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

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 has ring structure induced from renowned K-theory $$K^{\bullet }_{K}(K, _K^K)$$ maximal compact subgroup K. give geometric description representatives in terms equivalence classes certain correspondences and obtain an optimal set generators this ring. also establish various properties under some additional hypotheses on application to quantization q-Hamiltonian G-spaces appendix. suggest conjectures regarding relation positive energy representations LG are unitary noncompact case.