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 $\kappa_K$. 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 in T, and show how to recover from this the kernel of $\kappa_T$, as described by Tolman and Weitsman. (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 $\ker(\kappa_K)$ and $\ker(\kappa_T)$, where T is a maximal torus in K. An Appendix generalizes Theorem 1.8 to the case of singular values of $\kappa_T$.