Geometric K-homology and the Freed-Hopkins-Teleman theorem

We describe a map from the equivariant twisted K-homology of a compact, connected, simply connected Lie group $G$ to the Verlinde ring. Our map is described at the level of `D-cycles' for the geometric twisted K-homology of $G$, and is inverse to the Freed-Hopkins-Teleman isomorphism. As an application, we show that two possible definitions of the `quantization' of a Hamiltonian loop group space are compatible with each other.


Introduction
A remarkable theorem due to Freed, Hopkins and Teleman [17,19,18] relates the representation theory of the loop group LG of a compact Lie group G to the equivariant twisted K-theory of G. In the very special case of a connected, simply connected and simple Lie group, the theorem states that there is an isomorphism of rings R k (G) ≃ K G 0 (G, A (k+h ∨ ) ). Here R k (G) is the Verlinde ring of level k > 0 positive energy representations of the loop group, while K G 0 (G, A (k+h ∨ ) ) is the equivariant K-homology of G with twisting (Dixmier-Douady class) k + h ∨ ∈ Z = H 3 G (G, Z). The shift h ∨ is a Lie theoretic constant associated to G called the dual Coxeter number. Freed-Hopkins-Teleman work with twisted K-theory, which is related by a Poincare duality isomorphism [47].
In the proof Freed, Hopkins and Teleman construct a map from positive energy representations to twisted K-theory. The construction involves an interesting family of algebraic Dirac operators parametrized by the space of connections on a G-bundle over S 1 . Computing the equivariant twisted K-theory of G using techniques from algebraic topology, they are able to show that their map is an isomorphism.
It is less clear how to construct a map in the opposite direction, from twisted K-theory (or K-homology), to R k (G). One goal of this article is to describe such an inverse map. Theorem 1.1. Let G be connected, simply connected and simple. Let A be a twisting with Dixmier-Douady class DD(A) = k + h ∨ ∈ Z = H 3 G (G, Z), k > 0. We construct a map I : K G 0 (G, A) → R k (G), which is inverse to the Freed-Hopkins-Teleman isomorphism.
Let us briefly describe the construction of I (x), [x] ∈ K G 0 (G, A); for details see Section 5.7. As a first step we restrict x from G to a small tubular neighborhood U of a maximal torus T inside G. Over U there is a Morita equivalent Dixmier-Douady bundle A U which has an especially simple structure: its algebra of continuous sections can be presented as a twisted crossed product algebra Π ⋉ τ C 0 (U ), where τ is the twist and Π is the integer lattice. Applying some tools from KK-theory (a Green-Julg isomorphism and the analytic assembly map), we The proof of this theorem is in Section 6, but we give a brief outline here. As a first step we obtain a description of an analytic K-homology cycle (H, ρ, D) corresponding to the D-cycle (M, E, Φ, S). Here, the Hilbert space H is the space of L 2 sections of a smooth Hilbert bundle over M and D is a Dirac-type operator acting on smooth sections. Because the bundle has infinite dimensional fibres, D is not necessarily Fredholm, but the action of C 0 (A) along the fibres gives the needed control to make this a cycle. After passing to the Morita equivalent Dixmier-Douady bundle over U ⊃ T , the fibres are replaced with copies of L 2 (Π) (tensored with a finite dimensional bundle); using a correspondence that is well-known (for example in the context of Atiyah's L 2 -index theorem), the operator D then has an alternate interpretation as a Dirac-type operator on a Π-covering space of Φ −1 (U ). Applying the analytic assembly map gives the T -equivariant L 2 -index of this operator, and this leads to the description in the theorem.
Our initial motivation for this work was to understand the relationship between two approaches for 'quantizing' Hamiltonian LG-spaces; such a construction has interesting applications, for example to the Verlinde formula for moduli spaces of flat connections on Riemann surfaces, cf. [34].
Let (M, ω M , Φ M ) be a proper Hamiltonian LG-space, that is, a Banach manifold M equipped with a smooth LG-action, a symplectic form, and a proper moment map Φ M : M → Lg * (see Section 6.6). The subgroup ΩG ⊂ LG of based loops acts freely and properly on M, and the smooth quotient M = M/ΩG is an example of a quasi-Hamiltonian G-space [1]. It was shown in [2] ( [30] for an alternate proof) that quasi-Hamiltonian G-spaces naturally give rise to D-cycles for the twisted K-homology of G. Based on this, an approach for quantizing quasi-Hamiltonian G-spaces was introduced in [35] in terms of a push-forward in twisted K-homology. By the Freed-Hopkins-Teleman theorem, the result can be identified with a positive energy representation of the loop group, hence seems a reasonable candidate for the quantization of the corresponding LG-space.
In [30,31] we constructed a suitable finite dimensional spin-c manifold Y ⊂ M and then studied a type of index pairing involving the corresponding Dirac operator, with the result being a formal character of T (alternating under the action of the affine Weyl group). This suggests that alternately one might define the 'quantization' of M as the positive energy representation whose Weyl-Kac numerator equals this formal character.
A corollary of Theorems 1.1, 1.2 is that the two approaches described above agree with each other. Indeed, for x a D-cycle, the first-order elliptic operator of Theorem 1.2 representing I (x) coincides with the operator studied in [31], while Theorem 1.1 connects the index of this operator with the image of the twisted K-homology class under the Freed-Hopkins-Teleman isomorphism. We explain this in greater detail in Section 6.6.
Throughout the paper we have restricted ourselves to the very special case that G is connected, simply connected and simple. It seems likely to us that the methods will generalize and we hope to return to this in the future.
There is an overlap of some of our methods with the very interesting work of Doman Takata on Hamiltonian LT -spaces [45,46]. In particular, Takata also studies an analytic assembly map into the K-theory of a twisted group C * algebra of T × Π. Takata has built infinite dimensional analogues of several well-known objects from index theory/non-commutative geometry, in the setting of Hamiltonian LT -spaces. It would be interesting to explore further connections with his work.
The first three sections are introductory, and cover aspects of twisted K-homology, the Freed-Hopkins-Teleman theorem, twisted convolution algebras and generalized fixed-point algebras. In Section 5 we carry out the main construction of the map I , and show that it is inverse to the Freed-Hopkins-Teleman map. Section 6 is devoted to representing I (x) explicitly in case x is a D-cycle in the sense of Baum-Carey-Wang, and we also deduce that two definitions of the 'quantization' of a Hamiltonian loop group space are compatible with each other. In an appendix we discuss the semi-direct product group T ⋉ Π τ that plays an important role in the paper.
Acknowledgements. I especially thank Eckhard Meinrenken and Yanli Song for many helpful discussions about how the two approaches mentioned above for 'quantizing' a Hamiltonian LGspaces might be related. The work described in this paper is inspired by our joint work on spinor modules for Hamiltonian LG-spaces [30,31]. I also thank Nigel Higson and Shintaro Nishikawa for helpful suggestions and for answering many of my questions about KK-theory.
