Kirwan surjectivity in K-theory for Hamiltonian loop group quotients

TL;DR

This paper proves a surjectivity theorem in equivariant K-theory for Hamiltonian loop group quotients, extending previous cohomological results to a K-theoretic setting with novel proof techniques.

Contribution

It establishes a K-theoretic surjectivity result for Hamiltonian LG-spaces, using direct G-equivariant homotopy equivalences instead of the Borel construction.

Findings

Surjectivity of equivariant K-theory onto K-theory of quotients.

New proof techniques avoiding the Borel construction.

Foundation for K-theoretic results in quasi-Hamiltonian spaces.

Abstract

Let G be a compact Lie group and LG its associated loop group. The main result of this manuscript is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient. Our result is a K-theoretic analogue of previous work in rational Borel-equivariant cohomology of Bott, Tolman, and Weitsman. Our proof techniques differ from that of Bott, Tolman, and Weitsman in that they explicitly use the Borel construction, which we do not have at our disposal in equivariant K-theory; we instead directly construct G-equivariant homotopy equivalences to obtain the necessary isomorphisms in equivariant K-theory. The main theorem should also be viewed as a first step toward a similar theorem in K-theory for quasi-Hamiltonian G-spaces and their associated quasi-Hamiltonian quotients.