Symplectic and Poisson structures of certain moduli spaces. II. Projective representations of cocompact discrete planar 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 $\pi$ and yields (i) a symplectic structure on a certain smooth manifold $\text{Cal}M$ containing the space $\text{roman}Hom(\pi ,G)$ of homomorphisms and, furthermore, (ii) a hamiltonian $G$-action on $\text{Cal}M$ preserving the symplectic structure together with a momentum mapping in such a way that the reduced space equals the space $\text{roman}Rep(\pi ,G)$ 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 {\it stratified symplectic space/} in such a way that the strata have finite symplectic volume . For example, {\smc Mehta-Seshadri} moduli spaces of semistable holomorphic parabolic bundles with rational weights or spaces closely related to them arise in this way by {\it symplectic reduction in finite dimensions/}.