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.

Let F be a non-archimedean locally compact field of residue characteristic p = 2, let G = GLn(F ) and let H be an orthogonal subgroup of G. For π a complex smooth supercuspidal representation of G, we give a full characterization for the distinguished space HomH(π, 1) being non-zero and we further study its dimension as a complex vector space, which generalizes a similar result of Hakim for tame supercuspidal representations. As a corollary, the embeddings of π in the space of smooth functions on the set of symmetric matrices in G, as a complex vector space, is non-zero and of dimension four, if and only if the central character of π evaluating at −1 is 1.

Let F be a non-archimedean local field. The classification of the irreducible representations of GLn(F ), n ≥ 0 in terms of supercuspidal representations is one of the highlights of the Bernstein-Zelevinsky theory. We give an analogous classification for metaplectic coverings of GLn(F ), n ≥ 0.

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