Kirwan map and moduli space of flat connections

If K is a compact Lie group and g ≥ 2 an integer, the space K 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(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(β). Another method to construct classes in H∗ K(Φ (β)) is by using the so called universal bundle. When the group is SU(n) and β is a generator of the centre, these last classes are known to also generate the equivariant cohomology of Φ(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. In particular, I prove that the set of cohomology classes coming from the universal bundle is indeed a set of multiplicative generators for the cohomology of the moduli space. With K = SU(n), this is a new proof of the well-known construction of generators for the cohomology of the moduli space of semi-stable vector bundles with fixed determinant. 2000 Mathematics Subject Classification: 53D20 Index of notation Z Group of relative integers R Field of reals g Integer bigger than 1 F Free group on 2g generators x1, . . . , x2g R Element in F given by ∏g j=1[x2j−1, x2j ] Π Quotient of F by the relation ∏g j=1[x2j−1, x2j ]=1 Σ Closed Riemann surface of genus g Σ0 Σ with the interior of a disc removed K Simply connected compact Lie group Z(K) Centre of K Kc Quotient of K by its centre Z(K) 1 2 SÉBASTIEN RACANIÈRE