[Q,R]=0 and Verlinde series

The first part of the talk will be an introduction to Hamiltonian loop-group spaces, quasi-Hamiltonian spaces, their 'quantization', and the quantization-commutes-with-reduction ([Q,R]=0) theorem in this context. Combined with a vanishing result for higher cohomology, the [Q,R]=0 theorem implies the Verlinde formula for moduli spaces of flat connections on Riemann surfaces. In the second part of the talk, I will describe a new approach to the [Q,R]=0 theorem inspired by work of Paradan, Szenes-Vergne and Boysal-Vergne. The approach involves decomposing the fixed-point formula using a combinatorial identity, and re-organizing into a new formula with terms indexed by the critical values of the norm-square of the moment map. This is joint work with E. Meinrenken.