October, 2024 Symmetric periods for automorphic forms on unipotent groups
Nadir MATRINGE
Author Affiliations +
J. Math. Soc. Japan 76(4): 1187-1208 (October, 2024). DOI: 10.2969/jmsj/91279127

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.

Funding Statement

The author thanks the CNRS for granting us a “délégation” in 2022 from which this work benefited.

Citation

Download Citation

Nadir MATRINGE. "Symmetric periods for automorphic forms on unipotent groups." J. Math. Soc. Japan 76 (4) 1187 - 1208, October, 2024. https://doi.org/10.2969/jmsj/91279127

Information

Received: 26 April 2023; Published: October, 2024
First available in Project Euclid: 11 April 2024

Digital Object Identifier: 10.2969/jmsj/91279127

Subjects:
Primary: 11F70
Secondary: 22E50

Keywords: automorphic periods , unipotent groups

Rights: Copyright ©2024 Mathematical Society of Japan

Vol.76 • No. 4 • October, 2024
Back to Top