Supercuspidal representations of \(\text{GL}(n)\) distinguished by a unitary subgroup (Q1366636)

When \(E/F\) is a quadratic extension of \(p\)-adic fields, with \(p\neq 2\), and \(H'\) is a unitary similitude group in \(\text{GL}(n,E)\), it is shown that for every irreducible supercuspidal representation \(\pi\) of \(\text{GL}(n,E)\) of lowest level the space of \(H'\)-invariant linear forms has dimension at most one. The analogous fact for the corresponding unitary group \(H\) also holds, so long as \(n\) is odd or \(E/F\) is ramified. Then \(n\) is even and \(E/F\) is unramified, the space of \(H\)-invariant linear forms on the space of \(\pi\) may have dimension two.