Abstract
Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (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→L \(\mathfrak{g}\) * 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.
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Bott, R., Tolman, S. & Weitsman, J. Surjectivity for Hamiltonian loop group spaces . Invent. math. 155, 225–251 (2004). https://doi.org/10.1007/s00222-003-0319-2
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DOI: https://doi.org/10.1007/s00222-003-0319-2