Let X be a holomorphic curve of genus g and m the moduli space of rank n degree k stable holomorphic bundles on X with fixed determinant, n ≥ 2, (n, k) = 1. A generalisation of the construction proposed by M. Mumford and P.Newstead, M. S. Narasimhan and C. S. Seshadri when n = 2, k = 1, of a universal bundle on X × m is given. The space m can be seen as a quasi-hamiltonian reduction of the quasi-hamiltonian space SU(n) 2g . Using the previous construction of the universal bundle, some results are obtained about the image of the equivalent of the Kirwan map for quasi-hamiltonian spaces, the strongest results being when n = 2.