Notation. The C * algebras of bounded (resp. compact) operators on a Hilbert space H will be denoted B(H) (resp. K(H)).
If (V, g) is a finite dimensional real Euclidean vector space, Cliff(V ) denotes the complex Clifford algebra of V , the Z 2 graded complex algebra generated in degree 1 by the elements v ∈ V subject to the relation v 2 = v 2 . 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 irreducible representations of K, and R(K) is the representation ring. The formal completion R −∞ (K) of R(K) consists of formal infinite linear combinations of irreducibles π ∈ Irr(K) with coefficients in Z. When discussing a U (1) central extension Γ τ of a group Γ, we use the notation γ to denote some lift to Γ τ of an element γ ∈ Γ.
Throughout G denotes a compact, connected, simply connected, simple Lie group with Lie algebra g. Let T ⊂ G be a maximal torus with Lie algebra t. We fix a positive Weyl chamber t + , and let R + (resp. R − ) denote the positive roots (resp. negative roots). The half sum of the positive roots is denoted ρ, and h ∨ is the dual Coxeter number of G. Since G is simply connected, the integer lattice Π = ker(exp : t → T ) coincides with the coroot lattice. The dual Π * = 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 is 2. We often use the basic inner product to identify g ≃ g * , and we sometimes write B ♭ , B ♯ for the musical isomorphisms when we want to emphasize this. The basic inner product has the property that B(Π, Π) ⊂ Z, and thus B ♭ (Π) ⊂ Π * .

D-cycles and twisted K-homology
Here we give a brief introduction to twisted K-homology and D-cycles. Our discussion of twisted K-homology (using Dixmier-Douady bundles) and D-cycles is similar to [35,33,5] where one can find further details.
Let X be a locally compact space. A Dixmier-Douady bundle over X is a locally trivial bundle 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 is isomorphic 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 algebra structure on the fibres. The tensor product A 0 ⊗ A 1 of Dixmier-Douady bundles is again a Dixmier-Douady bundle.
A Morita morphism S : A 0 A 1 between Dixmier-Douady bundles over X is a bundle of A 1 ⊗ A op 0 modules S → X, locally modelled on the K(H 1 ) − K(H 0 ) bimodule K(H 0 , H 1 ). In the special case A 1 = C, S is called a Morita trivialization of A 0 . Any two Morita morphisms A 0 A 1 are related by tensoring with a line bundle; if the line bundle is trivial, one says the Morita morphisms are 2-isomorphic.
By a theorem of Dixmier and Douady [12], Morita isomorphism classes of Dixmier-Douady bundles are classified by a degree 3 integral cohomology class DD(A) ∈ H 3 (X, Z) known as the Dixmier-Douady class. The Dixmier-Douady class satisfies There are modest generalizations to the Z 2 -graded case, in which case the Dixmier-Douady class DD(A) ∈ H 3 (X, Z) ⊕ H 1 (X, Z 2 ). If X carries an action of a compact group G, one can define G-equivariant Dixmier-Douady bundles, which are classified up to G-equivariant Morita morphisms by classes in the analogous equivariant cohomology groups. The C * algebraic definition of twisted K-theory goes back to Donovan-Karoubi [13] (in the case of a torsion Dixmier-Douady class) and Rosenberg [41] (the general case); see also [4,24].
Let A be a Z 2 -graded, G-equivariant Dixmier-Douady bundle and C 0 (A) the Z 2 -graded G-C * algebra of continuous sections of A vanishing at infinity. One defines the G-equivariant A-twisted K-homology of X to be the G-equivariant analytic K-homology of this C * algebra: [25]). This definition is known to be equivalent to Atiyah-Segal's [4] description in terms of homotopy classes of continuous sections of bundles with typical fibre the Fredholm operators on a Hilbert space.
, and hence an isomorphism between the corresponding twisted Khomology 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 morphisms can lead to genuinely different maps. Any two Morita morphisms are related by tensoring with a Z 2 -graded line bundle, hence the set of Morita morphisms is a torsor for H 2 Baum, Carey and Wang [5] describe a 'geometric' approach to twisted K-homology, in the spirit of Baum-Douglas geometric K-homology [7] (see also [8]). Actually in [5] 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 cycles admit natural maps to the analytic K-homology group defined above. In this paper we will only discuss D-cycles, and only use the even case. The geometric twisted K-homology K G geo,i (X, A) of X is the set of D-cycles modulo an equivalence relation analogous to Baum-Douglas geometric K-homology (generated by suitable versions of 'disjoint union equals direct sum', 'bordism', and 'bundle modification'), see [5]. There is a functorial map (1) which is straight-forward to describe at the level of cycles. We will only use the even case i = 0 here; the odd case is similar. Let [D E ] ∈ KK G (Cl(M ), C) be the class of the de Rham-Dirac operator on M , coupled to the vector bundle E. The pair (Φ, S) defines a push-forward map given as the composition of the Morita morphism Cliff(T M ) Φ * A, with the map induced by the * -homomorphism The push-forward can alternately be expressed as a KK product where [S] ∈ KK G (C 0 (A), Cl(M )) is the class defined by the triple (C 0 (S), Φ * , 0).
Remark 2.5. A proof that the map (1) is an isomorphism has been announced by Baum, Joachim, Khorami and Schick, at least for the non-equivariant case.

The Freed-Hopkins-Teleman Theorem
Let G be a compact, connected, simply connected, simple Lie group acting on itself by conjugation. Then H 3 , where ξ ∈ g and θ L , θ R denote the left and right Maurer-Cartan forms. Thus G-equivariant Dixmier-Douady bundles over G are classified up to Morita equivalence by an integer ℓ ∈ Z, and moreover any two Morita morphisms are 2-isomorphic, see Remark 2.1. We use the notation A (ℓ) to denote a Dixmier-Douady bundle with Dixmier-Douady class corresponding to the integer ℓ. Although we will not use it, it is known that the twisted K-homology group K G 0 (G, A (ℓ) ) carries a ring structure, cf. [17,33]. Let LG denote the loop group of G. To obtain a Banach-Lie group, we take LG to consist of maps S 1 = R/Z → G of some fixed Sobolev level s > 1 2 . The basic inner product defines a central extension of the Lie algebra Lg by R, with cocycle This central extension integrates to a U (1) central extension LG bas of LG, that we will call the basic central extension. U (1) central extensions of LG are classified by an integer k known as the level of the central extension, and the basic central extension corresponds to the generator k = 1. For later reference, note that for G simply connected, the central extension LG bas has a canonical trivialization over the subgroup G ⊂ LG of constant loops.
