Inductive structure of Shalika germs and affine Springer fibers

TL;DR

This paper introduces an inductive algorithm to compute Shalika germs and affine Springer fibers, linking their structures to subgroups, combinatorics, and rational point counts, with applications to orbital integrals.

Contribution

It presents a novel inductive framework and algorithm for calculating Shalika germs and affine Springer fibers using subgroup data, combinatorics, and point counts.

Findings

Algorithm for computing Shalika germs and affine Springer fibers.

Connection between structures and rational point counts.

Applications to orbital integrals.

Abstract

This article has two parallel perspectives: to demonstrate an inductive structure of Shalika germs, and to show an analogous inductive structure for affine Springer fibers. More precisely, we give an algorithm to compute arbitrary Shalika germs (resp. affine Springer fibers up to stratification) in terms of three ingredients: Shalika germs (resp. affine Springer fibers) for twisted Levi subgroups, a finite list of combinatorial objects, and the numbers of rational points on varieties over the residue field (resp. varieties themselves) among an explicit finite list of such. We also discuss some formal applications of the algorithm to Shalika germs and orbital integrals.