Dirac operators on quasi-Hamiltonian G-spaces

We develop notions of twisted spinor bundle and twisted pre-quantum bundle on quasi-Hamiltonian G-spaces. The main result of this paper is that we construct a Dirac operator with index given by positive energy representation of loop group. This generalizes the quantization of Hamiltonian $G$-spaces to quasi-Hamiltonian G-spaces.Comment: All comments are welcom

Formal Verlinde Module

Let G be a compact, simple and simply connected Lie group and \A be an equivariant Dixmier-Douady bundle over G. For any fixed level k, we can define a G-C*-algebra C_{\A^{k+h}}(G) as all the continuous sections of the tensor power \A^{k+h} vanishing at infinity. A deep theorem by Freed-Hopkins-Teleman showed that the twisted K-homology KK^{G}(C_{\A^{k+h}}(G), \C) is isomorphic to the level k Verlinde ring R_{k}(G). By the construction of crossed product, we define a C*-algebra C^{*}(G,C_{\A^{k+h}}(G)). We show that the K-homology KK(C^{*}(G,C_{\A^{k+h}}(G)),\C) is isomorphic to the formal Verlinde module $R^{-\infty}(G) \otimes_{R(G)} R_{k}(G)$, where $R^{-\infty}(G) = Hom_{\Z}(R(G),\Z)$ is the completion of the representation ring.Comment: 23 pages, all comments are welcom

A K-homological approach to the quantization commutes with reduction problem

Kasparov defined a distinguished K-homology fundamental class, so called the Dirac element. We prove a localization formula for the Dirac element in K-homology of crossed product of C^{*}-algebras. Then we define the quantization of Hamiltonian G-spaces as a push-forward of the Dirac element. With this, we develop a K-homological approach to the quantization commutes with reduction theorem.Comment: Corrections made, title change

Equivariant indices of Spinc^c-Dirac operators for proper moment maps

We define an equivariant index of Spin$^c$-Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. The main result in this paper is that the index decomposes into irreducible representations according to the quantisation commutes with reduction principle.Comment: 60 pages, corrections, additions and streamlining based on referee's comment

On the Vergne conjecture

Consider a Hamiltonian action by a compact Lie group on a possibly noncompact symplectic manifold. We give a short proof of a geometric formula for decomposition into irreducible representations of the equivariant index of a Spin$^c$-Dirac operator in this context. This formula was conjectured by Mich\`ele Vergne in 2006 and proved by Ma and Zhang in 2014.Comment: 10 page

An equivariant index for proper actions I

Equivariant indices have previously been defined in cases where either the group or the orbit space in question is compact. In this paper, we develop an equivariant index without assuming the group or the orbit space to be compact. This allows us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In parts II and III of this series, we will explore some properties and applications of this index.Comment: 47 pages. The initial version was split into two parts. This is now the first part, the second part is arXiv:1602.02836. The last revision includes corrections after comments from a refere

Norm-square localization and the quantization of Hamiltonian loop group spaces

In an earlier article we introduced a new definition for the `quantization' of a Hamiltonian loop group space $\mathcal{M}$, involving the equivariant $L^2$-index of a Dirac-type operator $\mathscr{D}$ on a non-compact finite dimensional submanifold $\mathcal{Y}$ of $\mathcal{M}$. In this article we study a Witten-type deformation of this operator, similar to the work of Tian-Zhang and Ma-Zhang. We obtain a formula for the index with infinitely many non-trivial contributions, indexed by the components of the critical set of the norm-square of the moment map. This is the main part of a new proof of the $[Q,R]=0$ theorem for Hamiltonian loop group spaces.Comment: 36 pages, title changed, some corrections and additions in section 6.2 based on referee suggestion

An equivariant index for proper actions II: properties and applications

In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of this index. We prove that it has an induction property that can be used to deduce various other properties of the index. In the case of compact orbit spaces, we show how it is related to the analytic assembly map from the Baum-Connes conjecture, and an index used by Mathai and Zhang. We use the index to define a notion of K-homological Dirac induction, and show that, under conditions, it satisfies the quantisation commutes with reduction principle.Comment: 39 pages. The first version of preprint 1512.07575 was split up into two parts, this is the second par

Spinor modules for Hamiltonian loop group spaces

Let $LG$ be the loop group of a compact, connected Lie group $G$. We show that the tangent bundle of any proper Hamiltonian $LG$-space $\mathcal{M}$ has a natural completion $\overline{T}\mathcal{M}$ to a strongly symplectic $LG$-equivariant vector bundle. This bundle admits an invariant compatible complex structure within a natural polarization class, defining an $LG$-equivariant spinor bundle $\mathsf{S}_{\overline{T}\mathcal{M}}$, which one may regard as the Spin$_c$-structure of $\mathcal{M}$. We describe two procedures for obtaining a finite-dimensional version of this spinor module. In one approach, we construct from $\mathsf{S}_{\overline{T}\mathcal{M}}$ a twisted Spin$_c$-structure for the quasi-Hamiltonian $G$-space associated to $\mathcal{M}$. In the second approach, we describe an `abelianization procedure', passing to a finite-dimensional $T\subset LG$-invariant submanifold of $\mathcal{M}$, and we show how to construct an equivariant Spin$_c$-structure on that submanifold.Comment: 32 page

A geometric formula for multiplicities of KK-types of tempered representations

Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be compact. Under a condition on $K$, which holds in particular if $K$ is maximal compact, we give a geometric expression for the multiplicities of the $K$-types of any tempered representation (in fact, any standard representation) $\pi$ of $G$. This expression is in the spirit of Kirillov's orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of $\pi|_K$ obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of $\mathrm{SU}(p,1)$, $\mathrm{SO}_0(p,1)$ and $\mathrm{SO}_0(2,2)$ restrict multiplicity-freely to maximal compact subgroups.Comment: 48 pages. The initial version of preprint 1705.02088 was split into two parts; this is part 2. In the current version, applications to multiplicity-free restrictions were added. arXiv admin note: substantial text overlap with arXiv:1705.0208