Restriction map in a regular reduction of \(\mathbf{SU}(n)^{2g}\) (Q1405756)

Let \(\mu: G^{2g}\to G\) be the map \((a_1,b_1,\dots, a_g, b_g)\to \prod^g_{k=1} [a_k, b_k]\), and let \(\zeta\) be the \(n\)th root of unity and I the identity in \(\text{SU}(n)\). Then it is shown the restriction map \[ r: H^*_{\text{SU}(n)}(\text{SU}(n)^{2g})\to H^*_{\text{SU}(n)} (\mu^{-1}(\zeta{\mathbf I}) \] is given by \[ \begin{alignedat}{2} r(c_k) &= a_k,\quad &&k= 2,\dots, n,\\ r(\sigma_{k,j}) &= b_{k,j},\quad &&k=2,\dots, n,\;j= 1,\dots, 2g\end{alignedat} \] (Theorem 5.1). Here \(\{c_k,\sigma_{k,j}, 2\leq k\leq n, 1\leq j\leq g\}\) are the multiplicative generators for the equivariant cohomology of \(\text{SU}(n)^{2g})\) (Theorem 4.4 and 4.6) and \(\{a_k, b_{k,j}, d_k, 2\leq k\leq n, 1\leq j\leq 2g\}\) are the multiplicative generators of the equivariant cohomology of \(\mu^{-1}(\zeta{\mathbf I})\) (Theorem 3.4). Note that this right-hand side can be interpreted as the cohomology of moduli space by the theorem of Narashimhan-Seshadri. Let \(\beta= \zeta{\mathbf I}\) and \({\mathbf m}= S_\beta/\text{SU}(n)\), \(S_\beta = \{\rho\in \Hom(\pi, \text{SU}(n))\mid\rho(c)= \beta\}\). Then the equivariant cohomology of \(\mu^{-1}(\beta)\) is identified with \(H^*( {\mathbf m}_\beta)\). Since \({\mathbf m}_\beta\) is isomorphic to the moduli space of holomorphic semi-stable vector bundles of rank \(n\), degree \(d\), and fixed determinant over a compact Riemannian surface \(X\) of genus \(g\) [\textit{M. S. Narasimhan} and \textit{C. S. Seshadri}, Ann. Math. (2) 82, 540--567 (1965; Zbl 0171.04803)], and this space is obtained as the Marsden-Weinstein reduction of quasi-Hamiltonian space [\textit{A. Alekseev}, \textit{A. Malkin} and \textit{E. Meinrenken}, J. Differ. Geom. 48, 445--495 (1998; Zbl 0948.53045)], semistable bundle, quasi-Hamiltonian spaces and characteristic classes are reviewed in Section 2. The cohomology of \({\mathbf m}_\beta\) is compact in Section 3, via the construction of a universal bundle on \({\mathbf m}_\beta\). In Section 4, constructing a bundle over \((\text{SU} (n)^{2g})_{\text{SU} (n)}\times X'\), \(X'= X- \{x_0\}\), the equivariant cohomology of \(\text{SU}(n)^{2g}\) is computed. By using these results, above description of the restriction map from the equivariant cohomology of \(\text{SU} (n)^{2g}\) to the cohomology of the moduli space is proved in Section 5.