A fixed-point formula for Dirac operators on Lie groupoids

TL;DR

This paper derives a fixed-point formula for equivariant Dirac operators on Lie groupoids, extending classical index theorems to more general geometric structures with group actions.

Contribution

It introduces a fixed-point formula for Dirac operators on Lie groupoids using Getzler rescaling, generalizing known formulas to new settings.

Findings

Derived a fixed-point formula for equivariant Dirac operators on Lie groupoids

Reduced the formula to classical cases like the equivariant index on manifolds

Extended the approach to foliations and manifolds with divisors

Abstract

We study equivariant families of Dirac operators on the source fibers of a Lie groupoid with a closed space of units and equipped with an action of an auxiliary compact Lie group. We use the Getzler rescaling method to derive a fixed-point formula for the pairing of a trace with the K-theory class of such a family. For the pair groupoid of a closed manifold, our formula reduces to the standard fixed-point formula for the equivariant index of a Dirac operator. Further examples involve foliations and manifolds equipped with a normal crossing divisor.