Globalization of distinguished supercuspidal representations of GL(n)

An irreducible supercuspidal representation 𝜋 of 𝐺 = GL(n, 𝐹), where 𝐹 is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup 𝐻 of 𝐺 and a quasicharacter 𝒳 of 𝐻 if Hom𝐻(𝜋, 𝒳) ≠ 0. There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to GL(n). Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.