The loop group has a much-studied class of projective representations known as positive energy representations, which have a detailed theory parallel to the theory for compact groups cf. [23,38]. Let S 1 rot ⋉ LG denote the semi-direct product constructed from the action of S 1 on LG by rigid rotations. This action lifts to an action 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 1 rot ⋉ LG bas such that the weights of S 1 rot are bounded below. One can always tensor a positive energy representation by a 1-dimensional representation of S 1 rot , hence one often normalizes positive energy representations by requiring that the minimal S 1 rot weight is 0, and we always assume this.
For an irreducible positive energy representation, the central circle of LG bas acts by a fixed weight k ≥ 0 called the level. There are finitely many irreducible positive energy representations at 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 * ∩ ka, where a ⊂ t + is the fundamental alcove.
Let R k (G) denote the finite rank free abelian group generated by Z-linear combinations of the level k irreducible positive energy representations. There is an isomorphism (cf. [17]) where R(G) is the representation ring of G and I k (G) is the Verlinde ideal consisting of characters vanishing on the conjugacy classes of the elements The ring R k (G) is known as the level k Verlinde ring. Its rank as a Z-module is #(Π * k ). The following is a very special case of the Freed-Hopkins-Teleman theorem.
Theorem 3.1 (Freed-Hopkins-Teleman [17,19,18]). Let k > 0. The group K G 1 (G, A (k+h ∨ ) ) = 0, and there is an isomorphism of rings This is an appropriate place to mention an alternate description of R k (G) that will be crucial for us later on; this description plays a significant role in the proof of the Freed-Hopkins-Teleman Theorem as well. An element 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, scaled alcove (k + h ∨ )a − ρ. The latter is a fundamental domain for the ρ-shifted level (k + h ∨ ) action of the affine Weyl group W aff = W ⋉ Π, given by Thus, m has a unique extension to a map m : Π * → Z which is alternating under (4), i.e.
where l(w) is the length of the affine Weyl group element w. The extension of m vanishes on the boundary of the fundamental domain (k + h ∨ )a − ρ. This defines an isomorphism of abelian groups where the right hand side denotes the formal characters of T which are alternating under the action (4).
A more conceptual explanation of (5) comes from the Weyl-Kac character formula (cf. [23,38]). Positive energy representations have formal characters (elements of R −∞ (S 1 rot × T )) given by a formula similar to the Weyl character formula for compact Lie groups, but with the numerator and denominator both being formal infinite expressions. Multiplying the formal character of an element of R k (G) by the Weyl-Kac denominator and then restricting to 1 ∈ S 1 rot , one obtains the formal character on the right hand side of (5).

Crossed products and twisted K-homology
In this section we describe some general facts involving crossed product algebras, central extensions, and generalized fixed-point algebras. Throughout this section Γ, S, N are locally compact, second countable topological groups equipped with left Haar measure, and A is a separable C * algebra.

4.1.
Twisted crossed-products. Let Γ be a locally compact group with left invariant Haar measure, and let Γ τ be a U (1)-central extension: Normalize Haar measure on Γ τ such that the integral of a function over Γ τ is given by first averaging over U (1) (using normalized Haar measure) followed by integration over Γ. A choice of section Γ → Γ τ is not needed. In detail, for f ∈ C c (Γ τ ) let Thenf is a U (1)-invariant function on Γ so descends to a function on Γ, and Let A be a Γ-C * algebra. Note that A can be regarded as a Γ τ -C * algebra such that the central 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 homogeneous ideals 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 There is a * -homomorphism from C * (U (1)) into the multiplier algebra M (Γ τ ⋉ A) (cf. [11, II.10.3.10-12]) making Γ τ ⋉ A into a C * (U (1)) = C 0 (Z)-algebra, and the ideals (Γ τ ⋉ A) (n) are the fibres. The decomposition (8) is also not difficult to prove directly. A short calculation using (6) shows that the (Γ τ ⋉ A) (n) are 2-sided ideals, and hence one has a * -homomorphism from the right hand side of (8) to Γ τ ⋉ A. One also has a * -homomorphism in the opposite direction, given by 'taking Fourier coefficients'. For further details see for example [45], where a similar approach to twisted crossed products is taken.
We define the τ -twisted crossed product algebra Γ ⋉ τ A to be the ideal The special case A = C gives the twisted group C * algebra One often sees the twisted crossed-product algebra defined in terms of a cocycle for the central extension, cf. [32]. One can translate to this definition by choosing a section Γ → Γ τ . One reason we take the approach above is that later on we will consider the action of a second group S Γ ⋉ τ A, and it seems slightly awkward to describe this in terms of a section 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- Γ is representation of Γ τ such that the central circle acts with weight 1 (a τ -projective representation of Γ), and for all γ ∈ Γ τ , a ∈ A. The space L 2 (Γ τ ) splits into an ℓ 2 -direct sum of its homogeneous subspaces where Recall the left and right regular representations of Γ τ on L 2 (Γ τ ) are given, respectively, by Both actions preserve the decomposition (10).  (1) . The restriction of the right regular representation to L 2 τ (Γ) is the right (−τ )-twisted regular representation of Γ. (Note that under the right regular representation, the central circle of Γ τ acts on L 2 τ (Γ) with weight −1.)

4.2.
Dixmier-Douady bundles from crossed-products. Let X be a locally compact Hausdorff space with a continuous proper action of a locally compact group Γ. The quotient X/Γ equipped with the quotient topology is then also a locally compact Hausdorff space. Let Γ act on L 2 (Γ) by right translation, and on K := K(L 2 (Γ)) by the adjoint action. Define the algebra of sections of a field of C * -algebras over X/Γ, suggestively denoted C 0 (X × Γ K), consisting of Γ-equivariant continuous maps X → K vanishing at infinity in X/Γ. The algebra C 0 (X × Γ K) is an example of a generalized fixed-point algebra.
The following result is attributed to Rieffel  Proposition 4.4. Let X be a locally compact Hausdorff space with a continuous proper action of a locally compact group Γ, and let K = K(L 2 (Γ)). Then x ∈ X of compact operators defined by the family of integral kernels where µ : Γ → R >0 is the modular homomorphism of Γ.
Remark 4.6. Proposition 4.4 can be viewed as a generalization of the Stone-von Neumann theorem (obtained from the special case Γ = R acting on X = R by translations). More generally for X = Γ, Proposition 4.4 specializes to a well-known isomorphism Using equation (11) one verifies that the induced action of Γ on L 2 (Γ) under (12) is the left regular representation.
If the action of Γ on X is free, then X → X/Γ is a principal Γ-bundle, and the generalized fixed-point algebra is the algebra of continuous sections vanishing at infinity of the associated bundle This is a Dixmier-Douady bundle, with typical fibre K(L 2 (Γ)). In fact A is Morita trivial with Morita trivialization X × Γ L 2 (Γ).
To obtain something more interesting from the construction (13), we adjust it slightly in two ways. First we consider the equivariant situation, where a second group S acts on X and L 2 (Γ). It may happen that the Morita trivialization X × Γ L 2 (Γ) is not S-equivariant. Second, we replace Γ ⋉ C 0 (X) with a twisted crossed product algebra, as in Definition 4.1. This will be important later on, when central extensions of the loop group come into the picture.
