The K-theory of abelian symplectic quotients

Let T be a compact torus and (M,\omega) a Hamiltonian T-space. In a previous paper, the authors showed that the T-equivariant K-theory of the manifold M surjects onto the ordinary integral K-theory of the symplectic quotient M \mod T of M by T, under certain technical conditions on the moment map. In this paper, we use equivariant Morse theory to give a method for computing the K-theory of the symplectic quotient by obtaining an explicit description of the kernel of the surjection \kappa: K^*_T(M) \onto K^*(M \mod T). Our results are K-theoretic analogues of the work of Tolman and Weitsman for Borel equivariant cohomology. Further, we prove that under suitable technical conditions on the T-orbit stratification of M, there is an explicit Goresky-Kottwitz-MacPherson (``GKM'') type combinatorial description of the K-theory of a Hamiltonian T-space in terms of fixed point data. Finally, we illustrate our methods by computing the ordinary K-theory of compact symplectic toric manifolds, which arise as symplectic quotients of an affine space \C^N by a linear torus action.