Symmetric periods for automorphic forms on unipotent groups

Abstract

Let 𝑘 be a number field and 𝔸 be its ring of adeles. Let 𝑈 be a unipotent group defined over 𝑘, and 𝜎 a 𝑘-rational involution of 𝑈 with fixed points 𝑈⁺. As a consequence of the results of Moore, the space 𝐿²(𝑈(𝑘) ⧵ 𝑈_𝔸) is multiplicity free as a representation of 𝑈_𝔸. Setting 𝑝⁺ to be the period integral attached to 𝜎 on the space of smooth vectors of 𝐿²(𝑈(𝑘) ⧵ 𝑈_𝔸), we prove that if Π is a topologically irreducible subspace of 𝐿²(𝑈(𝑘) ⧵ 𝑈_𝔸), then 𝑝⁺ is nonvanishing on the subspace of smooth vectors in Π if and only if Π^∨ = Π^𝜎. This is a global analogue of local results of Benoist and the author, on which the proof relies.