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Local character expansions and asymptotic cones over finite fields

Dan Ciubotaru, Emile Okada

TL;DR

The paper tackles the problem of determining wave front sets for $p$-adic representations with positive depth by extending Barbasch–Moy test functions to $\mathbb{R}/\mathbb{Z}$-graded Lie algebras. It introduces graded generalised Gelfand-Graev representations and a family of test functions $f_{x,r,\underline{\mathcal{O}}^*}$ whose Fourier-analytic data control local character expansions, linking nonvanishing chars to rational asymptotic cones of Fourier supports. A central result provides a concrete formula for the wave front set: $\mathrm{WF}(\pi)=\max\bigcup_{x}( -\mathscr L_{x,-r}(\mathrm{cone}(\mathrm{supp}(\mathrm{FT}_{x,r}(\theta_{\pi,x,r})))) )$, with a bound and closed-form cone descriptions via Lusztig-Spaltenstein induction. The paper culminates with an explicit example of toral positive-depth supercuspidals, showing how the cone framework recovers DeBacker–Reeder genericity criteria and yields a precise geometric wave front set.

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.

Local character expansions and asymptotic cones over finite fields

TL;DR

The paper tackles the problem of determining wave front sets for -adic representations with positive depth by extending Barbasch–Moy test functions to -graded Lie algebras. It introduces graded generalised Gelfand-Graev representations and a family of test functions whose Fourier-analytic data control local character expansions, linking nonvanishing chars to rational asymptotic cones of Fourier supports. A central result provides a concrete formula for the wave front set: , with a bound and closed-form cone descriptions via Lusztig-Spaltenstein induction. The paper culminates with an explicit example of toral positive-depth supercuspidals, showing how the cone framework recovers DeBacker–Reeder genericity criteria and yields a precise geometric wave front set.

Abstract

We generalise Gelfand-Graev characters to -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 -graded Lie algebras. As an illustration, we compute the geometric wave front set of certain toral supercuspidal representations in a straightforward manner.
Paper Structure (14 sections, 27 theorems, 83 equations)

This paper contains 14 sections, 27 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Preliminaries
  4. The wave front set of characters in $\mathbb R/\mathbb Z$-graded Lie algebras
  5. Graded generalised Gelfand-Graev representations
  6. The wave front set
  7. The wave front set of $0$-graded characters
  8. Borho's theory of $G$-Sheets
  9. The asymptotic cone and rational points
  10. Wave front sets
  11. The wave front set of positive depth representations
  12. Preliminaries
  13. Main theorem
  14. Example

Key Result

Theorem 1.1

Let $r\ge 0$. For every $x\in \mathscr B(\mathbb G,F)$ and $\underline{\mathcal{O}}^*\in \mathcal{N}(\underline{\mathfrak g}^*_{x,-r})/\underline G_x$ the test functions $f_{x,r,\underline{\mathcal{O}}^*}$ defined in Definition d:test-function satisfy Note that for any $\mathcal{O}^*\in \mathcal{N}(\mathfrak g^*)/G$ there exists $x\in \mathscr B(\mathbb G,F)$ and $\underline{\mathcal{O}}^*\in \ma

Theorems & Definitions (59)

  • Theorem 1.1: Lemma \ref{['lem:eval']}, Proposition \ref{['prop:orbitalint']}
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5: Corollary \ref{['cor:bound']}
  • Theorem 1.6: Theorem \ref{['thm:ratwfungraded']}
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 49 more