Symplectic and Poisson structures of certain moduli spaces, II: Projective representations of cocompact planar discrete groups

Let G be a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a certain construction carried out in an earlier paper for the fundamental group of a closed surface may be extended to an arbitrary infinite orientation preserving cocompact planar discrete group of euclidean or non-euclidean motions π and yields (i) a symplectic structure on a certain smooth manifold M containing the space Hom(π, G) of homomorphisms and, furthermore, (ii) a hamiltonian G-action on M preserving the symplectic structure together with a momentum mapping in such a way that the reduced space equals the space Rep(π, G) of representations. More generally, the construction also applies to certain spaces of projective representations. For G compact, the resulting spaces of representations inherit structures of stratified symplectic space in such a way that the strata have finite symplectic volume. For example, Mehta-Seshadri moduli spaces of semistable holomorphic parabolic bundles with rational weights or spaces closely related to them arise in this way by symplectic reduction in finite dimensions.