Volume formula for N-fold reduced products

Abstract

Let G be a semisimple compact connected Lie group. An N-fold reduced product of G is the symplectic quotient of the Hamiltonian system of the Cartesian product of N coadjoint orbits of G under diagonal coadjoint action of G. Under appropriate assumptions, it is a symplectic orbifold. Using the technique of nonabelian localization and the residue formula of Jeffrey and Kirwan, we investigate the symplectic volume of an N-fold reduced product of G. Suzuki and Takakura gave a volume formula for the N-fold reduced product of \( \mathbf {SU}(3) \) in [25] by using geometric quantization and the Riemann–Roch formula. We compare our volume formula with theirs and prove that our volume formula agrees with theirs in the case of triple reduced products of \( \mathbf {SU}(3) \).