C ∗ algebra, withthe S action being the continuous extension of the S action on C c (Γ τ , A ) given by( s · a )( b γ ) = s.a ( s − b γs ) . (18)This applies in particular to the ( S ⋉ Γ)- C ∗ algebra C ( X ), and one has the following variationof Proposition 4.4. Proposition 4.8.
Consider a semi-direct product S ⋉ Γ , where S , Γ are locally compact groups,and the S -action preserves Haar measure on Γ . Let Γ τ be a U (1) -central extension, and assumethe action of S on Γ lifts to an action on Γ τ . Let ( L τ (Γ) , ρ ) be the right ( − τ ) -twisted regularrepresentation (Definition 4.3), extended to a representation of S ⋉ Γ τ as in equation (17) , andlet S ⋉ Γ τ act on K = K ( L τ (Γ)) by the adjoint action Ad( ρ ) . Let X be a locally compact S ⋉ Γ space, such that the Γ action is proper. There is an isomorphism of S - C ∗ algebras Γ ⋉ τ C ( X ) ≃ C ( X × Γ K ) . Proof.
This follows in a straight-forward manner from Proposition 4.4 applied to Γ τ . Theaction of Γ on X induces a proper action of Γ τ on X with the central circle acting trivially.Applying Proposition 4.4 to Γ τ ,Γ τ ⋉ C ( X ) ≃ C (cid:0) X × Γ τ K ( L (Γ τ )) (cid:1) = C (cid:0) X × Γ K ( L (Γ τ )) U (1) (cid:1) , (19)where for the second equality we use the fact that the central circle acts trivially on X . Thealgebra on the left hand side of (19) splits into a direct sum of its homogeneous idealsΓ τ ⋉ C ( X ) = M n ∈ Z (Γ τ ⋉ C ( X )) ( n ) . Decompose L (Γ τ ) into isotypic components for the action of the central circle, as in (13): L (Γ τ ) = M n ∈ Z L (Γ τ ) ( n ) . (20) EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 13
The subalgebra K ( L (Γ τ )) U (1) ⊂ K ( L (Γ τ )) is the set of compact operators preserving thedecomposition (20); hence K ( L (Γ τ )) U (1) = M n ∈ Z K ( L (Γ τ ) ( n ) ) . (21)We claim the isomorphism (19) restricts to an isomorphism(Γ τ ⋉ C ( X )) ( n ) → C (cid:0) X × Γ K ( L (Γ τ ) ( n ) ) (cid:1) . To see this let a ∈ C c (Γ τ ⋉ C c ( X )) ( n ) , and let K a be the corresponding family of operatorsdefined by the integral kernels k a in (14). We suppress the basepoint x ∈ X from the notationas it plays no role in the argument. The homogeneity of a (and U (1) invariance of µ ) implies(see (14)) k a ( b γ , z − b γ ) = z − n k a ( b γ , b γ ), z ∈ U (1). For f ∈ L (Γ τ ),( K a f )( b γ ) = Z Γ τ k a ( b γ , b γ ) f ( b γ ) d b γ . According to (9), (10) the integral over Γ τ can be carried out by first averaging with respectto the U (1) action, and then integrating over Γ. Note that Z U (1) k a ( b γ , z − b γ ) f ( z − b γ ) dz = k a ( b γ , b γ ) Z U (1) z − n f ( z − b γ ) dz. The integral over z ∈ U (1) gives the projection to the ( n )-isotypical component, hence K a iscontained in the ideal K ( L (Γ τ ) ( n ) ). In particular for n = 1Γ ⋉ τ C ( X ) ≃ C (cid:0) X × Γ K ( L (Γ τ ) (1) ) (cid:1) = C (cid:0) X × Γ K ( L τ (Γ)) (cid:1) . (cid:3) Assuming Γ acts on X freely, we can form the associated S -equivariant Dixmier-Douadybundle over X/ Γ A = X × Γ K , and Γ ⋉ τ C ( X ) ≃ C ( A ) as S - C ∗ algebras.4.3. An example: a Dixmier-Douady bundle A T over T . Let LG τ denote a U (1) centralextension of the loop group, corresponding to 0 < ℓ ∈ Z times the generator LG bas . Let T ⋉ Π τ denote the corresponding U (1) central extension of the subgroup T × Π (see Section 3).Carrying out the construction of the previous section with S = T , Γ τ = Π τ , X = t we obtaina Dixmier-Douady bundle over T = t / Π: Definition 4.9.
Let A T be the T -equivariant associated bundle A T = t × Π K (cid:0) L τ (Π) (cid:1) → t / Π = T. (22) A T is a T -equivariant Dixmier-Douady bundle over T .Recall the G -equivariant Dixmier-Douady bundle A described in Section 3.2: A = P G × LG K ( V ∗ ) → G where V is a level ℓ positive energy representation. The map t ֒ → P G, ξ γ ξ where γ ξ ( s ) = exp( sξ ) , s ∈ R / Z embeds t into P G , Π-equivariantly. Restricting to t ⊂ P G in (37), we obtain a Dixmier-Douadybundle A| T = t × Π K ( V ∗ ) , (23)over the maximal torus. The central circle in T ⋉ Π τ acts on both L τ (Π), V ∗ with weight − L τ (Π) we use the right regular representation ρ in Definition 4.3), hence thediagonal action of T ⋉ Π τ on the tensor product L τ (Π) ⊗ V descends to an action of T × Π. Define E = t × Π (cid:0) L τ (Π) ⊗ V (cid:1) , (24)a bundle of Hilbert spaces over T . By (22) and (37), E defines a T -equivariant Morita morphism A| T A T .4.4. A Green-Julg isomorphism.
For a compact group K , the Green-Julg theorem statesthat the K -equivariant K-theory of a K - C ∗ algebra A is isomorphic to the K-theory of thecrossed-product algebra K ⋉ A . There is a ‘dual’ version of the Green-Julg theorem (cf.[12, Theorem 20.2.7(b)]) which applies to discrete groups instead of compact groups and K-homology instead of K-theory. Proposition 4.10.
Let Γ be a discrete group, and let A be a Γ - C ∗ algebra. Then KK Γ ( A, C ) ≃ KK(Γ ⋉ A, C ) . More generally, suppose a locally compact group S acts on Γ preserving Haar measure. If A isan S ⋉ Γ - C ∗ algebra then KK S ⋉ Γ ( A, C ) ≃ KK S (Γ ⋉ A, C ) , where Γ ⋉ A is equipped with the S -action in equation (18) . The isomorphism is simple to describe at the level of cycles. Let (
H, π, F ) be a cycle repre-senting a class in KK(Γ ⋉ A, C ). We may assume π is non-degenerate. The universal propertyof the crossed product Γ ⋉ A guarantees π comes from a covariant pair ( π A , π Γ ). For the triple( H, π A , F ) to represent a class in KK Γ ( A, C ), one needs the operators π A ( a )(1 − F ) , [ F, π A ( a )] , π A ( a )(Ad π Γ ( γ ) F − F ) (25)to be compact, for all a ∈ A , γ ∈ Γ. The assumption that Γ is discrete means that A is a sub-algebra of Γ ⋉ A , so π A is simply the restriction of π to A , and the first two operators in(25) are compact. For the last operator, note that π A ( a )(Ad π Γ ( γ ) F − F ) = [ F, π ( a )] − [ F, π ( a ) π Γ ( γ )] π Γ ( γ ) − . (26)The operator π ( a ) π Γ ( γ ) ∈ π (Γ ⋉ A ), hence the compactness of both terms follows because( H, π, F ) is a cycle.The inverse map is similar: a triple (
H, π A , F ) representing a class in KK Γ ( A, C ) is sent tothe triple ( H, π, F ), where π : Γ ⋉ A → B ( H ) is the representation induced by the covariant pair( π A , π Γ ). The crossed product π (Γ ⋉ A ) contains a dense sub-algebra consisting of finite linearcombinations of operators of the form π ( a ) π Γ ( γ ) = π A ( a ) π Γ ( γ ). The operator π A ( a ) π Γ ( γ )( F − EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 15
1) = π Γ ( γ ) π A ( γ − · a )( F −
1) is compact, while the commutator [
F, π A ( a ) π Γ ( γ )] is compactusing (26) (multiply both sides by π Γ ( γ ) on the right).The maps are well-defined on homotopy classes because one can apply the same maps tocycles for the pair ( A, C ([0 , ⋉ A, C ([0 , Definition 4.11.
Let N be a locally compact group with U (1) central extension N τ . Let A , B be N - C ∗ algebras (trivial U (1) action). For n ∈ Z defineKK N τ ( A, B ) ( n ) to be the direct summand of KK N τ ( A, B ) generated by cycles where the central circle of N τ acts with weight n . Proposition 4.12.
