An equivariant index for proper actions III: the invariant and discrete series indices

TL;DR

This paper explores specialized equivariant indices for Spin$^c$-Dirac operators under noncompact group actions, establishing relations with the assembly map, quantization, and vanishing theorems, with applications to discrete series representations.

Contribution

It introduces new equivariant indices for noncompact group actions, linking them to representation theory and geometric quantization, with explicit operator decompositions.

Findings

Relation with the analytic assembly map

Quantization commutes with reduction results

Atiyah-Hirzebruch type vanishing theorems

Abstract

We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of Spin$^c$-Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit spaces. One special case is an index defined in terms of multiplicities of discrete series representations of semisimple groups, where we assume the Riemannian metric to have a certain product form. The other is an index defined in terms of sections invariant under a group action. We obtain a relation with the analytic assembly map, quantisation commutes with reduction results, and Atiyah-Hirzebruch type vanishing theorems. The arguments are based on an explicit decomposition of Spin$^c$-Dirac operators with respect to a global slice for the action.