1995/10/23 by K. Guruprasad, J. Huebschmann, L. Jeffrey +1
Mathematics · #dg-ga #math.DG #math.SG
published as Duke Math. J. 89 (1997), 377-412 · AMSTeX 2.1, 33 pages
arxiv created 1995/10/23 · arxiv updated 2009/11/30
Let G be a Lie group, with an invariant non-degenerate symmetric bilinear form on its Lie algebra, let π be the fundamental group of an orientable (real) surface M with a finite number of punctures, and let \bold C be a family of conjugacy classes in G, one for each puncture. A finite-dimensional construction used earlier to obtain a symplectic structure on the moduli space of flat G-bundles over compact M is extended to the punctured case. It yields a symplectic structure on a certain smooth manifold \Cal M\bold C containing the space \romanHom(π,G)\bold C of homomorphisms mapping the generators corresponding to the punctures into the corresponding conjugacy classes. It also yields a Hamiltonian G-action on \Cal M\bold C such that the reduced space equals the moduli space \romanRep(π,G)\bold C of representations. For G compact, each such space, obtained by finite-dimensional symplectic reduction, is a \it stratified symplectic space\/. For G=U(n) one gets moduli spaces of semistable holomorphic parabolic bundles or spaces closely related to them.