Quantization of Hamiltonian loop group spaces

We prove a Fredholm property for spin-c Dirac operators D on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group K⋉ Γ , with K compact and Γ discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing [X] ∩ [D] yields an element of the formal completion R^-∞(T) of the representation ring of a maximal torus T⊂ H; the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating [X] ∩ [D] corresponds to an element of the ring of projective positive energy representations of the loop group.