An analytic index for Lie groupoids

TL;DR

This paper introduces a new analytic index for Lie groupoids that facilitates pairing with cyclic cocycles, providing a more accessible way to extract numerical invariants from the $K$-theory of the associated $C^*$-algebra.

Contribution

It defines a novel analytic index morphism for Lie groupoids that maps into a group enabling pairings with cyclic cocycles, using tangent groupoid deformation.

Findings

New index allows pairing with cyclic cocycles

Uses tangent groupoid for index deformation

Simplifies extraction of numerical invariants

Abstract

For a Lie groupoid there is an analytic index morphism which takes values in the $K-$theory of the $C^*$-algebra associated to the groupoid. This is a good invariant but extracting numerical invariants from it, with the existent tools, is very difficult. In this work, we define another analytic index morphism associated to a Lie groupoid; this one takes values in a group that allows us to do pairings with cyclic cocycles. This last group is related to the compactly supported functions on the groupoid. We use the tangent groupoid to define our index as a sort of ''deformation''.