Consider a semi-direct product N = S ⋉ Γ , where S , Γ are locally compactgroups and Γ is discrete. Let Γ τ be a U (1) -central extension, and assume the action of S on Γ lifts to an action on Γ τ . Let A be an S ⋉ Γ - C ∗ algebra. The twisted crossed product Γ ⋉ τ A isan S - C ∗ algebra with action given by (18) , and KK S ⋉ Γ τ ( A, C ) (1) ≃ KK S (Γ ⋉ τ A, C ) . Proof.
Let (
H, π, F, π S ) represent a class in KK S (Γ ⋉ τ A, C ). We may assume π is non-degenerate. The universal property of Γ ⋉ τ A implies that there is a covariant pair ( π A , π Γ τ ).Define π S ⋉ Γ τ ( s, b γ ) = π S ( s ) π Γ τ ( b γ ). At the level of cycles, the map sends ( H, π, F, π S ) to( H, π A , F, π S ⋉ Γ τ ).We first check that π S ⋉ Γ τ is indeed a representation of S ⋉ Γ τ . The action of S on Γ ⋉ τ A extends to an action on the multiplier algebra M (Γ ⋉ τ A ). By non-degeneracy the representation π of Γ ⋉ τ A extends to M (Γ ⋉ τ A ), and one obtains a covariant pair extending ( π, π S ). For b γ ∈ Γ τ , the function u b γ ( b γ ′ ) = ( z if b γ ′ = z − b γ, z ∈ U (1)0 elselies in M (Γ ⋉ τ A ) and satisfies π ( u b γ ) = π τ Γ ( b γ ), u s b γs − ( b γ ′ ) = u b γ ( s − b γ ′ s ). By (18), π S ( s ) π τ Γ ( b γ ) π S ( s ) − = π S ( s ) π ( u b γ ) π S ( s ) − = π ( s · u b γ ) = π ( u s b γs − ) = π τ Γ ( s b γs − ) . (27)Equation (27) implies that π S ⋉ Γ τ is a representation of S ⋉ Γ τ .The algebra A can be regarded as a sub-algebra of Γ ⋉ τ A , via the embedding a e a , where e a ( b γ ) = ( za if b γ = z − Γ τ , z ∈ U (1)0 elseand π A ( a ) = π ( e a ). The argument that ( H, π A , F ) represents a class in KK S ⋉ Γ τ ( A, C ) is thensimilar to Proposition 4.10. For example, (26) now reads π A ( a )(Ad π S ⋉ Γ τ ( s, b γ ) F − F ) = [ F, π ( e a )] + π ( e a ) π τ Γ ( b γ )(Ad π S ( s ) F − F ) π τ Γ ( b γ ) − − [ F, π ( e a ) π τ Γ ( b γ )] π τ Γ ( b γ ) − (we have used (27)). Note π ( e a ) π τ Γ ( b γ ) ∈ π (Γ ⋉ τ A ), hence compactness of all three terms followsbecause ( H, π, F ) is a cycle.In the reverse direction, let (
H, π A , F, π S ⋉ Γ τ ) represent a class in KK S ⋉ Γ τ ( A, C ) (1) , and let π τ Γ (resp. π S ) be the restriction of π S ⋉ Γ τ to Γ τ (resp. S ). The representations ( π A , π τ Γ ) form a covariant pair as in (12), and the map sends ( H, π A , F, π S ⋉ Γ τ ) to ( H, π, F, π S ) where π is therepresentation of Γ ⋉ τ A guaranteed by the universal property. One checks that the result is acycle similar to before.The maps are well-defined on homotopy classes because one may apply the same maps tocycles for ( A, C ([0 , ⋉ τ A, C ([0 , (cid:3) The analytic assembly map.
Let X be a locally compact space with a proper actionof a locally compact group N . If the action of N is cocompact , i.e. X/N is compact, then thereis a map µ N : K N ( X ) = KK N ( C ( X ) , C ) → KK( C , C ∗ ( N )) = K ( C ∗ ( N )) , known as the analytic assembly map . If N is compact, the analytic assembly map is just theequivariant index: µ N ([( H, ρ, F )]) = [ker( F + )] − [ker( F − )] ∈ K ( C ∗ ( N )) ≃ R ( N ) . For non-compact N , the definition of the assembly map is more involved. We give a briefdescription here and refer the reader to e.g. [7], [43, Section 2], [17, Section 4.2], [28] fordetails.Let ( H, ρ, F ) be a cycle representing a class [ F ] ∈ KK N ( C ( X ) , C ). Assume the operator F is properly supported , in the sense that for any f ∈ C c ( X ) one can find an h ∈ C c ( X ) suchthat ρ ( h ) F ρ ( f ) = F ρ ( f ) (this can always be achieved by perturbing F , cf. [7, Section 3]). Todefine µ N , the first step is to define a C c ( N )-valued inner product ( − , − ) N on the subspace ρ ( C c ( X )) H ⊂ H , by ( f , f ) N ( n ) = ( f , n · f ) L . Complete ρ ( C c ( X )) H in the norm k f k N = k ( f, f ) N k / C ∗ ( N ) , where k − k C ∗ ( N ) denotes the normof the C ∗ algebra C ∗ ( N ), to obtain a Hilbert C ∗ ( N )-module H . Then F acts on ρ ( C c ( X )) H (here use that F is properly supported) and extends to an adjointable operator F on H . Thepair ( H , F ) represents a class in K ( C ∗ ( N )), and µ N ([ F ]) = [( H , F )] ∈ K ( C ∗ ( N )) . Since F commutes with the C ∗ ( N ) action, ker( F ± ) are C ∗ ( N )-modules, but unfortunately ingeneral they need not be finitely generated and projective, so that ‘[ker( F + )] − [ker( F − )]’ is nota K-theory class. If the range of F is closed and ker( F ± ) are finitely generated and projective,then indeed µ N ([ F ]) = [ker( F + )] − [ker( F − )] (cf. [22, Proposition 3.27]); more generally it isnecessary to perturb F to obtain such a description.There is another description of the analytic assembly map due to Kasparov that we brieflyrecall; see for example [17, Section 4.2] for a recent review, and [43, Section 2.4] for a discussionof the relation between the two descriptions of µ N (at least for N discrete). As the actionof N on X is cocompact, one can find a continuous compactly supported ‘cut-off function’ c : X → [0 , ∞ ) such that for all x ∈ X , Z N c ( n − .x ) = 1 . Define p c : G × X → [0 , ∞ ) by p c ( n, x ) = µ ( n ) − / c ( n − .x ) c ( x ) , EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 17 where µ is the modular homomorphism of N . The function p c defines a self-adjoint projectionin G ⋉ C ( X ), and hence an element [ c ] ∈ KK( C , N ⋉ C ( X )). Kasparov’s definition of theassembly map is as a Kasparov product µ N ([ F ]) = [ c ] ⊗ N ⋉ C ( X ) j N ([ F ]) , where j N : KK N ( C ( X ) , C ) → KK( N ⋉ C ( X ) , C ∗ ( N )) is the descent homomorphism.5. The group T ⋉ Π bas . In this section we collect results about the group T ⋉ Π bas and the K-theory of its group C ∗ algebra. For another discussion of the K-theory of C ∗ ( T ⋉ Π bas ) see [51]. Throughout G isassumed to be connected, simply connected, simple. Let T ⊂ G be a fixed maximal torus andwe identify t × t ∗ with the basic inner product, and hence Π is identified with a sublattice ofΠ ∗ .5.1. The group Π bas . Let Π bas denote the restriction to Π ⊂ LG of the basic central extension LG bas of the loop group. We give an explicit 2-cocycle σ for Π bas . Recall that the cocycle of a U (1) central extension associated to a splitting η ∈ Π b η ∈ Π bas is the function σ : Π × Π → U (1) defined by the equation b η b η = σ ( η , η ) d η η . (The group operation in Π is written multiplicatively.)Let β , ..., β r ∈ Π be a lattice basis for Π. It is known [44, Proposition 4.8.1], [30, Theorem3.2.1] that one can choose lifts b β , ..., b β r ∈ Π bas such that b β i b β j b β − i b β − j = ( − B ( β i ,β j ) , (28)where B is the basic inner product. For η = P n i β i ∈ Π let b η = b β n · · · b β n r r . (29)Define a bilinear map ǫ : Π × Π → Z by ǫ ( β i , β j ) = ( B ( β i , β j ) if i > j i ≤ j and extend bilinearly. Proposition 5.1.
