Symmetric periods for automorphic forms on unipotent groups

Let $k$ be a number field and $\mathbb{A}$ be its ring of adeles. Let $U$ be a unipotent group defined over $k$, and $\sigma$ a $k$-rational involution of $U$ with fixed points $U^+$. As a consequence of the results of C. Moore, the space $L^2(U(k)\backslash U_{\mathbb{A}})$ is multiplicity free as a representation of $U_{\mathbb{A}}$. Setting $p^+:\phi\mapsto \int_{U^+(k)\backslash {U}_{\mathbb{A}}^+} \phi(u)du$ to be the period integral attached to $\sigma$ on the space of smooth vectors of $L^2(U(k)\backslash U_{\mathbb{A}})$, we prove that if $\Pi$ is a topologically irreducible subspace of $L^2(U(k)\backslash U_{\mathbb{A}})$, then $p^+$ is nonvanishing on the subspace $\Pi^\infty$ of smooth vectors in $\Pi$ if and only if $\Pi^\vee=\Pi^\sigma$. This is a global analogue of local results due to Y. Benoist and the author, on which the proof relies.