Mathematical Research Letters
Volume 15 (2008)
The $K$-theory of abelian symplectic quotients
Pages: 57 – 72
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 $M \mod T$, 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 $M \mod T$ 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.