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Twisted character formula for toral supercuspidal representations

Masao Oi

TL;DR

This work extends the Adler–Spice and DeBacker–Spice framework to twisted endoscopy, deriving an explicit formula for twisted Harish-Chandra characters of toral supercuspidal representations in the twisted setting. By developing twisted normal $r$-approximations, twisted Heisenberg–Weil representations, and a twisted ADS-type descent, the authors express the twisted character as a sum over appropriate double cosets with head–tail separation and descent to centralizers. The results leverage Kaletha’s toral data, the descent of root data, and Gérardin-type formulas to produce a computable, case-sensitive character formula applicable to quasi-split groups with involutions. The findings facilitate twisted endoscopy applications and contribute to explicit local Langlands correspondences for classical groups in the twisted context.

Abstract

We establish an explicit formula for twisted Harish-Chandra characters of toral supercuspidal representations of p-adic reductive groups under several technical assumptions. Our setup especially includes the case of a quasi-split group equipped with a involution.

Twisted character formula for toral supercuspidal representations

TL;DR

This work extends the Adler–Spice and DeBacker–Spice framework to twisted endoscopy, deriving an explicit formula for twisted Harish-Chandra characters of toral supercuspidal representations in the twisted setting. By developing twisted normal -approximations, twisted Heisenberg–Weil representations, and a twisted ADS-type descent, the authors express the twisted character as a sum over appropriate double cosets with head–tail separation and descent to centralizers. The results leverage Kaletha’s toral data, the descent of root data, and Gérardin-type formulas to produce a computable, case-sensitive character formula applicable to quasi-split groups with involutions. The findings facilitate twisted endoscopy applications and contribute to explicit local Langlands correspondences for classical groups in the twisted context.

Abstract

We establish an explicit formula for twisted Harish-Chandra characters of toral supercuspidal representations of p-adic reductive groups under several technical assumptions. Our setup especially includes the case of a quasi-split group equipped with a involution.
Paper Structure (38 sections, 47 theorems, 266 equations)

This paper contains 38 sections, 47 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Notation and assumptions on $p$
  3. Notation
  4. $p$-adic fields
  5. Algebraic varieties and algebraic groups
  6. Centralizers and normalizers
  7. Reductive groups
  8. Bruhat--Tits theory
  9. Finite sets with Galois actions
  10. Finite fields
  11. Assumptions on $p$
  12. Twisted spaces
  13. Twisted spaces
  14. Twisted maximal torus
  15. Steinberg's result on the structure of descended groups
  16. ...and 23 more sections

Key Result

Theorem 1.1

Suppose that $p\gg0$. Let $\pi$ be a toral supercuspidal representation of $\mathbf{G}(F)$ corresponding to a toral pair $(\mathbf{S},\vartheta)$. Let $r\in\mathbb{R}_{\geq0}$ be the depth of $\pi$. For any elliptic regular semisimple element $\gamma\in \mathbf{G}(F)$ with a normal $r$-approximation where $\Phi_\pi$ denotes the normalized Harish-Chandra character of $\pi$.

Theorems & Definitions (111)

  • Theorem 1.1: Adler--DeBacker--Spice character formula
  • Theorem 1.2: Theorem \ref{['thm:TCF-final']}
  • Example 3.1
  • Definition 3.2: MW18
  • Lemma 3.3
  • proof
  • Example 3.4
  • Proposition 3.5
  • proof
  • Definition 3.6
  • ...and 101 more