Kirwan map and moduli space of flat connections

If $K$ is a compact Lie group and $g\ge 2$ an integer, the space $K^{2g}$ is endowed with the structure of a Hamiltonian space with a Lie group valued moment map $\mathrm{\Phi }$. Let $\beta$ be in the centre of $K$. The reduction ${\mathrm{\Phi }}^{-1}(\beta )\mathrm{/}K$ 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 ${\mathrm{\Phi }}^{-1}(\beta )$. Another method to construct classes in $H_K^{\ast }({\mathrm{\Phi }}^{-1}(\beta ))$ is by using the so called universal bundle. When the group is $\text{Sun}$ and $\beta$ is a generator of the centre, these last classes are known to also generate the equivariant cohomology of ${\mathrm{\Phi }}^{-1}(\beta )$. 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.