Equivariant cohomology of \(SU(n)^{2g}\) and the Kirwan map (Q2745766)

When a compact manifold \(M\) has a Hamiltonian action of a compact group \(G\) with momentum mapping \(\mu:M \to \mathfrak{g}^*\) then the restriction map on equivariant cohomology \(\rho:H^*_G(M) \to H^*_G(\mu^{-1}(0))\) is surjective. When the action of \(G\) is not Hamiltonian but quasi-Hamiltonian with momentum mapping \(\mu:M \to G\) then the restriction map \(r:H^*_G(M) \to H^*_G(\mu^{-1}(1))\) is not in general surjective. Recent work by \textit{R. Bott}, \textit{S. Tolman} and \textit{J. Weitsman} [Invent. Math. 155, No. 2, 225--251 (2004; Zbl 1067.53067), http://arxiv.org/abs/math.DG/0210036] provides information about the extent to which surjectivity fails in the quasi-Hamiltonian case. The paper under review predates the work by Bott, Tolman and Weitsman; it studies quasi-Hamiltonian action of \(SU(n)\) on \(SU(n)^{2g}\) by conjugation, with momentum mapping given by NEWLINE\[NEWLINE(A_1, \ldots, A_g,B_1, \ldots, B_g) \mapsto \zeta^{-1} \prod_{i=1}^g [A_i,B_i]NEWLINE\]NEWLINE where \(\zeta\) is a primitive \(n\)th root of unity. If \(\zeta=\exp(2\pi i d/n)\) and \(g \geq 2\) then the quasi-Hamiltonian reduction of this action can be identified with the moduli space \(\mathcal{M}\) of stable holomorphic vector bundles of rank \(n\) and degree \(d\) with fixed determinant over a compact Riemannian surface of genus \(g\), and the cohomology of this moduli space is isomorphic to \(H^*_G(\mu^{-1}(1))\). In this paper it is shown that the restriction map \(r\) cannot be surjective since \(H^2_G(M)=0\) whereas the induced symplectic form \(\omega\) represents a nonzero element of \(H^2(\mu^{-1}(1)/G) \cong H^2_G(\mu^{-1}(1))\); however when \(n=2\) it is shown that the composition of \(r\) with the quotient map to \(H^*_G(M)/<\omega >\) is surjective, providing an alternative derivation of the set of generators for the cohomology of the moduli space \(\mathcal{M}\) originally obtained by Atiyah and Bott.