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# Symmetric periods for automorphic forms on unipotent groups

Dec 2022

Let $k$ be a number field and $\mathbb{A}$ be its ring of adeles. Let $U$ bea 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, thespace $L^2(U(k)\backslash U_{\mathbb{A}})$ is multiplicity free as arepresentation of $U_{\mathbb{A}}$. Setting $p^+:\phi\mapsto\int_{U^+(k)\backslash {U}_{\mathbb{A}}^+} \phi(u)du$ to be the period integralattached to $\sigma$ on the space of smooth vectors of $L^2(U(k)\backslashU_{\mathbb{A}})$, we prove that if $\Pi$ is a topologically irreduciblesubspace of $L^2(U(k)\backslash U_{\mathbb{A}})$, then $p^+$ is nonvanishing onthe 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.

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