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 $L^2(U(k)\mathrm{\backslash }{U}_{\mathbb{A}})$ is multiplicity free as a representation of $U_{\mathbb{A}}$. Setting $p^{+}:\phi \mapsto {\int }_{U^{+}(k)\mathrm{\backslash }{U}_{\mathbb{A}}^{+}}\phi (u)du$ to be the period integral attached to $\sigma$ on the space of smooth vectors of $L^2(U(k)\mathrm{\backslash }{U}_{\mathbb{A}})$, we prove that if $\mathrm{\Pi }$ is a topologically irreducible subspace of $L^2(U(k)\mathrm{\backslash }{U}_{\mathbb{A}})$, then $p^{+}$ is nonvanishing on the subspace ${\mathrm{\Pi }}^{\mathrm{\infty }}$ of smooth vectors in $\mathrm{\Pi }$ if and only if ${\mathrm{\Pi }}^{\vee }={\mathrm{\Pi }}^{\sigma }$. This is a global analogue of local results due to Y. Benoist and the author, on which the proof relies.