Symmetric periods for automorphic forms on unipotent groups

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) UA) is multiplicity free as a representation of UA. Setting p⁺ to be the period integral attached to σ on the space of smooth vectors of L²(U(k) UA), we prove that if Π is a topologically irreducible subspace of L²(U(k) UA), then p⁺ is nonvanishing on the subspace of smooth vectors in Π if and only if Π^ = Πσ. This is a global analogue of local results of Benoist and the author, on which the proof relies.