Equivariant indices of Spin$^c$-Dirac operators for proper moment maps

TL;DR

This paper introduces an equivariant index for Spin$^c$-Dirac operators on noncompact manifolds with compact Lie group actions, demonstrating its decomposition aligns with the quantisation commutes with reduction principle.

Contribution

It extends the concept of equivariant indices to noncompact manifolds and establishes their decomposition in accordance with a fundamental geometric principle.

Findings

Index decomposes into irreducible representations

Supports the quantisation commutes with reduction principle

Applicable to noncompact manifolds with group actions

Abstract

We define an equivariant index of Spin$^c$-Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. The main result in this paper is that the index decomposes into irreducible representations according to the quantisation commutes with reduction principle.