KIRWAN SURJECTIVITY IN K-THEORY FOR HAMILTONIAN LOOP GROUP QUOTIENTS

Let G be a compact Lie group and LG be its associated loop group. The main result of this article 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 by R. Bott, S. Tolman and J. Weitsman, Surjectivity for Hamiltonian loop group spaces, Invent. Math.155 (2004), 225–251, math.DG/0210036. Our proof techniques differ from that of Bott et al. 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 towards a similar theorem in K-theory for quasi-Hamiltonian G-spaces and their associated quasi-Hamiltonian quotients.