The cocycle associated to the splitting (29) is σ ( η , η ) = ( − ǫ ( η ,η ) , η , η ∈ Π . Remark . The function ( − ǫ is the ‘off-diagonal’ part of what Kac [25, Section 7.8] callsan asymmetry function . Proof. If i > j then using (28) we have b β i b β j = ( − B ( β i ,β j ) b β j b β i = ( − B ( β i ,β j ) d β i β j , while if i ≤ j then b β i b β j = d β i β j . This verifies σ ( β i , β j ) = ( − ǫ ( β i ,β j ) for i, j = 1 , ..., r . On the other hand, using the definition of the lift (29) and the commutationrelation (28), one sees that σ is bimultiplicative: σ ( η + η , η ) = σ ( η , η ) σ ( η , η ) , σ ( η, η + η ) = σ ( η, η ) σ ( η, η ) . (cid:3) The group T ⋉ Π bas . Define a group homomorphism κ : Π → Hom(
T, U (1)) = Π ∗ , κ η ( t ) = t − B ♭ ( η ) . (30)It is known (cf. [21, Section 2.2], [44]) that in the subgroup T ⋉ Π bas ⊂ LG bas , elements t ∈ T and η ∈ Π bas satisfy the commutation relation b ηt b η − t − = κ η ( t ) . Moreover the data ( σ, κ ) determine the group T ⋉ Π bas (up to isomorphism). Let T ⋉ Π triv denote the analogous group defined by the data (1 , κ ), i.e. Π triv = Π × U (1) is the trivial centralextension, and the commutator map for T , Π triv is the same κ defined in (30). In detail, if weuse the section Π → Π bas defined in (29) to view Π bas (topologically) as a product Π × U (1),then the group multiplication in T ⋉ Π bas is( t , η , z )( t , η , z ) = ( t t , η + η , κ η ( t ) σ ( η , η ) z z ) (31)while in T ⋉ Π triv the group multiplication is( t , η , z )( t , η , z ) = ( t t , η + η , κ η ( t ) z z ) . (32)As we saw above, in general Π bas is not isomorphic to Π triv (Π bas need not be abelian).Perhaps surprisingly, the distinction between Π bas , Π triv disappears after taking semi-directproduct with T . Proposition 5.3.
The groups T ⋉ Π bas , T ⋉ Π triv are (non-canonically) isomorphic.Proof. We will show that the additional sign σ ( η , η ) can be absorbed into the phase κ η ( t ),by choosing an appropriate identification T ⋉ Π bas → T ⋉ Π triv .For η ∈ Π, define η ǫ = B ♯ ( ǫ ( − , η )) ∈ t , where here one views the contraction ǫ ( − , η ) as an element of t ∗ , and then uses B ♯ to convertthis to an element of t . The image of the map η ∈ Π η ǫ ∈ t is contained in B ♯ (Π ∗ ). Byconstruction exp( η ǫ ) B ♭ ( µ ) = e π i ǫ ( µ,η ) = σ ( µ, η ) , η, µ ∈ Π . (33)Define Ψ : T ⋉ Π bas → T ⋉ Π triv , Ψ( t, η, z ) = ( t exp( η ǫ ) , η, z ) . A short calculation using (33) shows that Ψ is a group homomorphism. (cid:3)
EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 19
The C ∗ algebra of T ⋉ Π bas . Using Proposition 5.3, T ⋉ Π bas ≃ T ⋉ Π triv . There is anobvious isomorphism ( t, η, z ) ∈ T ⋉ Π triv ( t, z, η ) ∈ T triv ⋊ Π, where T triv = T × U (1) is thetrivial central extension.If G ⋉ G is a semi-direct product of locally compact groups, then there is an isomorphism C ∗ ( G ⋉ G ) ≃ G ⋉ C ∗ ( G )induced by the natural map C c ( G × G ) → C c ( G , C c ( G )), cf. [55]. Thus C ∗ ( T triv ⋊ Π) ≃ C ∗ ( T triv ) ⋊ Π . The group T triv = T × U (1) is abelian, hence C ∗ ( T triv ) is isomorphic to C (Π ∗ × Z ) (thePontryagin dual). Thus C ∗ ( T triv ) ⋊ Π ≃ C (Π ∗ × Z ) ⋊ Π . If ξ ∈ Π ∗ and ℓ ∈ Z , the isomorphism C (Π ∗ × Z ) → C ∗ ( T triv ) sends δ ( ξ,ℓ ) ∈ C (Π ∗ × Z ) to itsFourier transform e ξ,ℓ ∈ C ∗ ( T triv ) , e ξ,ℓ ( t, z ) = t ξ z ℓ . Using the commutation relation in T ⋉ Π triv ≃ T triv ⋊ Π, the action of η ∈ Π on e ξ,ℓ is( η · e ξ,ℓ )( t, z ) = e ξ,ℓ ( t, t η z ) = t ξ + ℓη z ℓ . This corresponds to the action of Π on the Pontryagin dual Π ∗ × Z by η · ( ξ, ℓ ) = ( ξ + ℓη, ℓ ) . (34)We see that C (Π ∗ × Z ) ⋊ Π = M ℓ ∈ Z C (Π ∗ ) ⋊ ℓ Πwhere C (Π ∗ ) ⋊ ℓ Π denotes the crossed product formed using the ‘level ℓ ’ action (34). Thesub-algebra C ∗ ( T ⋉ Π bas ) ( ℓ ) corresponds to the ℓ th summand.For ℓ = 0 the action of Π on Π ∗ is trivial, hence C (Π ∗ ) ⋊ Π ≃ C (Π ∗ ) ⊗ C ∗ (Π) ≃ C (Π ∗ × T ∨ )where T ∨ = t ∗ / Π ∗ is the Pontryagin dual of Π. For ℓ = 0, the algebra C (Π ∗ ) ⋊ ℓ Π is isomorphicto a direct sum of finitely many copies of the compact operators on L (Π), indexed by the finitequotient Π ∗ /ℓ Π. One can deduce this from the Takai duality theorem, but it is also not difficultto argue directly as follows. One has a faithful Schr¨odinger-type representation of C (Π ∗ ) ⋊ ℓ Πon L (Π ∗ ), where C (Π ∗ ) acts by multiplication operators, and Π acts by translations as in(34). The decomposition of Π ∗ into cosets [ ξ ] = ξ + ℓ Π for the Π action gives a direct sumdecomposition L (Π ∗ ) = M [ ξ ] ∈ Π ∗ /ℓ Π L ([ ξ ]) , (35)and the action of C (Π ∗ ) ⋊ ℓ Π preserves this decomposition. For η ∈ Π and µ ∈ Π ∗ let θ η,µ ∈ C c (Π , C c (Π ∗ )) = C c (Π × Π ∗ ) be the delta function at ( η, µ ) ∈ Π × Π ∗ . As an elementof C (Π ∗ ) ⋊ n Π, θ η,µ acts on L (Π ∗ ) by the rank 1 linear transformation mapping δ µ to δ µ + ℓη .Such rank 1 operators generate the algebra of all compact operators on L (Π ∗ ) that preservethe direct sum decomposition (35), and thus C (Π ∗ ) ⋊ ℓ Π ≃ M [ ξ ] ∈ Π ∗ /ℓ Π K ( L ([ ξ ])) . We summarize these observations with a proposition.
Proposition 5.4.