Let Γ, S be locally compact groups, and let X be a locally compact Γ × S-space, such that the action of Γ on X is proper. Suppose we are given a U (1) central extension of S × Γ of the form where Γ τ is a U (1) central extension of Γ, and the action of S on Γ τ is possibly non-trivial. Let κ : Γ → Hom(S, U (1)), γ → κ γ ∈ Hom(S, U (1)) be the group homomorphism determined by the commutator map: 3) extends to a representation of S ⋉ Γ τ (such that the central circle acts with weight −1) according to The adjoint action Ad(ρ) on K = K(L 2 τ (Γ)) descends to an action of S × Γ, and the generalized fixed point algebra C 0 (X × Γ K) is a S-C * algebra.
Remark 4.7. For later reference note that the left τ -twisted regular representation (L 2 τ (Γ), λ) (Definition 4.3) also extends to a representation of S ⋉ Γ τ (such that the central circle acts with weight 1) according to This applies in particular to the (S × Γ)-C * algebra C 0 (X), and one has the following variation of Proposition 4.4.
Proposition 4.8. Let S, Γ be locally compact groups, and let X be a locally compact S × Γ space, such that the Γ action is proper. Let S ⋉ Γ τ be a U (1) central extension of S × Γ. Let (L 2 τ (Γ), ρ) be the right (−τ )-twisted regular representation (Definition 4.3), extended to a representation of S ⋉ Γ τ as in equation (14), and let S ⋉ Γ τ act on K = K(L 2 τ (Γ)) by the adjoint action Ad(ρ). There is an isomorphism of S-C * algebras Proof. This follows in a straight-forward manner from Proposition 4.4 applied to Γ τ . The action 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 , where for the second equality we use the fact that the central circle acts trivially on X. The algebra on the left hand side of (16) splits into a direct sum of its homogeneous ideals Decompose L 2 (Γ τ ) into isotypic components for the action of the central circle, as in (10): The subalgebra K(L 2 (Γ τ )) U (1) ⊂ K(L 2 (Γ τ )) is the set of compact operators preserving the decomposition (17); hence We claim the isomorphism (16) restricts to an isomorphism To see this let a ∈ C c (Γ τ ⋉ C c (X)) (n) , and let K a be the corresponding family of operators defined by the integral kernels k a in (11). We suppress the basepoint x ∈ X from the notation as it plays no role in the argument. The homogeneity of a (and U (1) invariance of µ) implies (see (11) According to (6), (7) the integral over Γ τ can be carried out by first averaging with respect to the U (1) action, and then integrating over Γ. Note that The integral over z ∈ U (1) gives the projection to the (n)-isotypical component, hence K a is contained in the ideal K(L 2 (Γ τ ) (n) ). In particular for n = 1 Assuming Γ acts on X freely, we can form the associated S-equivariant Dixmier-Douady bundle over X/Γ A = X × Γ K, and Γ ⋉ τ C 0 (X) ≃ C 0 (A) as S-C * algebras.

4.3.
A Green-Julg isomorphism. For a compact group K, the Green-Julg theorem states that the K-equivariant K-theory of a K-C * algebra A is isomorphic to the K-theory of the crossed-product algebra K ⋉ A. There is a 'dual' version of the Green-Julg theorem (cf. [10, Theorem 20.2.7(b)]) which applies to discrete groups instead of compact groups and Khomology instead of K-theory. Proposition 4.9. Let Γ be a discrete group, and let A be a Γ-C * algebra. Then The isomorphism is simple to describe at the level of cycles. Let (H, π, F ) be a cycle representing a class in KK(Γ ⋉ A, C). We may assume π is non-degenerate. The universal property of 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 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 (19) are compact. For the last operator, note that 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 to the 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 linear combinations of operators of the form π(a)π Γ (γ) = π A (a)π Γ (γ). The operator π A (a)π Γ (γ)( is compact using (20) (multiply both sides by π Γ (γ) on the right).
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.
The twisted crossed product Γ ⋉ τ A is an S-C * algebra with action given by (15), and Proof. Let (H, π, F ) 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 τ Γ ). At the level of cycles, the map sends (H, π, F ) to (H, π A , F ). For γ 0 ∈ Γ τ , the function determines an element in the multiplier algebra M (Γ ⋉ τ A) with π(u γ ) = π τ Γ ( γ). Moreover, the representation π and the action of S on Γ ⋉ τ A extend to M (Γ ⋉ τ A) in such a way that one obtains a covariant pair extending (π, π S ). Hence from (15), It follows from (21) that π S⋉Γ τ (s, γ) := π S (s)π τ Γ ( γ) defines a representation of S ⋉ Γ τ on H such that the central circle acts with weight 1.
The algebra A can be regarded as a sub-algebra of Γ ⋉ τ A, via the embedding a → a, where and π A (a) = π( a). The argument that (H, π A , F ) represents a class in KK S⋉Γ τ (A, C) is then similar to Proposition 4.9. For example, (20) now reads (21)). Note π( a)π τ Γ ( γ) ∈ π(Γ ⋉ τ A), hence compactness of all three terms follows because (H, π, F ) is a cycle.
In the reverse direction, let (H, π A , F ) 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 (9), and the map sends (H, π A , F ) to (H, π, F ) where π is the representation of Γ⋉ τ A guaranteed by the universal property. To see why the action of S on Γ ⋉ τ A is as in (15), let s ∈ S, a ∈ C c (Γ τ , A) and v ∈ H, then and this equals π(s · a)π S (s)v if we set (s · a)( γ) = κ γ (s) −1 s.a( γ). One checks that the result is a cycle similar to before. The maps are well-defined on homotopy classes because one may apply the same maps to cycles for (A, C([0, 1])) (resp. (Γ ⋉ τ A, C([0, 1]))).

The map I
Let A be a G-equivariant Dixmier-Douady bundle over G, with Dixmier-Douady class ℓ ∈ Z = H 3 G (G, Z) and ℓ > 0. In this section we construct a map I : K G 0 (G, A) → R −∞ (T ), and show that in a suitable sense it is an inverse of the Freed-Hopkins-Teleman isomorphism.
5.1. Dixmier-Douady bundles over G. We begin by fixing a convenient model for the Dixmier-Douady bundle A. Let ℓ > 0 and let LG τ denote the central extension of LG corresponding to ℓ times the basic inner 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 with weight 1. The dual space V * carries a negative energy representation such that the central circle acts with weight −1. Let P G denote the space of quasi-periodic paths in G of Sobolev level s > 1 2 , that is, P G is the space of paths γ : R → G such that γ(t)γ(t + 1) −1 is a fixed element of G, independent of t ∈ R. The group LG × G acts on P G, with LG acting by right multiplication, and G by left multiplication (cf. [30] for further discussion). The map q : γ ∈ P G → γ(t)γ(t + 1) −1 ∈ G makes P G into a G-equivariant principal LG-bundle over G. The algebra of compact operators K(V * ) carries an action of LG, and the associated bundle is a G-equivariant Dixmier-Douady bundle over G such that DD(A) = ℓ, cf. [33].

