SUPERCUSPIDAL REPRESENTATIONS OF GL_n(F) DISTINGUISHED BY A UNITARY INVOLUTION

Let F/F₀ be a quadratic extension of non-Archimedean locally compact fields of residue characteristic p 6 ≠ 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 subgroup of G with respect to F/F₀, 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 was first a conjecture proposed by Jacquet and was proved in the 2010s by Feigon–Lapid–Offen by using global methods. Our proof is local and works for both complex representations and l-modular representations with l 6= p. We further study the dimension of Hom_Gτ (π, 1) and show that it is at most 1.