The group C ∗ algebra C ∗ ( T ⋉ Π bas ) is an infinite direct sum of its homo-geneous ideals C ∗ ( T ⋉ Π bas ) ( ℓ ) , ℓ ∈ Z . A choice of group isomorphism T ⋉ Π bas ∼ −→ T ⋉ Π triv determines isomorphisms of C ∗ algebras C ∗ ( T ⋉ Π bas ) (0) ∼ −→ C (Π ∗ × T ∨ ) , and for ℓ = 0 C ∗ ( T ⋉ Π bas ) ( ℓ ) ∼ −→ M [ ξ ] ∈ Π ∗ /ℓ Π K ( L ([ ξ ])) , where [ ξ ] = ξ + ℓ Π ⊂ Π ∗ is a coset for the ‘level ℓ ’ action of Π on Π ∗ . The map K ( C ∗ τ ( T × Π)) → R −∞ ( T ) ℓ Π . Let τ be some integer multiple 0 = ℓ ∈ Z of thebasic central extension of LG , and let T ⋉ Π τ denote the restriction of LG τ to the subgroup T × Π ⊂ LG . For ℓ = 1 this is precisely the group T ⋉ Π bas considered above. Elements t ∈ T and b η ∈ Π τ satisfy the commutation relation b ηt b η − t − = κ ℓη ( t ) = t − ℓB ♭ ( η ) ∈ U (1) , see equation (30).The structure of C ∗ τ ( T × Π) follows immediately from Proposition 5.4, and in particular itsK-theory is K ( C ∗ τ ( T × Π)) ≃ M [ ξ ] ∈ Π ∗ /ℓ Π K (cid:0) K ( L ([ ξ ])) (cid:1) . The K-theory of K ( L ([ ξ ])) is a copy of the integers, generated by the finitely generated,projective module L ([ ξ ]). Let R −∞ ( T ) ℓ Π denote the subspace of R −∞ ( T ) consisting of formalcharacters invariant under the ‘level ℓ ’ action of Π, that is, formal sums X ξ ∈ Π ∗ a ξ e ξ , e ξ ( t ) = t ξ where the coefficients satisfy a ξ + ℓη = a ξ for all η ∈ Π (we identify Π with a sublattice of Π ∗ using the basic inner product). There is a mapK (cid:0) K ( L ([ ξ ])) (cid:1) → R −∞ ( T ) ℓ Π sending the generator L ([ ξ ]) to its formal T -character: L ([ ξ ]) X η ∈ Π e ξ + ℓη . Put differently this formal character has multiplicity function given by the indicator functionof the coset [ ξ ] in Π ∗ . It is clear that this map gives an isomorphism of abelian groups:K (cid:0) C ∗ τ ( T × Π) (cid:1) ∼ −→ R −∞ ( T ) ℓ Π . (36) EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 21 The map I : K G ( G, A ) → R −∞ ( T ) W aff − anti , ℓ Let A be a G -equivariant Dixmier-Douady bundle over G , with Dixmier-Douady class ℓ ∈ Z ≃ H G ( G, Z ) and ℓ >
0. In this section we construct a map I : K G ( G, A ) → R −∞ ( T ) W aff − anti , ℓ and show that in a suitable sense it is an inverse of the Freed-Hopkins-Teleman isomorphism.We begin by fixing a model for A as in Section 3.2: A = P G × LG K ( V ∗ ) , (37)where V is a level ℓ positive energy representation of LG bas . Let LG τ denote the centralextension of LG corresponding to ℓ times the generator LG bas , thus V is a representation of LG τ such that the central circle acts with weight 1.Let U be a small N ( T )-invariant tubular neighborhood of T in G , with projection map π T : U → T . A neighborhood U can be described explicitly: for ǫ > B ǫ ( t ⊥ ) an ǫ -ball in t ⊥ ⊂ g , the map T × B ǫ ( t ⊥ ) , ( t, ξ ) t exp( ξ ) , is a N ( T )-equivariant diffeomorphism onto its image, which we may take to be U , with π T theprojection to the first factor. The first stage in the definition of I is the restriction mapK G ( G, A ) → K T ( U, A| U ) (38)induced by the ‘extension by 0’ algebra homomorphism C ( A| U ) ֒ → C ( A ).Recall the Dixmier-Douady bundle A T → T constructed in Section 4.3. Let A U = π ∗ T A T . Bypullback of (24) we obtain a Morita equivalence A| U A U , and hence also an isomorphismK T ( U, A| U ) ∼ −→ K T ( U, A U ) . (39)There is a canonical identification t ⊥ ≃ g / t . The complexification ( g / t ) C ≃ n + ⊕ n − , where n + (resp. n − ) is the direct sum of the positive (resp. negative) root spaces. We choose acomplex structure on g / t such that ( g / t ) , = n − . This choice of complex structure determinesa Bott-Thom isomorphism K T ( U, A U ) ∼ −→ K T ( T, A T ) . (40)By equation (22) and Proposition 4.8, the algebra of sections C ( A T ) has an alternate descrip-tion as a twisted crossed product algebra Π ⋉ τ C ( t ). The isomorphism C ( A T ) ∼ −→ Π ⋉ τ C ( t )yields an isomorphism of K-homology groupsK T ( T, A T ) ∼ −→ K T (Π ⋉ τ C ( t )) . (41)By Proposition 4.12 there is a Green-Julg isomorphismK T (Π ⋉ τ C ( t )) ∼ −→ K T ⋉ Π τ ( C ( t )) (1) . (42)Since Π (hence also T ⋉ Π τ ) acts cocompactly on t , we can apply the analytic assembly map:K T ⋉ Π τ ( C ( t )) → K ( C ∗ ( T ⋉ Π τ )) . (43) Restricted to K T ⋉ Π τ ( C ( t )) (1) , the image of the assembly map is contained in the direct sum-mand isomorphic to K ( C ∗ τ ( T × Π)), and the latter is isomorphic to R −∞ ( T ) ℓ Π by (36). Com-posing the maps (38)—(43) completes the construction of the desired map I : K G ( G, A ) → R −∞ ( T ) ℓ Π . We verify in the next two subsections that the range is the subspace R −∞ ( T ) W aff − anti , ℓ . Remark . The vector space t is a classifying space for proper actions of T ⋉ Π τ . The Baum-Connes conjecture says that the assembly map (43) is an isomorphism. The conjecture hasbeen proved for a very large class of groups including, for example, all amenable groups, ofwhich T ⋉ Π τ is an example (we thank Shintaro Nishikawa for pointing this out). Consequently,each of the maps in the definition of I except the first (38) are isomorphisms. Remark . There are slight variations in the order of the maps in the definition of I that areequivalent. For example, let U ≃ t × B ǫ ( t ⊥ ) be the fibre product t × T U , and for x ∈ K G ( G, A )let x U denote the class in KK T ⋉ Π τ ( C ( U ) , C ) (1) obtained by applying the compositionK G ( G, A ) → K T ( U, A| U ) ∼ −→ K T ( U, A U ) ∼ −→ KK T (Π ⋉ τ C ( U ) , C ) ∼ −→ KK T ⋉ Π τ ( C ( U ) , C ) (1) similar to the definition of I given above. Then, identifying C ( U ) ≃ C ( B ǫ ( t ⊥ )) ⊗ C ( t ), wehave I ( x ) = µ T ⋉ Π τ ( β ⊗ C ( B ǫ ( t ⊥ )) [ x U ]) = [ c ] ⊗ S ⋉ C ( t ) j S ( β ⊗ C ( B ǫ ( t ⊥ )) x U ) , (44)where β ∈ K T ( B ǫ ( t ⊥ )) is the Bott-Thom element, S = T ⋉ Π τ , and for the second equality weuse Kasparov’s description of the assembly map (Section 4.5).6.1. Weyl group symmetry.
The subgroup N ( T ) ⊂ G normalizes Π τ inside LG τ . It followsthat there is an action of N ( T ) by conjugation on T ⋉ Π τ , L τ (Π), and Π ⋉ τ C ( t ) ≃ C ( A T ).Hence each of the C ∗ algebras appearing in the definition of I is in a natural way an N ( T )- C ∗ algebra. There is only one aspect of the definition which is not N ( T )-equivariant, namely theBott-Thom map.Let N be a locally compact group and let H be the connected component of the identity in N . Assume N is unimodular for simplicity. Let A be an N - C ∗ algebra, with α A : N → Aut( A )the action map. For n ∈ N we can view α A ( n ) as an isomorphism of H - C ∗ algebras A → A ( n ) ,where A ( n ) denotes the C ∗ algebra A equipped with the conjugated H -action α A ( n ) ( n ′ ) := α A ( nn ′ n − ). Thus if A , B are N - C ∗ algebras then any n ∈ N induces a mapKK H ( A, B ) → KK H ( A ( n ) , B ( n ) ) . Composing with the ‘restriction homomorphism’ ([27, Definition 3.1]) for the automorphismAd n ∈ Aut( H ), we obtain an automorphism θ n : KK H ( A, B ) → KK H ( A, B ) . See [36, Appendix A] for details (note that the notation in [36, Appendix A] is different).The automorphism θ n acts trivially on elements in the image of the restriction map fromKK N ( A, B ), and only depends on the class of the element n in the component group N/H .Let A be an N - C ∗ algebra. A group element n ∈ N gives rise to an algebra automorphism θ An : H ⋉ A → H ⋉ A, EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 23 defined on the dense subspace C c ( H, A ) by the formula θ An ( f )( h ) = n − .f (Ad n h ). In [36,Appendix A] we show that the corresponding element θ An ∈ KK( H ⋉ A, H ⋉ A ) intertwines θ n and the descent homomorphism; more precisely j H ( θ n ( x )) = θ An ⊗ j H ( x ) ⊗ ( θ Bn ) − , (45)for any x ∈ KK H ( A, B ).As a special case of the above, consider H = T ⊂ N ( T ) = N . As the automorphism θ n (resp. θ An , θ Bn ) only depends on the class w = [ n ] ∈ N ( T ) /T = W , we denote it θ w (resp. θ Aw , θ Bw ). The Bott-Thom element β ∈ K T ( t ⊥ ) is not N ( T )-equivariant, but instead satisfies ([36,Proposition 4.8]) θ w ( β ) = ( − l ( w ) C ρ − wρ ⊗ β, (46)where ρ is the half sum of the positive roots, and l ( w ) is the length of the Weyl group element w . This is a simple consequence of the fact that (1) Ad n | t ⊥ reverses orientation (hence grading)according to the length of w , (2) the weight decomposition for ∧ n − is not symmetric under theWeyl group.To simplify notation let S = T ⋉ Π τ . By (44) and using an argument similar to that givenin [36, Section 4.5], we have I ( x ) ⊗ ( θ C w ) − = [ c ] ⊗ j S ( β ⊗ x U ) ⊗ ( θ C w ) − = [ c ] ⊗ ( θ C ( t ) w ) − ⊗ θ C ( t ) w ⊗ j S ( β ⊗ x U ) ⊗ ( θ C w ) − = [ c ] ⊗ ( θ C ( t ) w ) − ⊗ j S ( θ w ( β ⊗ x U ))= ( − l ( w ) [ c ] ⊗ j S ( β ⊗ x U ) ⊗ C ρ − wρ . In third line we used (45). In the fourth line we used (46), the N ( T )-equivariance of x U (it liesin the image of the restriction map from KK N ( T ) ⋉ Π τ ( C ( U ) , C )), and the fact that the cut-offfunction c : t → [0 , ∞ ) may be chosen to be N ( T )-invariant, which implies [ c ] ⊗ ( θ C ( t ) w ) − = [ c ].In the last line we are also using that K ( C ∗ ( T ⋉ Π τ )) is an R ( T )-module. Corollary 6.3.
