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2019 Supercuspidal representations of ${\rm GL}_n({\rm F})$ distinguished by a Galois involution
Vincent Sécherre
Algebra Number Theory 13(7): 1677-1733 (2019). 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 πσπ (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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Vincent Sécherre. "Supercuspidal representations of ${\rm GL}_n({\rm F})$ distinguished by a Galois involution." Algebra Number Theory 13 (7) 1677 - 1733, 2019.


Received: 19 July 2018; Revised: 19 March 2019; Accepted: 25 May 2019; Published: 2019
First available in Project Euclid: 16 January 2020

zbMATH: 07110519
MathSciNet: MR4009674
Digital Object Identifier: 10.2140/ant.2019.13.1677

Primary: 22E50
Secondary: 11F70 , 11F85

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

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.13 • No. 7 • 2019
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