Volume formula for \(N\)-fold reduced products (Q6081633)

Consider the adjoint action of a compact semisimple connected Lie group \(G\) (think of \(G=\mathrm{SU}(3)\)) on its lie algebra \({g}\). For any \(\xi \in {g}\) denote \(O_\xi \subset {g}\) the orbit. Define \(M=O_{\xi_1} \times O_{\xi_2} \times O_{\xi_3}\) the product of three orbits. One can prove that \(M\) is a symplectic manifold equipped with the KKS symplectic form, and the action of \(G\) on \(M\) is Hamiltonian with momentum map \[ \mu_G: M \to {g}, (\eta_1,\eta_2,\eta_3) \mapsto \eta_1+\eta_2+\eta_3 \, . \] The space \(M_{red}=\mu_G^{-1}(0)/G\) is called a triple reduced product of \(G\). Under some technical assumptions one can prove that \(M_{red}\) is a symplectic orbifold or symplectic manifold. In this article the authors study the symplectic volume of triple reduced products of the group \(\mathrm{SU}(3)\), giving an explicit formula to compute it; then they go on to generalize the result for \(N\)-fold reduced products on \(\mathrm{SU}(3)\), and finally to \(N\)-fold reduced products of general semisimple compact and connected Lie groups. The techniques they use to compute the symplectic volume are based on non-abelian localization and the residue formula developed by the first author and \textit{F. C. Kirwan} [Topology 36, No. 3, 647--693 (1997; Zbl 0876.55007)]. They compare their formula to the formula obtained by \textit{T. Suzuki} and \textit{T. Takakura} [Tokyo J. Math. 31, No. 1, 1--26 (2008; Zbl 1157.53045)] using a Riemann-Roch based argument.