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 structure of a Hamiltonian space with a Lie group valued moment map $Φ$. Let $β$ be in the centre of $K$. The reduction $Φ^{-1}(β)/K$ 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 $Φ^{-1}(β)$. Another method to construct classes in $H^*_K(Φ^{-1}(β))$ is by using the so called universal bundle. When the group is $\Sun$ and $β$ is a generator of the centre, these last classes are known to also generate the equivariant cohomology of $Φ^{-1}(β)$. The aim of this paper is to compare the classes constructed using the result of Bott, Tolman and Weitsman and the ones using the universal bundle.