Cartan Motion Group and Orbital Integrals

TL;DR

This paper investigates how orbital integrals, which are traces on group algebras, behave under deformations, showing continuity and invariance properties for regular elements while highlighting variations at singular elements.

Contribution

It demonstrates the continuity of orbital integrals under deformation groupoids and establishes invariance of pairings with K-theory for regular elements, revealing behavior at singular points.

Findings

Orbital integrals are continuous under deformation.

Pairings with K-theory are invariant for regular elements.

Pairings vary at singular elements.

Abstract

In this short note, we study the variation of orbital integrals, as traces on the group algebra $G$, under the deformation groupoid. We show that orbital integrals are continuous under the deformation. And we prove that the pairing between orbital integrals and $K$-theory element of $C^*_r(G)$ stays constant with respect to the deformation for regular group elements, but vary at singular elements.