The K-theory of symplectic quotients

Let G be a compact connected Lie group, and (M,ω) a compact Hamiltonian Gspace with moment map μ. We give a surjectivity result which expresses the K-theory of the symplectic quotient M//G in terms of the equivariant K-theory of the original manifold M , under certain technical conditions on μ. This result is a natural K-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. In the process, we prove that the equivariant K-theory Euler class of a G-bundle is not a zerodivisor, provided that an S subgroup fixes precisely the zero section. This is the K-theoretic version of a lemma due to Atiyah and Bott, which plays a fundamental role in the symplectic geometry of Hamiltonian G-spaces.