PhD advisor: Eckhard Meinrenken
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In this thesis we prove norm-square localization formulas for two invariants of Hamiltonian loop group spaces: twisted Duistermaat-Heckman measures and a K-theoretic "quantization". The terms in the formulas are indexed by the components of the critical set of the norm-square of the moment map. These results are analogous to results proved by Paradan in the case of Hamiltonian G-spaces.
An important application of the norm-square localization formula is to prove that the multiplicity of the fundamental level k representation in the quantization is a quasi-polynomial function of k. This is closely related to the [Q,R]=0 theorem of Alekseev-Meinrenken-Woodward for Hamiltonian loop group spaces.