Supercuspidal representations of GLn(F) distinguished by a Galois involution

Sécherre, Vincent

doi:10.2140/ant.2019.13.1677

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Supercuspidal representations of ${\rm GL}_n({\rm F})$ distinguished by a Galois involution

Vincent Sécherre

Vol. 13 (2019), No. 7, 1677–1733

DOI: 10.2140/ant.2019.13.1677

Abstract

Let $F ∕ F_{0}$ be a quadratic extension of nonarchimedean locally compact fields of residual characteristic $p \neq 2$ and let $σ$ denote its nontrivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$ . To any cuspidal representation $π$ of ${G L}_{n} (F)$ , with coefficients in $R$ , such that $π^{σ} ≃ π^{\lor}$ (such a representation is said to be $σ$ -selfdual) we associate a quadratic extension $D ∕ D_{0}$ , where $D$ is a tamely ramified extension of $F$ and $D_{0}$ is a tamely ramified extension of $F_{0}$ , together with a quadratic character of $D_{0}^{\times}$ . When $π$ is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for $π$ to be ${G L}_{n} (F_{0})$ -distinguished. When the characteristic $ℓ$ of $R$ is not $2$ , denoting by $ω$ the nontrivial $R$ -character of $F_{0}^{\times}$ trivial on $F ∕ F_{0}$ -norms, we prove that any $σ$ -selfdual supercuspidal $R$ -representation is either distinguished or $ω$ -distinguished, but not both. In the modular case, that is when $ℓ > 0$ , we give examples of $σ$ -selfdual cuspidal nonsupercuspidal representations which are not distinguished nor $ω$ -distinguished. In the particular case where $R$ is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan, Kable and Tandon, when $p \neq 2$ .

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Keywords

cuspidal representation, distinguished representation, Galois involution, modular representation, p-adic reductive group

Mathematical Subject Classification 2010

Primary: 22E50

Secondary: 11F70, 11F85

Milestones

Received: 19 July 2018

Revised: 19 March 2019

Accepted: 25 May 2019

Published: 21 September 2019

Authors

Vincent Sécherre
Laboratoire de Mathématiques de Versailles, UVSQ, CNRS Université Paris-Saclay 78035 Versailles France

Vol. 13, No. 7, 2019