The Kirwan map , equivariant Kirwan maps , and their kernels

Consider a Hamiltonian action of a compact Lie group K on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a regular value of the moment map κK . We start with the case when K is a torus T : we determine the kernel of the equivariant Kirwan map (defined by Goldin in [Go]) corresponding to a generic circle S ⊂ T , and show how to recover from this the kernel of κT , as described by Tolman and Weitsman in [To-We]. (In the situation when the fixed point set of the torus action is finite, similar results have been obtained in our previous papers [Je], [Je-Ma]). For a compact nonabelian Lie group K we will use the “non-abelian localization formula” of [Je-Ki1] and [Je-Ki2] to establish relationships — some of them obtained by Tolman and Weitsman in [To-We] — between Ker(κK) and Ker(κT ), where T ⊂ K is a maximal torus. In the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of μT .