Quantising proper actions on Spin$^c$-manifolds

TL;DR

This paper extends the quantisation commutes with reduction principle to noncompact groups and manifolds in the Spin$^c$ setting, providing new index formulas and results for both cocompact and non-cocompact actions.

Contribution

It generalizes existing results to noncompact groups and manifolds, and establishes new index formulas for Spin$^c$-Dirac operators in this broader context.

Findings

Quantisation commutes with reduction for noncompact groups and manifolds.

Index formulas for Spin$^c$-Dirac operators twisted by vector bundles.

Results expressed in K-theory and numerical indices.

Abstract

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to Spin$^c$-manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The result for cocompact actions is stated in terms of $K$-theory of group $C^*$-algebras, and the result for non-cocompact actions is an equality of numerical indices. In the non-cocompact case, the result generalises to Spin$^c$-Dirac operators twisted by vector bundles. This yields an index formula for Braverman's analytic index of such operators, in terms of characteristic classes on reduced spaces.