2007-04-27

Megumi Harada, McMaster University

The K-theory of symplectic quotients

Let G be a compact connected Lie group, and (M,\omega) a Hamiltonian G-space with proper moment map \mu. A classical theorem of Kirwan states that there is a surjective ring map \kappa from the G-equivariant rational cohomology of M surjects onto the ordinary rational cohomology ring of the symplectic quotient M//G. The Kirwan surjectivity theorem, in addition to computations of the kernel of \kappa, give powerful methods for explicit computations of the cohomology rings of symplectic quotients.

We present integral K-theoretic analogues of this theory which therefore gives methods for computing the integral K-theory of symplectic quotients. More specifically: (1) we prove a K-theoretic Kirwan surjectivity theorem; (2) give a relationship between the kernel of the Kirwan map \kappa_G for a nonabelian Lie group and the kernel of the Kirwan map \kappa_T for its maximal torus (thus allowing us to reduce computations to the abelian case; and (3) in the abelian case, give methods for explicit computations of the kernel of \kappa_T. Our results are K-theoretic analogues of the rational-cohomological theory developed by Kirwan, Martin, Tolman-Weitsman, and others.