A Dixmier-Douady bundle
A T over T . The product group T × Π may be viewed as a subgroup of LG, where T is embedded as constant loops and Π as exponential loops: η ∈ Π corresponds to the loop θ ∈ [0, 2π] → exp(θη) ∈ T . The restriction of the central extension LG τ to T × Π is a central extension It is known (cf. [19, Section 2.2], [38]) that elements t ∈ T and η ∈ Π τ satisfy the commutation relation Carrying out the construction in Section 4.2 with S = T , Γ = Π, X = t we obtain a Dixmier-Douady bundle over T = t/Π.
embeds t into P G, Π-equivariantly. Restricting to t ⊂ P G in (22), we obtain a Dixmier-Douady bundle over the maximal torus. The central circle in T ⋉ Π τ acts on both L 2 τ (Π), V * with weight −1 (recall that for L 2 τ (Π) we use the right regular representation ρ in Definition 4.3), hence the diagonal action of T ⋉ Π τ on the tensor product a bundle of Hilbert spaces over T . By (23) and (24), E defines a T -equivariant Morita morphism A| T A T , and hence an isomorphism

5.4.
A tubular neighborhood of T . 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 > 0 sufficiently small, and B ǫ (t ⊥ ) an ǫ-ball in t ⊥ ⊂ g, the map is a N (T )-equivariant diffeomorphism onto its image, which we may take to be U , with π T the projection to the first factor. Let A U = π * T A T . By pullback of (25) we obtain a Morita equivalence A| U A U , and hence also an isomorphism (26) The choice of a complex structure on t ⊥ ≃ g/t determines a Bott-Thom isomorphism see the beginning of Section 6.4 for details.
5.5. The analytic assembly map. Let X be a locally compact space with a proper action of a locally compact group N . If the action of N is cocompact, i.e. X/N is compact, then there is a map N )), known as the analytic assembly map. If N is compact, the analytic assembly map is just the equivariant index: For non-compact N , the definition of the assembly map is more involved. We give a brief description here and refer the reader to e.g. [6], [ Let (H, ρ, F ) be a cycle representing a class [F ] ∈ KK N (C 0 (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) such that ρ(h)F ρ(f ) = F ρ(f ) (this can always be achieved by perturbing F , cf. [6, Section 3]). To define 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 Complete ρ(C c (X))H in the norm f N = (f, f ) N 1/2 C * (N ) , where − C * (N ) denotes the norm of 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. The pair (H, F) represents a class in K 0 (C * (N )), and Since F commutes with the C * (N ) action, ker(F ± ) are C * (N )-modules, but unfortunately in general they need not be finitely generated and projective, so that '[ker(F + )]−[ker(F − )]' is not a K-theory class. If the range of F is closed and ker(F ± ) are finitely generated and projective, then (cf. [20,Proposition 3.27]) There is another description of the analytic assembly map due to Kasparov that we briefly recall; see for example [15,Section 4.2] for a recent review, and [37, Section 2.4] for a discussion of the relation between the two descriptions of µ N (at least for N discrete). As the action of N on X is cocompact, one can find a continuous compactly supported 'cut-off function' where µ is the modular homomorphism of N . The function p c defines a self-adjoint projection in G ⋉ C 0 (X), and hence an element [c] ∈ KK(C, N ⋉ C 0 (X)). Kasparov's definition of the assembly map reads: . For a closely related discussion of the K-theory of C * τ (T × Π) see [45].
Recall that we are using the basic inner product to identify t = t * , and hence Π is identified with a sub-lattice of Π * . For ℓ = 0, the algebra C * τ (T × Π) ≃ C * (T × Π) is abelian, hence isomomorphic to the algebra of continuous functions vanishing at infinity on the Pontryagin dual Π * × T ∨ , where T ∨ = t * /Π * is the Pontryagin dual of Π.
Use this to define a representation of T ⋉ Π τ on L 2 (Π * ) by where z is the phase factor appearing in the decomposition (28) for η. The induced representation of C * τ (T × Π) turns out to be faithful, with image equal to the block diagonal sub-algebra: where [ξ] = ξ + ℓΠ ⊂ Π * denotes a coset for the action of ℓΠ (viewed as a sub-lattice of Π * ) on Π * by translation.
Remark 5.2. The reason for the slightly awkward definition involving a lattice basis and (28) is that for general G the central extension Π τ pulled back from LG τ is non-trivial; this means there is a small operator ordering ambiguity, and so to get a representation we must choose an ordering. In the appendix we describe a slightly surprising fact: although Π τ is not isomorphic to Π × U (1) in general, after taking semi-direct product with T , it is as if it were: T ⋉ Π τ ≃ T ⋉ (Π × U (1)), although the isomorphism is not canonical. With this in hand one easily deduces a description of C * (T ⋉ Π τ ), see the appendix.
Thus, from (29), The K-theory of K(L 2 ([ξ])) is a copy of the integers, generated by the finitely generated, projective module L 2 ([ξ]). There is a map sending the generator L 2 ([ξ]) to its 'formal T -character': The resulting formal character has multiplicity function given by the indicator function of the coset [ξ] in Π * . Thus, there is an isomorphism 5.7. The definition of I . We now give the definition of I using notation from Sections 5.1-5.6. Let DD(A) = ℓ > 0 where A is as in Section 5.1. There is a restriction map The Morita morphism (26), composed with the Bott-Thom isomorphism (27) is an isomorphism By equation (23) and Proposition 4.8, the algebra of sections C 0 (A T ) has an alternate description as a twisted crossed product algebra Π ⋉ τ C 0 (t). The isomorphism C 0 (A T ) ∼ − → Π ⋉ τ C 0 (t) yields an isomorphism of K-homology groups By Proposition 4.11 there is a Green-Julg isomorphism Since Π (hence also T ⋉ Π τ ) acts cocompactly on t, we can apply the analytic assembly map: Restricted to K 0 T ⋉Π τ (C 0 (t)) (1) , the image of the assembly map is contained in the direct summand isomorphic to K 0 (C * τ (T × Π)), and the latter is isomorphic to R −∞ (T ) ℓΠ by (30). Composing the maps (31)- (35) gives the desired map The vector space t is a classifying space for proper actions of T ⋉ Π τ , hence by a known case of the Baum-Connes conjecture, the map (35) is an isomorphism (we thank Shintaro Nishikawa for pointing this out). Consequently, each of the maps in the definition of I except the first (31) are isomorphisms.
