Two explorations in symplectic geometry: I. Moduli spaces of parabolic vector bundles over curves II. Characters of quantisations of Hamiltonian actions of compact Lie groups on symplectic manifolds

In Part I we study the moduli space of holomorphic parabolic vector bundles over a curve, using combinatorial techniques to obtain information about the structure of the cohomology ring. We consider the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincare duals to these Chern classes have simple geometric representatives, and use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.;In Part II we study the geometric quantisation of symplectic manifolds with Lie group actions, and the characters of the resulting virtual representations. In particular, let K ⊂ G be compact connected Lie groups with common maximal torus T. Let (M, &ohgr;) be a prequantisable compact connected symplectic manifold with a Hamiltonian G-action. Geometric quantisation gives a virtual representation of G; we give a formula for the character &khgr; of this virtual representation as a quotient of virtual characters of K. When M is a generic coadjoint orbit our formula agrees with the Gross-Kostant-Ramond-Sternberg formula. We then derive a generalisation of the Guillemin-Prato multiplicity formula which, for lambda a dominant integral weight of K, gives the multiplicity in &khgr; of the irreducible representation of K of highest weight lambda.