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 C. Moore, the space $L^2(U(k)\backslash U_{\mathbb{A}})$ is multiplicity free as a representation of $U_{\mathbb{A}}$. Setting $p^{+}:\varphi \mapsto {\int }_{U^{+}(k)\backslash U_{\mathbb{A}}^{+}}\varphi (u)du$ 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.

The proof of Proposition 3.2 on the convergence of the Richardson intertwining operators has been corrected and expanded. We added a paragraph on the very strong rigidity property of automorphic representations of unipotent groups with a corollary concerning distinction