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Exceptional supercuspidal representations in small residue characteristic

Yiannis Fam

Abstract

In this paper, in residue characteristic 2 and 3, we extend the construction of epipelagic representations of Reeder--Yu to produce new supercuspidals of higher depth, building on work of Gastineau. In particular, we produce examples of epipelagic representations that do not arise from the construction of Reeder--Yu.

Exceptional supercuspidal representations in small residue characteristic

Abstract

In this paper, in residue characteristic 2 and 3, we extend the construction of epipelagic representations of Reeder--Yu to produce new supercuspidals of higher depth, building on work of Gastineau. In particular, we produce examples of epipelagic representations that do not arise from the construction of Reeder--Yu.
Paper Structure (14 sections, 35 theorems, 102 equations)

This paper contains 14 sections, 35 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notation and Conventions
  4. Characters of some compact open subgroups
  5. Notation and review of Bruhat--Tits theory
  6. The abelianisation of $G(k)_{x,+:1}$
  7. Compactly induced representations and supercuspidals
  8. Generalities on intertwining
  9. Epipelagic representations
  10. Towards higher depth
  11. Examples
  12. $G=G_2$, $q=2$
  13. $G=B_n$, $q=2$
  14. $G=G_2$, $p=3$

Key Result

Theorem 1.1

Fix a nontrivial additive character $\chi: \mathbb{F}_q \to \mathbb{C}^\times$. Let $\lambda \in V(x)^\vee$ be an $\mathbb{F}_q$-stable functional and let $\chi_\lambda$ denote the inflation of $\chi \circ \lambda$ to a character of $G(k)_{x,+}$. The compactly induced representation is a finite direct sum of irreducible supercuspidal representations of $G(k)$.

Theorems & Definitions (76)

  • Theorem 1.1: Theorem \ref{['thm:main']}
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • ...and 66 more