There are slight variations in the order of the maps in the definition of I that are equivalent. For example, let U ≃ t × B ǫ (t ⊥ ) be the fibre product t × T U , and for x ∈ K G 0 (G, A) let x U denote the class in KK T ⋉Π τ (C 0 (U ), C) (1) obtained by applying the composition (1) similar to the definition of I given above. Then where β ∈ K 0 T (B ǫ (t ⊥ )) is the Bott-Thom element. Equivalently, using Kasparov's description of the assembly map (Section 5.5), 5.8. Weyl group symmetry. In this section we provide a sketch of how, with a little additional effort, it is possible to keep track of the Weyl group action. The group G and hence also the normalizer N (T ) of T in G can be viewed as a subgroup of constant loops in LG τ (we recall that for G simply connected the restriction of any central extension to the subgroup of constant loops has a canonical trivialization), and as such it also normalizes the subgroup Π τ ⊂ LG τ . It follows that there is an action of N (T ) by conjugation on T ⋉ Π τ , L 2 τ (Π), and Π ⋉ τ C 0 (t) ≃ C 0 (A T ). Hence each of the C * algebras appearing in the definition of I is in a natural way a N (T )-C * algebra. There is only one aspect of the definition which is not N (T )-equivariant, namely the Bott-Thom element.
Let H be a closed normal subgroup of a group N . Assume N is unimodular for simplicity. If A, B are N -C * algebras, then any n ∈ N induces an automorphism built by combining the action maps (by n) for A, B with the 'restriction homomorphism' ([25, Definition 3.1]) for the automorphism Ad n ∈ Aut(H); see [31,Appendix A] for details (note that in [31] we use the notation τ σ instead of θ n ). The automorphism θ n acts trivially on elements in the image of the restriction map from KK N (A, B).
Let A be an N -C * algebra. A group element n ∈ N gives rise to an algebra automorphism defined on the dense subspace C c (H, A) by the formula θ A n (f )(h) = n −1 .f (Ad n h). In [31, Appendix A] we show that the corresponding element θ A n ∈ KK(H ⋉ A, H ⋉ A) intertwines θ n and the descent homomorphism; more precisely for any x ∈ KK H (A, B). As a special case of the above, take H = T ⊂ N (T ) = N . For any N (T )-C * algebras A, B the automorphism θ n (resp. θ A n , θ B n ) only depends on the connected component of n ∈ N (T ), 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 the Weyl group.
To simplify notation let S = T ⋉ Π τ . Using (36) and arguing similar to [31,Section 4.5], In third line we used (37). In the fourth line we used (38), the N (T )-equivariance of x U (it lies in the image of the restriction map from KK N (T )⋉Π τ (C 0 (U ), C)), and the fact that the cut-off In the last line we are also using that K 0 (C * (T ⋉ Π τ )) is an R(T )-module.
It follows from this calculation that the image of I is contained in the subset R −∞ (T ) W aff −anti, ℓ consisting of formal characters that are alternating for the ρ-shifted level ℓ action of the affine Weyl group (4). Thus we obtain the following small refinement.
Corollary 5.4. The image of I is contained in R −∞ (T ) W aff −anti, ℓ , the space of formal characters that are alternating under the ρ-shifted level ℓ action (4) of the affine Weyl group.

5.9.
Inverse of the Freed-Hopkins-Teleman map. For ease of notation set ℓ = k + h ∨ . Let ι : {e} ֒→ G be the inclusion of the identity element in G. The Hilbert space V * gives a (canonical) G-equivariant Morita trivialization of ι * A. Freed-Hopkins-Teleman [17,19,18] construct an isomorphism R k (G) → K G 0 (G, A), which moreover fits into a commutative diagram where the top horizontal arrow is the quotient map to R(G)/I k (G). This implies K G 0 (G, A) has a particularly simple Z-basis given by Kasparov triples where R λ is the finite-dimensional irreducible representation of G with highest weight λ ∈ Π * k , and ι * : C 0 (A) → A e ≃ K(V * ) is restriction of a section of A to the fibre over the identity e ∈ G. The corresponding element of It is easy to determine I (x λ ). Let R T λ denote the Z 2 -graded representation of T corresponding to the numerator of the Weyl character formula for R λ , thus R T λ has character w∈W (−1) l(w) e w(λ+ρ)−ρ , and, as elements of R(T ), The image of x λ under restriction to U ⊂ G, followed by the Bott-Thom map is Applying the Morita morphism A| T A T to (42) swaps L 2 τ (Π) for V * . The Green-Julg map followed by the assembly map send this element to the class of the C * τ (T × Π)-module where T ⋉ Π τ acts on L 2 τ (Π) (see Remarks 4.6, 4.7) by Since the formal character of (43) is exactly (41), we have proven the following.
Proposition 5.5. Let k > 0 and let A be a Dixmier-Douady bundle over G with Dixmier- intertwines I with the inverse of the Freed-Hopkins-Teleman isomorphism.

I as an index map
Let A be a G-equivariant Dixmier-Douady bundle over G, with DD(A) = ℓ ∈ Z = H 3 G (G, Z), ℓ > h ∨ . Let (M, E, Φ, S) be a D-cycle representing the class x = (Φ, S) * [D E ] ∈ K G 0 (G, A). In this section we exhibit I (x) as the T -equivariant L 2 -index of a 1 st order elliptic operator. Applied to a Hamiltonian loop group space, this result implies that two proposals (see [35,31]) for defining its 'quantization' are compatible with each other. 6.1. A cycle for the K-homology push-forward. As a first step, in this section we determine a cycle representing x ∈ K G 0 (G, A). To put this in context, one should keep the standard example 2.2 in mind. The construction in fact works more generally, with the target space G replaced by any compact Riemannian G-manifold X.
More invariantly, the operator D (viewed as an operator in L 2 (M, Cliff(T M ))) is given by the composition Recall that S is a right Cliff(T M )-module, and let denote the action. Let denote the action with a 'twist' coming from the grading. Choose G-invariant Hermitian connections ∇ E and ∇ S on S, and let ∇ S⊗E denote the induced connection on S ⊗ E. Assume moreover that ∇ S is chosen satisfying i.e. ∇ S is a Clifford connection (cf. [9,Definition 3.39]). Such a connection can be constructed as in the case of a finite dimensional Clifford module. In short, one constructs the connection locally 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 Hermitian connection 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 composition Proposition 6.2. The operator D E defined in (49) is essentially self-adjoint. The triple (L 2 (M, S ⊗ E), ρ, D E ) is a cycle for an element of K G 0 (X, A). Proof. The presence of a vector bundle E does not alter the proof, so we set E = C to simplify notation. The condition that ∇ S is a Clifford connection ensures D is symmetric, as for a finite dimensional Clifford module (cf. [29,Proposition 5.3]). It is possible to extend certain proofs of the essential self-adjointness of a Dirac operator on a finite dimensional vector bundle over a compact manifold quite directly to the case of a smooth Hilbert bundle, cf. [14, Proposition 2.16] for details.
