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

Abstract

Let F/F0 be a quadratic extension of non-archimedean locally compact fields of residue characteristic p = 2. Let R be an algebraically closed field of characteristic different from p. For π a supercuspidal representation of G = GLn(F ) over R and G τ a unitary group in n variables contained in G, we prove that π is distinguished by G τ if and only if π is Galois invariant. When R = C and F is a p-adic field, this result first as a conjecture proposed by Jacquet was proved in 2010's by Feigon-Lapid-Offen by using global method. Our proof is local which works for both complex case and l-modular case with l = p. We further study the dimension of HomGτ (π, 1) and show that it is at most one.

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

“…as an anti-involution on M n (F ). Then we may use the same argument in [Zou19], Proposition 5.19, with σ t in loc. cit.…”

“…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.…”

“…The novelty of our argument is first to consider a special involution τ 0 , and then to regard τ as another involution which differs from τ 0 up to a G-conjugation. Thus we choose (J, Λ) to be a τ 0 -selfdual simple type and, using the general results built up by the author in [Zou19], we can still study those J-G τ double cosets contributing to the distinction. If one wants to fit the method in the above cases to a general involution τ , one major problem encountered is to construct a τ -selfdual simple type, which, as we explained, may be impossible if τ does not contribute to the distinction.…”

“…In this subsection, we follow the introduction of the simple type theory given in [Zou19], section 3 summarizing results of [BK93], [BH96], [BH14]. Since it seems redundant to repeat the same words again, we simply recall the necessary notation and refer to [Zou19] for more details.…”

“…In this section, we follow the strategy in [Zou19], section 5 and [AKM + 19], section 4 to prove the following theorem: Theorem 4.1. For π and τ as above, there exists a maximal simple stratum [a, β] and a simple character θ ∈ C(a, β) contained in π, such that…”