Computations of orbital integrals and Shalika germs

TL;DR

This paper introduces a novel method for computing orbital integrals in reductive groups over non-archimedean fields, enabling the calculation of Shalika germs and local character expansions for certain supercuspidal representations.

Contribution

It presents a new approach to compute orbital integrals and Shalika germs using geometric techniques related to Hessenberg varieties, expanding the tools available for representation theory.

Findings

Computed orbital integrals for very elliptic elements.

Expressed Shalika germs in terms of rational points on Hessenberg covers.

Connected Shalika germs to Harish-Chandra local character expansions.

Abstract

For a reductive group $G$ over a non-archimedean local field, with some assumptions on (residue) characteristic we give an method to compute certain orbital integrals using a method close to that of Goresky-Kottiwitz-MacPherson but in a different language. These orbital integrals allow us to compute the Shalika germs at some ``very elliptic'' elements in terms of number of rational points on some quasi-finite covers of the Hessenberg varieties of GKM, which are subvarieties of (partial) flag varieties. Such values of Shalika germs determine the Harish-Chandra local character expansions of the so-called very supercuspidal representations.