It suffices to check that for a dense set of a ∈ Γ ∞ (A), (1) the commutator [D, ρ(a)] is bounded, and (2) the operator ρ(a)(1 + D 2 ) −1 is compact. Since the underlying space X is compact, 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, and the action ρ of A| U on S| U is given by the defining representation of K(H) on the first factor in H ⊗ F . Using a partition of unity subordinate to the cover, we can assume a has support contained 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 The first term is bounded since f is smooth. The second term is bounded because on U , D = D 0 + A, where D 0 is defined in the same way as D but using the trivial connection on U (hence [D 0 , 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. The range of the operator (1 + D 2 ) −1 is contained in the Sobolev space H 2 (M, S), hence the range of ρ(a)(1 + D 2 ) −1 is contained in the space f · H 2 (U, ran(b) ⊗ F ). It follows that the operator ρ(a)(1 + D 2 ) −1 factors through the inclusion Since ran(b)⊗F is finite dimensional, the Rellich Lemma implies that this inclusion is compact. Proof. The presence of a vector bundle E does not alter the proof, so we set E = C to simplify notation. We have shown that the triple (L 2 (M, S), ρ, D) represents a class in K G 0 (X, A) with the correct Hilbert space and representation. Thus it suffices to check the product criterion in unbounded KK-theory [27], which involves checking a 'connection condition' and a 'semiboundedness condition'. The semi-boundedness condition is automatically satisfied, because the operator in the triple representing [S] is 0.
For s ∈ C 0 (S), let T s denote the map The 'connection condition' says that for a dense set of s ∈ C 0 (S) the operators extend to bounded operators from L 2 (M, Cliff(T M )) to L 2 (M, S). Let ϕ ∈ Γ ∞ (Cliff(T M )). From Proposition 6.1, and T s (ϕ) = c(ϕ)s.
Calculating in terms of a local orthonormal frame and using (48) we have The second term is bounded (in ϕ). For the first term recall that c is a right action, hence Thus, using (45), the first term is The argument for T * s is similar.
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 Sections 5.3, 5.4 gives a Morita mor- The pullback A t = exp * A T = t × K(L 2 τ (Π)) has a canonical T ⋉ Π τ -equivariant Morita trivialization A t C given by the A op t -module t × L 2 τ (Π) * . Let q Y : Y → Y be the pullback bundle. Hence, we have a pullback diagram Composing with the Morita trivialization of A t , we obtain a Morita trivialization or in other words, a spinor module for Cliff(T Y). Thus S is a finite dimensional T ⋉ Π τ -equivariant Z 2 -graded Hermitian vector bundle over Y, together with an isomorphism c : Cliff(T Y) ∼ − → End(S). The central circle in Π τ acts on L 2 τ (Π), S with opposite weight (for the action on L 2 τ (Π) we use the right regular representation, for which the weight of the central circle action is −1), and hence the diagonal Π τ action on L 2 τ (Π) ⊗ S descends to an action of Π. By construction, Let [V] ∈ KK T (C 0 (A U ), Cl(Y )) denote the corresponding KK-element. The action of C 0 (A U ) on the right hand side in (54) is as follows. Given a ∈ C 0 (A U ), the pullback q * Y Φ * a is a Π-invariant map Y → K(L 2 (Π)), hence acts on the first factor of L 2 (Π) ⊗ S by the defining representation for K(L 2 τ (Π)). This action preserves the space of Π-invariant sections of L 2 τ (Π)⊗ S, hence descends to an action ρ of C 0 (A U ) on C 0 (V). The action of Cl(Y ) on the right hand side in (54) can be described in similar terms.
The restriction of the fundamental class [D] of M to Y is the fundamental class of Y , and we 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 . This cycle has an alternate interpretation as the class represented by a Dirac operator on the covering space Y. The correspondence between differential operators on Y and Y that we make use of is well-known, cf. [42,Section 7.5], [3,44] for further details.
Proposition 6.4. There is a N (T )-equivariant isomorphism of Hilbert spaces intertwining the Clifford actions and preserving the subspaces of smooth compactly supported sections. Under this isomorphism the operator D E in L 2 (Y, V ⊗ E) corresponds to the Dirac operator in L 2 (Y, S ⊗ E).
Proof. Let s ∈ C ∞ c (Y, S) be a smooth compactly supported section of S, and let δ ∈ L 2 τ (Π) denote the function (This element plays the role of the delta function of L 2 (Π) supported at 1 Π .) Define a smooth section s of the bundle of Hilbert spaces L 2 τ (Π) ⊗ S over Y by 'averaging over Π': where here we use the fact that Π acts on L 2 τ (Π) ⊗ S (the summand on the right could also be written η.δ ⊗ η.s(η −1 .y), for any lift η ∈ Π τ of η). The section s is Π-invariant, hence descends to a section of V, which is again smooth and compactly supported. The map intertwines the L 2 norms, hence extends to a unitary mapping. It's clear that the map intertwines the Clifford actions, and hence also the corresponding Dirac-type operators.
Abusing notation slightly, we continue to write D E (resp. ρ) for the Dirac operator on the covering space Y acting on sections of S⊗E (resp. the representation of C 0 (A U ) on L 2 (Y, S⊗E) induced by the isomorphism in Proposition 6.4). 6.4. The Bott-Thom map. The isomorphism t ⊥ ≃ n − determines a complex structure on t ⊥ such that the weights of the T -action in the adjoint representation are the negative roots. The Bott-Thom class in K 0 T (t ⊥ ) is represented by the triple (C 0 (t ⊥ ) ⊗ ∧n − , ρ, θ), where θ : t ⊥ → End(∧n − ) is the bundle endomorphism given at ξ ∈ t ⊥ by the Clifford action of ξ on the spinor module ∧n − for Cl(t ⊥ ).
Choose a diffeomorphism B ǫ (t ⊥ ) ∼ − → t ⊥ which we use to pull the Bott element back to an element β ∈ K 0 T (B ǫ (t ⊥ )). Taking the external product with the identity element in KK T (C 0 (A T ), C 0 (A T )) and using the isomorphism we obtain an invertible element, still denoted β, in the group The next step is to describe a cycle representing the product We studied a similar product in [31,Section 4.7], and the product here can be handled similarly. The result is the triple , D E β = γD + θ where γ is the grading operator for ∧n − , and θ is the pullback, via the map of the bundle endomorphism θ : t ⊥ → End(∧n − ) described above.
6.5. The analytic assembly map and the index. In [31,Section 4.7] we verified that the operator D E β = γD E +θ is T -Fredholm, i.e. the multiplicity of each irreducible representation of [31,Section 2.5].
