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 D0 . 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 F0 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. 1.9.The starting point of our strategy for proving Theorem 1.2 is to use Bushnell-Kutzko's construction of cuspidal representations of G via compact induction. This construction, elaborated in the complex case by Bushnell and Kutzko [10], has been extended to the modular case by Vignéras [46] and Minguez-Sécherre [36]. There is a family of pairs pJ, λq, made of certain compact mod centre open subgroups J of G and certain irreducible representation λ of J, such that:-for any such pair pJ, λq, the compact induction of λ to G is irreducible and cuspidal; -any irreducible cuspidal representation of G occurs in this way, for a pair pJ, λq uniquely determined up to G-conjugacy.Such pairs are called extended maximal simple types in [10], which we will abbreviate to types for simplicity. We need to give more details about the structure of these types:(1) the group J has a unique maximal compact subgroup J, and a unique maximal normal pro-p-group J 1 ;(2) there is a group isomorphism J{J 1 » GL m plq for some divisor m of n and finite extension l of the residual field k of F;(3) the restriction of λ to J 1 is made of copies of a single irreducible representation η, which extends (not uniquely, nor canonically) to J;(4) given a representation κ of J extending η, there is a unique irreducible representation ρ of J trivial on J 1 such that λ is isomorphic to κ b ρ; such a ρ restricts irreducibly to J; (5) the representation of GL m plq obtained by restricting ρ to J is cuspidal.The integer m, called the relative degree of π, is uniquely determined by π. There is another type-theoretical invariant called the tame param...

Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by an orthogonal involution

“…Let η be the Heisenberg representation of J 1 associated to θ, we have the following result as in [Séc19], Proposition 6.12 and [Zou19], Proposition 6.13: Proposition 5.11. Given g ∈ G, we have…”

“…Last but not least, it should also be pointed out that the method we use in this article is not new. It has first been initiated by Sécherre to solve the similar problem where τ is a Galois involution [AKM + 19], [Séc19], and then used and developed by the author for the the case where τ is a unitary involution [Zou19], and then used by Sécherre for the case where τ is an inner involution [Séc20] (there G can also be an inner form of GL n (F )). The sketches of the proof in different cases are similar, but one major difference in the current case is worth to be mentioned, that is, we need to consider those involution τ not contributing to the distinction.…”

“…Remark 5.10. Noting that in [Séc19] and [Zou19], the corresponding double coset lemma says that γ ∈ JB × J if and only if g ∈ JB × G τ . However in our case if we assume ε = ε 0 and γ = τ (g)g −1 ∈ JB × J, then it is possible that g is not in JB × G τ .…”

Galois self-dual cuspidal types and Asai local factors

“…Our first main result is the following. For further use in another context (see [47]), we state it and prove it for cuspidal representations of G with coefficients not necessarily in C, but more generally in an algebraically closed field R of characteristic different from p. For Bushnell-Kutzko's theory in this more general context, in particular the description of cuspidal R-representations by compact induction of extended maximal simple types, see [54,39].…”

Endo-parameters for p-adic classical groups

“…Endo-classes (endo-equivalence classes of ps-characters) have subsequently been extended to inner forms of general linear groups [5], and have proved fundamental in understanding fine properties of the local Langlands correspondence [10,11] and the Jacquet-Langlands correspondence [19,37], as well as in the study of Galois-distinguished cuspidal representations [1,33] and in Bernstein decompositions of the category of smooth representations over fields of positive characteristic [36].…”