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

Abstract

Let F{F0 be a quadratic extension of non-Archimedean locally compact fields of residual characteristic p ‰ 2, and let σ denote its non-trivial automorphism.Let R be an algebraically closed field of characteristic different from p. To any cuspidal representation π of GLnpFq, with coefficients in R, such that π σ » π _ (such a representation is said to be σ-selfdual) we associate a quadratic extension D{D0, where D is a tamely ramified extension of F and D0 is a tamely ramified extension of F0, together with a quadratic character of D 0 .When π is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for π to be GLnpF0q-distinguished.When the characteristic ℓ of R is not 2, denoting by ω the non-trivial R-character of F 0 trivial on F{F0-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 ą 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-Tandon, when p ‰ 2.

Locations

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