Geometric orbital integrals and the center of the enveloping algebra

TL;DR

This paper extends the explicit geometric evaluation of semisimple orbital integrals from the Casimir operator to all elements in the center of the enveloping algebra, broadening the scope of orbital integral analysis.

Contribution

It provides a generalized method for evaluating orbital integrals for arbitrary central elements, building upon previous work on the Casimir operator.

Findings

Extended explicit geometric evaluation to arbitrary central elements.

Unified approach for orbital integrals across different central elements.

Enhanced understanding of the center of the enveloping algebra.

Abstract

The purpose of this paper is to extend the explicit geometric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of the enveloping algebra.