  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.