The image of I is contained in R −∞ ( T ) W aff − anti , ℓ , the space of formal char-acters that are alternating under the ρ -shifted level ℓ action (5) of the affine Weyl group. Inverse of the Freed-Hopkins-Teleman map.
The commutative diagram (8) in theFreed-Hopkins-Teleman theorem implies K G ( G, A ) has a particularly simple Z -basis obtainedby pushforward from K G (pt) (together with the Morita morphism V ∗ : A| E C ). Theseelements are represented by Kasparov triples x λ with trivial operator F = 0: x λ = [( V ∗ ⊗ R λ , ι ∗ ⊗ id R λ , R λ ∈ R ( G ) is the finite-dimensional irreducible representation of G with highest weight λ ∈ Π ∗ k , and ι ∗ : C ( A ) → A e ≃ K ( V ∗ ) is restriction of a section of A to the fibre over theidentity e ∈ G , so that ι ∗ ⊗ id R λ : C ( A ) → B ( V ∗ ⊗ R λ ) is a representation of C ( A ) on the Hilbertspace V ∗ ⊗ R λ , with range contained in the compact operators. By (8) the correspondingelement of R k ( G ) is the image [ R λ ] ∈ R k ( G ) of R λ ∈ R ( G ) under the quotient map. Under theisomorphism (6), [ R λ ] is sent to the formal character X w ∈ W aff ( − l ( w ) e w • ℓ λ ∈ R −∞ ( T ) . (48) It is easy to determine I ( x λ ). Let R Tλ denote the Z -graded representation of T correspond-ing to the numerator of the Weyl character formula for R λ , thus R Tλ has character X w ∈ W ( − l ( w ) e w ( λ + ρ ) − ρ . By the Weyl character formula the characters χ ( R λ | T ), χ ( R Tλ ) are related by χ ( R Tλ ) = χ ( R λ | T ) · χ ( ∧ n − )where ∧ n − denotes the Z -graded representation of T with character Y α ∈R − (1 − e α ) . In defining the Bott-Thom map we used a complex structure on g / t such that ( g / t ) , = n − .It follows that the image of x λ under restriction to U ⊂ G , followed by the Bott-Thom map is[( V ∗ ⊗ R Tλ , ι ∗ ⊗ id R Tλ , . (49)Applying the Morita morphism A| T A T to (49) swaps L τ (Π) for V ∗ . The Green-Julg mapfollowed by the assembly map send this element to the class of the C ∗ τ ( T × Π)-module L τ (Π) ⊗ R Tλ (50)in K ( C ∗ τ ( T × Π)), where T ⋉ Π τ acts on L τ (Π) (see Remarks 4.6, 4.7) by( t, b η ) · f ( b η ′ ) = κ η ′ ( t ) − f ( b η − b η ′ ) = t ℓη ′ f ( b η − b η ′ ) . Since the formal character of (50) is exactly (48), we have proven the following.
Proposition 6.4.
Let k > and let A be a Dixmier-Douady bundle over G with Dixmier-Douady class ℓ = k + h ∨ ∈ Z ≃ H G ( G, Z ) . The isomorphism R k ( G ) ≃ R −∞ ( T ) W aff − anti , ℓ intertwines I with the inverse of the Freed-Hopkins-Teleman isomorphism.Remark . Without using the Freed-Hopkins-Teleman theorem, the arguments above showthat the map I : K G ( G, A ) → R −∞ ( T ) W aff − anti , ℓ is at least surjective.7. Specialization to geometric cycles
Throughout this section let A be a G -equivariant Dixmier-Douady bundle over G withDD( A ) = ℓ = k + h ∨ ∈ Z ≃ H G ( G, Z ), with k >
0. Let (
M, E, Φ , S ) be a D-cycle representingthe class x = (Φ , S ) ∗ [ D E ] ∈ K G ( G, A ). In this section we exhibit I ( x ) as the T -equivariant L -index of a 1 st -order elliptic operator on a non-compact manifold. EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 25
A cycle for the K-homology push-forward.
As a first step we describe an analyticcycle representing x ∈ K G ( G, A ). To put this in context, one should compare the standardexample 2.2. The result will be a cycle given in terms of a ‘Dirac operator’ acting on sectionsof a Clifford module, except that the module will have infinite rank (since S has infinite rank).The action of the C ∗ algebra C ( A ) plays an essential role in making the result a well-definedanalytic cycle. The construction works more generally, with the target space G replaced byany compact Riemannian G -manifold X .The push-forward (Φ , S ) ∗ [ D E ] is given by the KK-product [ S ] ⊗ [ D E ], see (2) and (3). TheHilbert space of the KK-product is described by: Proposition 7.1.
There is an isomorphism C ( S ) b ⊗ Cl( M ) L ( M, Cliff(
T M ) ⊗ E ) ≃ L ( M, S ⊗ E ) of Z -graded representations of C ( A ) . The proof is essentially the same as for the standard example 2.2. For the reader’s benefit weinclude a proof in the appendix.Recall that S is a right Cliff( T M )-module, and let c : Cliff( T M ) → End( S ) (51)denote the action. Let b c : Cliff( T M ) → End( S ) , b c ( v ) s = ( − deg( s ) c ( v ) s, v ∈ T M (52)denote the action with a ‘twist’ coming from the grading. Choose G -invariant Hermitianconnections ∇ E and ∇ S on S , and let ∇ S⊗ E denote the induced connection on S ⊗ E . Assumemoreover that ∇ S is chosen satisfying ∇ S v ( c ( ϕ ) s ) = c ( ∇ v ϕ ) s + c ( ϕ ) ∇ S v s, (53)i.e. ∇ S is a Clifford connection (cf. [11, Definition 3.39]). Such a connection can be constructedas in the case of a finite dimensional Clifford module. In short, one constructs the connectionlocally and then patches the local definitions together with a partition of unity. Locally on U ⊂ M one can find a spin structure S spin , and S| U ≃ S spin ⊗ S ′ as Cliff( T M )-modules,with S ′ = Hom Cliff(
T M ) ( S spin , S| U ). Using the spin connection on S spin and any Hermitianconnection on S ′ produces a Clifford connection on S| U .The candidate Dirac-type operator D E acting on smooth sections of S ⊗ E is the compositionΓ ∞ ( S ⊗ E ) ∇ S⊗ E −−−−→ Γ ∞ ( T ∗ M ⊗ S ⊗ E ) g ♯ −→ Γ ∞ ( T M ⊗ S ⊗ E ) b c −→ Γ ∞ ( S ⊗ E ) . (54) Proposition 7.2.
The operator D E defined in (64) is essentially self-adjoint. The triple ( L ( M, S ⊗ E ) , ρ, D E ) is an unbounded cycle for an element of K G ( X, A ) .Proof. The presence of a vector bundle E does not alter the proof, so we set E = C to simplifynotation. The condition that ∇ S is a Clifford connection ensures D is symmetric, as for a finitedimensional Clifford module (cf. [32, Proposition 5.3]). It is possible to extend certain proofsof the essential self-adjointness of a Dirac operator on a finite dimensional vector bundle overa compact manifold quite directly to the case of a smooth Hilbert bundle, cf. [16, Proposition1.16] for details. It suffices to check that for a dense set of a ∈ Γ ∞ ( A ), (1) the commutator [ D , ρ ( a )] isbounded, and (2) the operator ρ ( a )(1 + D ) − is compact. Since the underlying space X iscompact, we can find a finite open cover such that for each U in the cover, A| U ≃ U × K ( H )for some Hilbert space H , S| U ≃ U × ( H ⊗ F ) with F a finite dimensional vector space, andthe action ρ of A| U on S| U is given by the defining representation of K ( H ) on the first factorin H ⊗ F . Using a partition of unity subordinate to the cover, we can assume a has supportcontained in a single U , and moreover that a is of the form a = f b where f ∈ C ∞ c ( U ) and b ∈ K ( H ) is a constant operator. For the first assertion, note that[ D , ρ ( f b )] = b c ( g ♯ ( df )) ρ ( b ) + f [ D , ρ ( b )] . The first term is bounded since f is smooth. The second term is bounded because on U , D = D + A , where D is defined in the same way as D but using the trivial connection on U (hence [ D , ρ ( b )] = 0), and A is a bounded bundle endomorphism.For the second assertion, it is convenient to assume that b also has finite constant rank. Therange of the operator (1+ D ) − is contained in the Sobolev space H ( M, S ) of sections with twoderivatives in L , hence the range of ρ ( a )(1+ D ) − is contained in the space f ·H ( U, ran( b ) ⊗ F ).It follows that the operator ρ ( a )(1 + D ) − factors through the inclusion f · H ( U, ran( b ) ⊗ F ) ֒ → L ( U, ran( b ) ⊗ F ) . Since ran( b ) ⊗ F is finite dimensional, the Rellich Lemma implies this inclusion is compact. (cid:3) Theorem 7.3.
