Kirwan map and moduli space of flat connections

Abstract

If $K$ is a compact Lie group and $g\geq 2$ an integer, the space $K^{2g}$ is endowed with the structureof a Hamiltonian space with a Lie group valued moment map $\Phi$. Let $\beta$ be in the centre of $K$. Thereduction $\Phi^{-1}(\beta)/K$ is homeomorphic to a moduli space of flat connections.When $K$ is simply connected, a direct consequence of a recent paper of Bott,Tolman and Weitsman is to give a set of generators for the $K$-equivariant cohomology of $\Phi^{-1}(\beta)$.Another method to construct classes in $H^*_K(\Phi^{-1}(\beta))$ is by using the so called universal bundle. Whenthe group is $\Sun$ and $\beta$ is a generator of the centre, these last classes are known to also generate theequivariant cohomology of $\Phi^{-1}(\beta)$. The aim of this paper is to compare the classes constructedusing the result of Bott, Tolman and Weitsman and the ones using the universal bundle.