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 $μ: X \to L\fg^*$ is proper. We consider the function $|\mu {|}^2:X\to \mathbb{R}$, and use a version of Morse theory to show that the inclusion map $j:{\mu }^{-1}(0)\to X$ induces a surjection $j^{\ast }:H_G^{\ast }(X)\to H_G^{\ast }({\mu }^{-1}(0))$, 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.

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