The cycle ( L ( M, S ⊗ E ) , ρ, D E ) represents the class [ S ] ⊗ [ D E ] ∈ K G ( X, A ) . The proof is essentially the same as the standard example 2.2, see the appendix.7.2.
The Morita morphism A| U A U . Recall from Section 6 that U denotes an N ( T )-invariant tubular neighborhood of T in G , and π T : U → T the projection map. Let Y = Φ − ( U ) ⊂ M, Φ T = π T ◦ Φ . The restriction of the Morita morphism Cliff(
T M ) Φ ∗ A to Y is a morphism Cliff( T Y ) Φ ∗ A| U . Composing with the morphism A| U A U of Section 6 gives a Morita morphism V : Cliff( T Y ) Φ ∗ A U . (55)The pullback A t = exp ∗ A T = t × K ( L τ (Π)) has a canonical T ⋉ Π τ -equivariant Moritatrivialization A t C given by the A op t -module t × L τ (Π) ∗ . Hence, we have a pullbackdiagram Y Φ t −−−−→ t q Y y y exp Y Φ T −−−−→ T and the pullback of V to Y is a Morita morphism q ∗ Y V : Cliff( q ∗ Y T Y ) ≃ Cliff( T Y ) Φ ∗ t A t . (56)Composing (56) with the Morita trivialization of A t , we obtain a T ⋉ Π τ -equivariant Moritatrivialization S : Cliff( T Y ) C , EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 27 or in other words, a T ⋉ Π τ -equivariant spinor module for Cliff( T Y ). Thus S is a finitedimensional T ⋉ Π τ -equivariant Z -graded Hermitian vector bundle over Y , together with anisomorphism c : Cliff( T Y ) ∼ −→ End( S ).The central circle in Π τ acts on L τ (Π), S with opposite weight (for the action on L τ (Π) weuse the right regular representation, for which the weight of the central circle action is − τ action on L τ (Π) ⊗ S descends to an action of Π. By construction,the Φ ∗ A U -Cliff( T Y ) bimodule V is the quotient V = ( L τ (Π) ⊗ S ) / Π . (57)Let [ V ] ∈ KK T ( C ( A U ) , Cl( Y )) denote the corresponding KK-element defined by the pair(Φ | Y , V ). The action of C ( A U ) on the right hand side in (57) is as follows. Given a ∈ C ( A U ),the pullback q ∗ Y Φ ∗ a is a Π-invariant map Y → K ( L (Π)), hence acts on the first factor of L (Π) ⊗ S by the defining representation for K ( L τ (Π)). This action preserves the space ofΠ-invariant sections of L τ (Π) ⊗ S , hence descends to an action ρ of C ( A U ) on C ( V ). Theaction of Cl( Y ) on the right hand side in (57) can be described in similar terms.The restriction of the fundamental class [ D ] of M to Y is the fundamental class of Y , andwe will abuse notation slightly denote it by [ D ] also. By functoriality of the Kasparov product,the image of (Φ , S ) ∗ [ D E ] | U under the Morita morphism A| U A U equals the KK-product[ V ] ⊗ [ D E ] ∈ KK T ( C ( A U ) , C ) . The Dirac operator on Y . Choose a complete N ( T )-invariant Riemannian metric on Y . The Kasparov product [ V ] ⊗ [ D E ] ∈ KK T ( C ( A U ) , C ) is represented by a cycle ( H, ρ, D E )similar to Section 7.1, with now H = L ( Y, V ⊗ E ). This cycle has an alternate interpretationas the class represented by a Dirac operator on the covering space Y . The correspondencebetween differential operators on Y and Y that we make use of is well-known, cf. [48, Section7.5], [4, 50] for further details. Proposition 7.4.
There is a N ( T ) -equivariant isomorphism of Hilbert spaces L ( Y, V ⊗ E ) ≃ L ( Y , S ⊗ E ) , intertwining the Clifford actions and preserving the subspaces of smooth compactly supportedsections. Under this isomorphism the operator D E in L ( Y, V ⊗ E ) corresponds to the Diracoperator in L ( Y , S ⊗ E ) .Proof. Let s ∈ C ∞ c ( Y , S ) be a smooth compactly supported section of S , and let δ ∈ L τ (Π)denote the function δ ( b γ ) = ( z if b γ = z − Γ τ L (Π) supported at 1 Π .) Define a smoothsection e s of the bundle of Hilbert spaces L τ (Π) ⊗ S over Y by ‘averaging over Π’: e s ( y ) = X η ∈ Π η. (cid:0) δ ⊗ s ( η − .y ) (cid:1) where here we use the fact that Π acts on L τ (Π) ⊗ S (the summand on the right could also bewritten b η.δ ⊗ b η.s ( η − .y ), for any lift b η ∈ Π τ of η ). The section e s is Π-invariant, hence descendsto a section of V , which is again smooth and compactly supported. The map intertwines the L norms, hence extends to a unitary mapping. It’s clear that the map intertwines the Cliffordactions, and hence also the corresponding Dirac operators. (cid:3) Abusing notation slightly, we continue to write D E (resp. ρ ) for the Dirac operator on thecovering space Y acting on sections of S ⊗ E (resp. the representation of C ( A U ) on L ( Y , S ⊗ E )induced by the isomorphism in Proposition 7.4). Corollary 7.5.
The product [ V ] ⊗ [ D E ] is the class [ D E ] represented by the triple ( L ( Y , S ⊗ E ) , ρ, D E ) . The Bott-Thom map.
Recall we chose a complex structure on t ⊥ such that ( t ⊥ ) , = n − ;thus the complex weights of the T -action on t ⊥ in the adjoint representation are the negativeroots. The Bott-Thom class [ β ∈ K T ( t ⊥ ) is represented by the triple ( C ( t ⊥ ) ⊗ ∧ n − , ρ, β ),where β : t ⊥ → End( ∧ n − ) is the bundle endomorphism given at ξ ∈ t ⊥ by the Clifford actionof ξ on the spinor module ∧ n − for Cl( t ⊥ ).Choose a diffeomorphism B ǫ ( t ⊥ ) ∼ −→ t ⊥ which we use to pull the Bott element backto an element of K T ( B ǫ ( t ⊥ )). Taking the external product with the identity element inKK T ( C ( A T ) , C ( A T )) and using the isomorphism C ( A U ) ≃ C ( B ǫ ( t ⊥ )) ⊗ C ( A T )we obtain an invertible element, still denoted [ β ], in the groupKK T ( C ( A T ) , C ( A U )) . The Bott-Thom isomorphism KK T ( C ( A U ) , C ) ∼ −→ KK T ( C ( A T ) , C ) is given by Kasparov prod-uct with this element.The next step is to describe a cycle representing the product[ β ] ⊗ [ D E ] ∈ KK T ( C ( A T ) , C ) . We studied a similar product in [36, Section 4.7], and we simply state the result. The operator D E is extended to sections of ∧ n − b ⊗ S ⊗ E (we use the same symbol for the extension) suchthat D E ( α b ⊗ σ ) = ( − deg( α ) α b ⊗ D E σ whenever α ∈ ∧ deg( α ) n − is constant and σ is a section of S b ⊗ E . The product is represented bythe triple ( L ( Y , ∧ n − b ⊗ S ⊗ E ) , ρ ◦ π ∗ T , D Eβ ) , D Eβ = D E + β Y where β Y is the pullback, via the map Y π −→ Y Φ −→ U ≃ T × B ǫ ( t ⊥ ) ≃ T × t ⊥ pr −−→ t ⊥ , of the odd bundle endomorphism β : t ⊥ → End( ∧ n − ) described above. EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 29
The analytic assembly map and the index.
