The Ring Structure of Twisted Equivariant KK-Theory for Noncompact Lie Groups

Abstract

Let G be a connected semi-simple Lie group with torsion-free fundamental group. 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.