Spinor modules for Hamiltonian loop group spaces

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Let LG be the loop group of a compact, connected Lie group G. We show that the tangent bundle of any proper Hamiltonian LG-space M has a natural completion TM to a strongly symplectic LG-equivariant vector bundle. This bundle admits an invariant compatible complex structure within a natural polarization class, defining an LG-equivariant spinor bundle STM, which one may regard as the Spinc-structure of M. We describe two procedures for obtaining a finite-dimensional version of this spinor module. In one approach, we construct from STM a twisted Spinc-structure for the quasi-Hamiltonian G-space associated to M. In the second approach, we describe an `abelianization procedure', passing to a finite-dimensional T⊂ LG-invariant submanifold of M, and we show how to construct an equivariant Spinc-structure on that submanifold.