Geometric quantization for proper actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin–Sternberg geometric quantization conjecture holds if M is symplectic with a Hamiltonian action of G that is proper and cocompact. This essentially solves a conjecture of Hochs and Landsman.