Symplectic Forms on Moduli Spaces of Flat Connections on 2-Manifolds

Let G be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms ω on ‘twisted’ moduli spaces of representations of the fundamental group π of a 2-manifold Σ (the smooth analogues of Hom(π1(Σ), G)/G) and on relative character varieties of fundamental groups of 2-manifolds. We extend this construction to exhibit a symplectic form on the extended moduli space [J1] (a Hamiltonian G-space from which these moduli spaces may be obtained by symplectic reduction), and compute the moment map for the action of G on the extended moduli space.