The Kirwan Map for Singular Symplectic Quotients

Let M be a Hamiltonian K-space with proper moment map µ.The symplectic quotient X = µ^–1(0)/K is a singularstratified space with a symplectic structure on the strata.In this paper we generalise the Kirwan map, which maps the Kequivariant cohomology of µ^–1(0) to the middle perversityintersection cohomology of X, to this symplectic setting. The key technical results which allow us to do this are Meinrenken'sand Sjamaar's partial desingularisation of singular symplecticquotients and a decomposition theorem, proved in Section 2 ofthis paper, exhibiting the intersection cohomology of a ‘symplecticblowup’ of the singular quotient X along a maximal depthstratum as a direct sum of terms including the intersectioncohomology of X.