# The ring structure of twisted equivariant $KK$-theory for noncompact Lie groups

## TL;DR

This paper explores the ring structure of twisted equivariant KK-theory for noncompact Lie groups, providing geometric descriptions, generators, and applications to quantization and representation theory.

## Contribution

It establishes a ring structure on twisted equivariant KK-theory for noncompact Lie groups, with geometric representatives and connections to positive energy representations.

## Key findings

- Ring structure induced from maximal compact subgroup
- Geometric description of KK-theory representatives
- Application to quantization of q-Hamiltonian G-spaces

## Abstract

Let $G$ be a connected semisimple Lie group with its maximal compact subgroup $K$ being simply-connected. We show that the twisted equivariant $KK$-theory $KK^{\bullet}_{G}(G/K, \tau_G^G)$ of $G$ has a ring structure induced from the renowned ring structure of the twisted equivariant $K$-theory $K^{\bullet}_{K}(K, \tau_K^K)$ of a maximal compact subgroup $K$. We give a geometric description of representatives in $KK^{\bullet}_{G}(G/K, \tau_G^G)$ in terms of equivalence classes of certain equivariant correspondences and obtain an optimal set of generators of this ring. We also establish various properties of this ring under some additional hypotheses on $G$ and give an application to the quantization of $q$-Hamiltonian $G$-spaces in an appendix. We also suggest conjectures regarding the relation to positive energy representations of $LG$ that are induced from certain unitary representations of $G$ in the noncompact case.