Surjectivity for Hamiltonian loop group spaces

Let G be a compact Lie group, and let LG denote the corresponding loop group. Lei (X, ω) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X, ω), and assume that the moment map μ : X → Lg* is proper. We consider the function |μ|² : X → ℝ, and use a version of Morse theory to show that the inclusion map j : μ^-1 (0) → X induces a surjection j* : H* _G(X) → H*_G (μ^-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.