Local character expansions and asymptotic cones over finite fields

TL;DR

This paper extends the theory of Gelfand-Graev characters to graded Lie algebras over finite fields, introducing new test functions that facilitate the analysis of local character expansions and their relation to asymptotic cones.

Contribution

It generalizes Gelfand-Graev characters to graded Lie algebras and develops new test functions to analyze local character expansions in positive depth.

Findings

New test functions effectively compute leading terms of local character expansion.

Established connection between local character expansion and asymptotic cones.

Computed geometric wave front set of certain supercuspidal representations.

Abstract

We generalise Gelfand-Graev characters to $\mathbb R/\mathbb Z$-graded Lie algebras and lift them to produce new test functions to probe the local character expansion in positive depth. We show that these test functions are well adapted to compute the leading terms of the local character expansion and relate their determination to the asymptotic cone of elements in $\mathbb Z/n$-graded Lie algebras. As an illustration, we compute the geometric wave front set of certain toral supercuspidal representations in a straightforward manner.