In equivariant symplectic geometry, it is often easier to work with actions of a torus rather than a compact Lie group. In many cases, after understanding the torus case, there is an additional step to pass from a compact Lie group to its maximal torus. In this talk we show how to compute the K-theory of the symplectic reduction with respect to a Lie group G in terms of the reduction with respect to its maximal torus T. This is the K-theoretic version of results of Shaun Martin in the symplectic case, or Michel Brion in the algebraic geometry setting. The known arguments for the analogous result in rational or real cohomology use techniques including integral formulae and averaging over the Weyl group, neither of which works in the presence of torsion. Our K-theory argument bypasses such torsion constraints, using the equivariant K-theory Kunneth spectral sequence to relate G-equivariant and T-equivariant K-theory, as well as (surprisingly) the Weyl character formula.
This is joint work with M. Harada.