Symmetric periods for automorphic forms on unipotent groups

Abstract

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 Moore, the space $L^{2}(U(k) \backslash U_{\mathbb{A}})$ is multiplicity free as a representation of $U_{\mathbb{A}}$. Setting $p^+$ 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 of smooth vectors in $\Pi$ if and only if $\Pi^{\vee} = \Pi^{\sigma}$. This is a global analogue of local results of Benoist and the author, on which the proof relies.