Geometric quantization for proper actions

TL;DR

This paper introduces an invariant index for G-equivariant elliptic operators on manifolds with proper cocompact group actions, extending Kawasaki's orbifold index, and proves a Guillemin-Sternberg type quantization conjecture in this setting.

Contribution

It generalizes Kawasaki's orbifold index to proper cocompact actions and proves a Guillemin-Sternberg conjecture for symplectic manifolds with Hamiltonian group actions.

Findings

Invariant index generalizes Kawasaki's orbifold index.

Proves the Guillemin-Sternberg conjecture in the proper cocompact setting.

Solves a conjecture of Hochs and Landsman.

Abstract

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.