Via the isomorphism Proposition 6.6. The image of the class [D E β ] under the composition and let H be the Hilbert C * (N )-module obtained as the completion of C c (t)H with respect to the norm defined by the C * (N )-valued inner product (s 1 , s 2 ) C * (N ) (n) = (s 1 , n · s 2 ) L 2 as in Section 5.5. This inner product takes values in the ideal C * (N ) (1) ⊂ C * (N ). Let χ : R → [−1, 1] be a smooth normalizing function, that is, χ is an odd function, χ(t) > 0 for t > 0 and lim t→±∞ χ(t) = ±1. We can moreover choose χ to have compactly supported Fourier transform. The operator F = χ(D E β ) is then a bounded, properly supported operator on H, with the same T -index as D E β , see [22,Chapter 10]. F preserves the subspace C c (t)H, and its restriction extends to a bounded operator F on H. The image of [D E β ] under the analytic assembly map µ N is the class in K 0 (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 compact operators on L 2 (Π): where [ξ] ⊂ Π * is viewed as a coset of the action of ℓΠ on Π * . There is in particular a faithful representation ρ : with image the block diagonal subalgebra (55) of K(L 2 (Π * )). For s 1 , s 2 ∈ C c (t)H, a short calculation shows that Tr(ρ(f )) = (s 1 , s 2 ) L 2 , f = (s 1 , s 2 ) C * (N ) .
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 ρ(f ) 1/2 , where f = (s, s) C * (N ) . Using (56) and since f is a positive element, one has ρ(f ) ≤ Tr(ρ(f )) = s 2 L 2 . It follows that H ֒→ H, and corresponds to 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: with H [ξ] a Hilbert K(L 2 ([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 2 ([ξ])) ∼ C, any countably generated Hilbert K(L 2 ([ξ]))-module can be realized as a direct summand of K(V ), for some V . The generalized Fredholm operator F [ξ] can be extended by the identity to K(V ), giving a generalized Fredholm operator 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 operators is naturally identified with B(V ) acting by left multiplication, while the space of generalized compact operators is K(V ) ⊂ B(V ) [48]. Thus the generalized Fredholm operators, in the sense of Hilbert modules, on K(V ), are precisely the operators given by left multiplication by a Fredholm operator on V in the ordinary sense. It follows from Atkinson's theorem that a generalized 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 finitedimensional, K(V, ker(F V )) ≃ V ⊗ ker(F V ) is a finitely generated, projective K(V )-module, and also a Hilbert space; moreover, the Hilbert space inner product is given by the composition of the K(V )-valued inner product with the trace.
By the above generalities, the generalized Fredholm operator F [ξ] on H [ξ] must have closed range, and hence the same is true for F. Moreover with ker(F ± ) being Hilbert spaces, with the inner product given by the composition of the K(L 2 (Π * ))-valued inner product with the trace. But the latter agrees with the L 2 -inner product in H by (56), hence ker(F ± ) ⊂ H. On H the operator F coincides with F , so this completes the proof.
A level k prequantization of M is a LG bas -equivariant prequantum line bundle L → M, such that the central circle in LG bas acts with weight k. See for example [36,1] for further background on Hamiltonian loop group spaces. The subgroup ΩG ⊂ LG acts freely on M, hence the quotient M = M/ΩG is a smooth finite-dimensional G-manifold fitting into a pullback diagram where the vertical maps are the quotient maps by ΩG. The quotient M is a quasi-Hamiltonian (or q-Hamiltonian) G-space, and the pullback diagram above gives a 1-1 correspondence between proper Hamiltonian LG-spaces and compact q-Hamiltonian G-spaces [1]. Let G be compact and connected. It was shown in [2] (see [30] for a simpler construction) that every q-Hamiltonian G-space gives rise, in a canonical way, to a D-cycle (M, C, Φ, S) for K G 0 (G, A) for a suitable Dixmier-Douady bundle A over G; the Morita morphism S is referred to as a twisted spin-c structure in [2,35,30]. For G simple and simply connected, the Dixmier-Douady class of A is h ∨ ∈ Z = H 3 G (G, Z), and we denote it by A (h ∨ ) . We will assume G is simple and simply connected below.
A level k prequantization [35] of a q-Hamiltonian space is a Morita morphism where DD(A (k) ) = k ∈ Z = H 3 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 [35,43] and references therein. Let x = (M, C, Φ, S ⊗ E) be the D-cycle for K G 0 (G, A (k+h ∨ ) ) obtained by taking the tensor product of the Morita morphisms S and E. The level k quantization of (M, E) was defined in [35] as the image of the D-cycle x in the analytic twisted K-homology group: ). In light of the Freed-Hopkins-Teleman theorem, as well as the 1-1 correspondence between q-Hamiltonian G-spaces and Hamiltonian LG-spaces, it would seem reasonable to define the level k quantization of the prequantized loop group space (M, ω M , Φ M , L) as the element of R k (G) corresponding to (Φ, S ⊗ E) * [D] under the Freed-Hopkins-Teleman isomorphism. This definition satisfies many desirable properties. For example, the quantization of a prequantized integral coadjoint orbit is the corresponding irreducible positive energy representation. Also, the definition satisfies a 'quantization commutes with reduction' principle, see [35].
In [31], building on constructions in [30], we suggested an alternative definition of the quantization of a Hamiltonian loop group space in terms of a certain index pairing [X ] ∩ [D L ] ∈ KK(C * (T ), C) ≃ R −∞ (T ). We showed that the pairing lies in the subgroup R −∞ (T ) W aff −anti, (k+h ∨ ) , and defined the quantization of M to be its image under the isomorphism between this subgroup and the Verlinde ring R k (G). We also proved that [X ] ∩ [D L ] equals the T -index of the operator D L β on Y. By Corollary 6.7 this coincides with I (x). Hence these two definitions of the quantization of M, one based on a push-forward in twisted Khomology and the other based on the T -index of a 1 st -order elliptic operator on Y, agree with each other.
Appendix A. The group T ⋉ Π τ .
In this section G is a simple, simply connected compact Lie group and Π bas denotes the restriction to Π ⊂ LG of the basic central extension LG bas of the loop group. We first give an explicit 2-cocycle τ for Π bas . Recall that the cocycle of a U (1) central extension associated to a splitting η ∈ Π → η ∈ Π bas is the function τ : Π × Π → U (1) defined by the equation (We write the group operation in Π multiplicatively here.) Let β 1 , ..., β r ∈ Π be a lattice basis for Π. For G simple and simply connected it is known [38,28] that one can choose lifts β 1 , ..., β r ∈ Π bas such that where B is the basic inner product. For η = n i β i ∈ Π let η = β n 1 1 · · · β nr r .
Remark A.2. The function (−1) ǫ is the 'off-diagonal' part of what Kac [23, Section 7.8] calls an asymmetry function, which he uses to give a concise definition of the Lie bracket for simple and simply laced Lie algebras.
Perhaps surprisingly, the distinction between Π bas , Π triv disappears after taking semi-direct product with T .
A short calculation using (62) shows that Ψ is a group homomorphism.