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

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.

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cuspidal representation, distinguished representation, Galois involution, modular representation, p-adic reductive group
Mathematical Subject Classification 2010
Primary: 22E50
Secondary: 11F70, 11F85
Received: 19 July 2018
Revised: 19 March 2019
Accepted: 25 May 2019
Published: 21 September 2019
Vincent Sécherre
Laboratoire de Mathématiques de Versailles, UVSQ, CNRS
Université Paris-Saclay
78035 Versailles