Surjectivity for Hamiltonian Loop Group Spacees

Abstract

Let $G$ be a compact Lie group, and let $LG$ denote the corresponding loop group. Let $(X,\omega)$ be a weakly symplectic Banach manifold. Consider a Hamiltonian action of $LG$ on $(X,\omega)$, and assume that the moment map $\mu: X \to L\fg^*$ is proper. We consider the function $|\mu|^2: X \to \R$, and use a version of Morse theory to show that the inclusion map $j:\mu^{-1}(0)\to X$ induces a surjection $j^*:H_G^*(X) \to H_G^*(\mu^{-1}(0))$, in analogy with Kirwan's surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian $G$-spaces.