Twisted K-Homology and Group-Valued Moment Maps

Let G be a compact, simply connected Lie group. We develop a ‘quantization functor’ from prequantized quasi-Hamiltonian G-spaces (M,ω,Φ) at level k to the fusion ring (Verlinde algebra) Rk(G). The quantization $\mathcal Q(M)\in R_k(G)$ is defined as a push-forward in twisted equivariant K-homology. It may be computed by a fixed point formula, similar to the equivariant index theorem for Spinc-Dirac operators. Using the formula, we calculate $\mathcal Q(M)$ in several examples.