In [36, Section 4.7] we verified that theoperator D Eβ = D E + β Y is T -Fredholm, i.e. the multiplicity of each irreducible representation of T in the L -kernel ker( D Eβ ) is finite. Thus D Eβ has a well-defined ‘ T -index’ denoted index( D Eβ ) ∈ R −∞ ( T ), see [36, Section 2.5].Via the isomorphism KK T ( C ( A T ) , C ) ≃ KK T ⋉ Π τ ( C ( t ) , C ) (1) the element [ D Eβ ] is identified with an element [ D Eβ ] ∈ KK T ⋉ Π τ ( C ( t ) , C ) (1) . Proposition 7.6.
The image of the class [ D Eβ ] under the composition KK T ⋉ Π τ ( C ( t ) , C ) (1) µ T ⋉ Π τ −−−−→ KK( C , C ∗ τ ( T × Π)) ≃ R −∞ ( T ) ℓ Π is the formal character index( D Eβ ) .Proof. Let N = T ⋉ Π τ . Let H = L ( Y , ∧ n − b ⊗ S ⊗ E ) and let H be the Hilbert C ∗ ( N )-moduleobtained as the completion of C c ( t ) H with respect to the norm defined by the C ∗ ( N )-valuedinner product ( s , s ) C ∗ ( N ) ( n ) = ( s , n · s ) L as in Section 4.5. This inner product takes values in the ideal C ∗ ( N ) (1) ⊂ C ∗ ( N ). Let χ : R → [ − ,
1] be a smooth normalizing function , that is, χ is an odd function, χ ( t ) > t > t →±∞ χ ( t ) = ±
1. We can moreover choose χ to have compactly supported Fouriertransform. The operator F = χ ( D Eβ ) is then a bounded, properly supported operator on H ,with the same T -index as D Eβ , see [24, Chapter 10]. F preserves the subspace C c ( t ) H , andits restriction extends to a bounded operator F on H . The image of [ D Eβ ] under the analyticassembly map µ N is the class in K ( C ∗ ( N ) (1) ) represented by the pair ( H , F ).Recall that the ideal C ∗ ( N ) (1) is isomorphic to a finite direct sum of copies of the compactoperators on L (Π): C ∗ ( N ) (1) ≃ M [ ξ ] ∈ Π ∗ /ℓ Π K ( L ([ ξ ])) , (58)where [ ξ ] ⊂ Π ∗ is viewed as a coset of the action of ℓ Π on Π ∗ . There is in particular a faithfulrepresentation ρ : C ∗ ( N ) (1) → K ( L (Π ∗ ))with image the block diagonal subalgebra (58) of K ( L (Π ∗ )). For s , s ∈ C c ( t ) H , a shortcalculation shows that Tr( ρ ( f )) = ( s , s ) L , f = ( s , s ) C ∗ ( N ) . (59)The norm of an element f ∈ C ∗ ( N ) (1) is equal to the operator norm of ρ ( f ). Thus for s ∈ C c ( t ) H , its norm in H is k ρ ( f ) k / , where f = ( s, s ) C ∗ ( N ) . Using (59) and since f is apositive element, one has k ρ ( f ) k ≤ Tr( ρ ( f )) = k s k L . It follows that H ֒ → H , and correspondsto the subspace of s ∈ H such that ρ ( f ) is trace class, where f = ( s, s ) C ∗ ( N ) .The Hilbert C ∗ ( N ) (1) -module H splits into a finite direct sum: H = M [ ξ ] ∈ Π ∗ /ℓ Π H [ ξ ] , H [ ξ ] = H · K ( L ([ ξ ])) with H [ ξ ] a Hilbert K ( L ([ ξ ]))-module. The operator F commutes with the C ∗ ( N ) (1) action,hence preserves this decomposition, and induces a generalized Fredholm operator F [ ξ ] on each H [ ξ ] . By the strong Morita equivalence K ( L ([ ξ ])) ∼ C , any countably generated Hilbert K ( L ([ ξ ]))-module can be realized as a direct summand of K ( V ), for some V . The generalizedFredholm operator F [ ξ ] can be extended by the identity to K ( V ), giving a generalized Fredholmoperator F V on K ( V ).Let V be an infinite dimensional Hilbert space and K ( V ) the compact operators. When K ( V ) is viewed as a right Hilbert K ( V )-module, the space of (bounded) adjointable operatorsis naturally identified with B ( V ) acting by left multiplication, while the space of generalizedcompact operators is K ( V ) ⊂ B ( V ) [54]. Thus the generalized Fredholm operators, in thesense of Hilbert modules, on K ( V ), are precisely the operators given by left multiplication bya Fredholm operator on V in the ordinary sense. It follows from Atkinson’s theorem that ageneralized Fredholm operator F V on K ( V ) has closed range. If F V is left multiplication by F V ∈ B ( V ) then ran( F ) = K ( V, ran( F V )) while ker( F ) = K ( V, ker( F V )). As ker( F V ) is finite-dimensional, K ( V, ker( F V )) ≃ V ⊗ ker( F V ) is a finitely generated, projective K ( V )-module, andalso a Hilbert space; moreover, the Hilbert space inner product is given by the composition ofthe K ( V )-valued inner product with the trace.By the above generalities, the generalized Fredholm operator F [ ξ ] on H [ ξ ] must have closedrange, and hence the same is true for F . Moreover µ N ([ D Eβ ]) = [ker( F + )] − [ker( F − )] ∈ K ( C ∗ ( N ) (1) ) , with ker( F ± ) being Hilbert spaces, with the inner product given by the composition of the K ( L (Π ∗ ))-valued inner product with the trace. But the latter agrees with the L -inner productin H by (59), hence ker( F ± ) ⊂ H . On H the operator F coincides with F , so this completesthe proof. (cid:3) Corollary 7.7.
Let ℓ > and let A be a Dixmier-Douady bundle on G with DD( A ) = ℓ ∈ Z ≃ H G ( G, Z ) . Let x = (Φ , S ) ∗ [ D E ] ∈ K G ( G, A ) be the class represented by a D-cycle ( M, E, Φ , S ) .The formal character I ( x ) ∈ R −∞ ( T ) W aff − anti , ℓ is given by the T -index of a st order ellipticoperator D Eβ acting on sections of a vector bundle ∧ n − b ⊗ S ⊗ E over the space Y = t × T Φ − ( U ) ,where U ⊃ T is a tubular neighborhood of the maximal torus. Application to Hamiltonian loop group spaces.
A proper Hamiltonian LG -space( M , ω M , Φ M ) is a Banach manifold M with a smooth action of LG , equipped with a weaklynon-degenerate LG -invariant closed 2-form ω M , and a proper LG -equivariant mapΦ M : M → L g ∗ satisfying the moment map condition ι ( ξ M ) ω M = − d h Φ M , ξ i , ξ ∈ L g . A level k prequantization of M is a LG bas -equivariant prequantum line bundle L → M , suchthat the central circle in LG bas acts with weight k . See for example [42, 1] for further back-ground on Hamiltonian loop group spaces. EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 31
The subgroup Ω G ⊂ LG acts freely on M , hence the quotient M = M / Ω G is a smoothfinite-dimensional G -manifold fitting into a pullback diagram M Φ M −−−−→ L g ∗ y y M Φ −−−−→ G (60)where the vertical maps are the quotient maps by Ω G . The quotient M is a quasi-Hamiltonian (or q-Hamiltonian ) G - space , and the pullback diagram above gives a 1-1 correspondence be-tween proper Hamiltonian LG -spaces and compact q-Hamiltonian G -spaces [1].Let G be compact and connected. It was shown in [2] (see [35] for a simpler construction)that every q-Hamiltonian G -space gives rise, in a canonical way, to a D-cycle ( M, C , Φ , S spin )for K G ( G, A ) for a suitable Dixmier-Douady bundle A over G ; the Morita morphism S spin isreferred to as a twisted spin-c structure in [2, 41, 35]. For G simple and simply connected, theDixmier-Douady class of A is h ∨ ∈ Z ≃ H G ( G, Z ), and we denote it by A (h ∨ ) . We will assume G is simple and simply connected below.A level k prequantization [41] of a q-Hamiltonian space is a Morita morphism E : C Φ ∗ A ( k ) where DD( A ( k ) ) = k ∈ Z ≃ H G ( G, Z ). Isomorphism classes of level k prequantizations E of M are in 1-1 correspondence with isomorphism classes of level k prequantum line bundles L over M , see [41, 49] and references therein.Let S = S spin ⊗E , then ( M, C , Φ , S ) is a D-cycle for K G ( G, A ( k +h ∨ ) ). The level k quantization of ( M, E ) was defined by Meinrenken in [41] as the image of the D-cycle ( M, C , Φ , S ) in theanalytic twisted K-homology group:(Φ , S ) ∗ [ D ] ∈ K G ( G, A ( k +h ∨ ) ) . (61)In light of the Freed-Hopkins-Teleman theorem, as well as the 1-1 correspondence betweenq-Hamiltonian G -spaces and Hamiltonian LG -spaces, it would seem reasonable to define thelevel k ‘quantization’ of the prequantized loop group space ( M , ω M , Φ M , L ) as the elementof R k ( G ) corresponding to (Φ , S ) ∗ [ D ] under the Freed-Hopkins-Teleman isomorphism. Thisdefinition satisfies many desirable properties. For example, the quantization of a prequantizedintegral coadjoint orbit is the corresponding irreducible positive energy representation. Also,the definition satisfies a ‘quantization commutes with reduction’ principle, see [41].In [36], building on constructions in [35], we suggested an alternative definition of the quanti-zation of a Hamiltonian loop group space in terms of the T -equivariant L -index of a Dirac-typeoperator on a non-compact spin-c submanifold of M . The latter submanifold and operator canbe identified, respectively, with the manifold Y and the operator D β that we discussed in Sec-tion 7; see [36] for details. As mentioned earlier, we proved in [36, Section 4.7] that D β hasa well-defined T -equivariant L -index, with formal character lying in R −∞ ( T ) W aff − anti , ( k +h ∨ ) ,and proposed that the quantization of M be defined as the corresponding element the Verlindering R k ( G ). The following is now an immediate consequence of Corollary 7.7 and Proposition6.4. Corollary 7.8.
