The index of geometric operators on Lie groupoids

TL;DR

This paper extends the cohomological index theorem to Lie groupoids, providing new index formulas for geometric operators like the signature and Dirac operator using Lie algebroid cohomology.

Contribution

It proves a Thom isomorphism for Lie algebroids, enabling a reformulation of the index theorem's topological side for Lie groupoids.

Findings

Index formulas for Lie groupoid analogues of geometric operators

Thom isomorphism for Lie algebroids established

Reformulation of the index theorem in Lie algebroid cohomology

Abstract

We revisit the cohomological index theorem for elliptic elements in the universal enveloping algebra of a Lie groupoid previously proved by the authors. We prove a Thom isomorphism for Lie algebroids which enables us to rewrite the "topological side" of the index theorem. This results in index formulae for Lie groupoid analogues of the familiar geometric operators on manifolds such as the signature and Dirac operator expressed in terms of the usual characteristic classes in Lie algebroid cohomology.