The transverse index theorem for proper cocompact actions of Lie groupoids

TL;DR

This paper develops a higher index pairing for proper cocompact Lie groupoid actions, providing a cohomological index formula and illustrating its significance through examples.

Contribution

It introduces a new higher index pairing for invariant elliptic operators on Lie groupoids and derives a cohomological index formula using algebraic index theory.

Findings

Established a cohomological index formula for the pairing

Applied the van Est map in the index computation

Demonstrated the index pairing in specific examples

Abstract

Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples the meaning of the index pairing and our index formula.