The two definitions of the quantization of M agree, that is, under the identi-fication R −∞ ( T ) W aff − anti , ( k +h ∨ ) ≃ R k ( G ) , the T -equivariant L − index( D β ) coincides with theimage of (Φ , S ) ∗ [ D ] ∈ K G ( G, A ( k +h ∨ ) ) under the Freed-Hopkins-Teleman isomorphism. Our principal motivation in [36] was to give a definition amenable to study with the Wittendeformation/non-abelian localization, and using this to obtain a new proof of the quantization-commutes-with-reduction theorem for Hamiltonian loop group spaces. This was mostly carriedout in [37] (combined with certain results of [33] or [34]). Thus a consequence of Corollary 7.8is that this new proof applies also to Meinrenken’s [41] definition (61).
Appendix A. The KK -product In this appendix we use the same notation as Section 7: (
M, E, Φ , S ) is a D-cycle representinga class x = [ S ] ⊗ [ D E ] ∈ K G ( X, A ). We provide proofs of two results that were omitted. Theseresults are well-known at least in the case of a finite-rank Clifford module, as in the standardexample 2.2, and the proofs are essentially the same as that case. The first is Proposition 7.1,which we restate here for the reader’s convenience. Proposition A.1.
There is an isomorphism C ( S ) b ⊗ Cl( M ) L ( M, Cliff(
T M ) ⊗ E ) ≃ L ( M, S ⊗ E ) of Z -graded representations of C ( A ) .Proof. Let C ( S ) ⊙ Cl( M ) L ( M, Cliff(
T M ) ⊗ E ) denote the algebraic graded tensor product ofCl( M )-modules. Define a pre-inner product on C ( S ) ⊙ Cl( M ) L ( M, Cliff(
T M ) ⊗ E ) by theformula: h s ⊙ ϕ , s ⊙ ϕ i = (cid:0) ϕ , ( s , s ) Cl( M ) · ϕ (cid:1) L , (62)where ( s , s ) Cl( M ) denotes the Cl( M )-valued inner product of the right Hilbert Cl( M )-module C ( S ), and ( − , − ) L denotes the ordinary Hilbert space inner product on L ( M, Cliff(
T M ) ⊗ E ).Dividing by elements of length 0 for the corresponding norm and then completing, we obtaina Hilbert space, usually denoted C ( S ) ⊗ Cl( M ) L ( M, Cliff(
T M ) ⊗ E ). Using the action ofCliff( T M ) on S , there is a map C ( S ) ⊙ Cl( M ) L ( M, Cliff(
T M ) ⊗ E ) → L ( M, S ⊗ E )with dense range. The map intertwines (62) with the inner product on L ( M, S ⊗ E ), henceextends to an isomorphism from the completion to L ( M, S ⊗ E ). (cid:3) Kasparov’s fundamental class [ D ] ∈ KK G (Cl( M ) , C ) is the class defined by the operator D = d + d ∗ in the Hilbert space L ( M, ∧ T ∗ M ) (cf. [27, Definition 4.2]). Identify T M ≃ T ∗ M using the Riemannian metric. In terms of a local orthonormal frame e i , i = 1 , ..., dim( M ), theoperator d + d ∗ is given by X i ( ǫ ( e i ) − ι ( e i )) ∇ e i where ∇ is the Levi-Civita connection, and ǫ ( v ) denotes exterior multiplication by v [32, Lemma5.13]. There is a (unique) isomorphism of left Cliff( T M )-modulesCliff(
T M ) → ∧ T ∗ M EOMETRIC K-HOMOLOGY AND THE FREED-HOPKINS-TELEMAN THEOREM 33 which sends 1 to 1 and intertwines left multiplication on Cliff(
T M ) by v ∈ T M with ǫ ( v ) + ι ( v ) ∈ End( ∧ T ∗ M ). Under this isomorphism ǫ ( v ) − ι ( v ) ∈ End( ∧ T ∗ M ) corresponds to theendomorphism b v of Cliff( T M ) given by (cf. [23, Section 1.11,1.12]) b vϕ = ( − deg( ϕ ) ϕv. Note that b v = −k v k . Hence D = d + d ∗ corresponds to the operator in L ( M, Cliff(
T M ))given in terms of a local orthonormal frame by the expression X i b c ( e i ) ∇ e i . (63)More invariantly, the operator D (viewed as an operator in L ( M, Cliff(
T M ))) is given by thecomposition Γ ∞ (Cliff( T M )) ∇ −→ Γ ∞ ( T ∗ M ⊗ Cliff(
T M )) b c −→ Γ ∞ (Cliff( T M )) . Recall from Section 7 that the candidate Dirac operator D E acting on smooth sections of S ⊗ E is the compositionΓ ∞ ( S ⊗ E ) ∇ S⊗ E −−−−→ Γ ∞ ( T ∗ M ⊗ S ⊗ E ) g ♯ −→ Γ ∞ ( T M ⊗ S ⊗ E ) b c −→ Γ ∞ ( S ⊗ E ) . (64) Theorem A.2.
The cycle ( L ( M, S ⊗ E ) , ρ, D E ) represents the class [ S ] ⊗ [ D E ] ∈ K G ( X, A ) .Proof. The presence of a vector bundle E does not alter the proof, so we set E = C to simplifynotation. We have shown that the triple ( L ( M, S ) , ρ, D ) represents a class in K G ( X, A ) (seeProposition 7.2) with the correct Hilbert space and representation. Thus it suffices to checkthe product criterion in unbounded KK-theory [29], which involves checking a ‘connection con-dition’ and a ‘semi-boundedness condition’. The semi-boundedness condition is automaticallysatisfied, because the operator in the triple representing [ S ] is 0.For s ∈ C ( S ), let T s denote the map ϕ ∈ L ( M, Cliff(
T M )) s ⊗ ϕ ∈ C ( S ) ⊗ Cl( M ) L ( M, Cliff(
T M )) . The ‘connection condition’ says that for a dense set of s ∈ C ( S ) the operators D ◦ T s − ( − deg( s ) T s ◦ D , T ∗ s ◦ D − ( − deg( s ) D ◦ T ∗ s (65)extend to bounded operators from L ( M, Cliff(
T M )) to L ( M, S ). Let ϕ ∈ Γ ∞ (Cliff( T M )).From Proposition A.1, C ( S ) ⊗ Cl( M ) L ( M, Cliff(
T M )) ≃ L ( M, S ) , (66)and T s ( ϕ ) = c ( ϕ ) s. Calculating in terms of a local orthonormal frame and using (53) we have D ◦ T s ( ϕ ) = X i b c ( e i ) ∇ S e i ( c ( ϕ ) s )= ( − deg( s )+deg( ϕ ) X i c ( e i ) ∇ S e i ( c ( ϕ ) s )= ( − deg( s )+deg( ϕ ) X i c ( e i ) c ( ∇ e i ϕ ) s + c ( e i ) c ( ϕ ) ∇ S e i s. The second term is bounded (in ϕ ). For the first term recall that c is a right action, hence( − deg( ϕ ) c ( e i ) c ( ∇ e i ϕ ) = ( − deg( ϕ ) c (cid:0) ( ∇ e i ϕ ) e i (cid:1) = c ( b e i ∇ e i ϕ ) . Thus, using (63), the first term is( − deg( s ) c ( D ϕ ) s = ( − deg( s ) T s ◦ D ( ϕ ) . The argument for T ∗ s is similar. (cid:3) References
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