Geometry and Topology of Lie group valued moment maps

A Hamiltonian G-space is a symplectic manifold which carries an action of a connected Lie group G preserving the symplectic  structure and generated by a moment map with values in the dual of the Lie algebra of G. The classical moment map theory is built around several extraordinary  results including the Marsden-Weinstein Reduction Theorem, the Atiyah-Guillemin-Sternberg  and Kirwan Convexity Theorems, the Duistermaat-Heckman localization and the Guillemin-Sternberg "Quantization commutes with Reduction"  principle.

In this talk, we shall introduce the theory of Lie group valued moment maps where the moment map takes values in the group G (instead of the dual of its Lie algebra) and the 2-form on the manifold is relatively closed (rather than closed). Surprisingly, this theory preserves all the good features of the classical moment map theory while relaxing a number of topological restrictions. Interesting examples are provided by moduli spaces of flat connections on orientable 2-dimensional manifolds.