Vol. 13, No. 7, 2019

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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 FF0 be a quadratic extension of nonarchimedean locally compact fields of residual characteristic p2 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 GLn(F), with coefficients in R, such that πσ π (such a representation is said to be σ-selfdual) we associate a quadratic extension DD0, where D is a tamely ramified extension of F and D0 is a tamely ramified extension of F0, together with a quadratic character of D0×. When π is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for π to be GLn(F0)-distinguished. When the characteristic of R is not 2, denoting by ω the nontrivial R-character of F0× trivial on FF0-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 > 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 p2.

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