The Freed-Hopkins-Teleman Theorem asserts a canonical link between the equivariant twisted K-theory of a compact Lie group equipped with the conjugation action by itself and the representation theory of its loop group. Motivated by this, we will present results on the equivariant twisted KK-theory of a noncompact semisimple Lie group G. We will give a geometric description of generators of the equivariant twisted KK-theory of G with equivariant correspondences, which are applied to formulate the geometric quantization of quasi-Hamiltonian manifolds with proper G-actions. We will also show that the Baum-Connes assembly map for the C^*-algebra of sections of the Dixmier-Douady bundle which realizes the twist is an isomorphism, and discuss a conjecture on representations of the loop group LG. This talk is based on joint work with Mathai Varghese.

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