Spinor modules for hamiltonian loop group spaces

Let LG be the loop group of a compact, connected Lie group G. We show that the tangent bundle of any proper Hamiltonian LGspace 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 S_TM, which one may regard as the Spin_c-structure of M. We describe two procedures for obtaining a finite-dimensional version of this spinor module. In one approach, we construct from S_TM a twisted Spin_c-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 Spin_c-structure on that submanifold.