Surjectivity for Hamiltonian Loop Group Spacees

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