Distinction inside L-packets of SL(n)

Anandavardhanan, U. K.; Matringe, Nadir

doi:10.2140/ant.2023.17.45

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Distinction inside $\mathrm{L}$-packets of $\mathrm{SL}(n)$

U. K. Anandavardhanan and Nadir Matringe

Vol. 17 (2023), No. 1, 45–82

DOI: 10.2140/ant.2023.17.45

Abstract

If $E ∕ F$ is a quadratic extension $p$ -adic fields, we first prove that the ${SL}_{n} (F)$ -distinguished representations inside a distinguished unitary $L$ -packet of ${SL}_{n} (E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N (E) ∕ N (F)$ . Then we establish a global analogue of this result. For this, let $E ∕ F$ be a quadratic extension of number fields, and let $π$ be an ${SL}_{n} (𝔸_{F})$ -distinguished square-integrable automorphic representation of ${SL}_{n} (𝔸_{E})$ . Let $(σ, d)$ be the unique pair associated to $π$ , where $σ$ is a cuspidal representation of ${GL}_{r} (𝔸_{E})$ with $n = d r$ . Using an unfolding argument, we prove that an element of the $L$ -packet of $π$ is distinguished with respect to ${SL}_{n} (𝔸_{F})$ if and only if it has a degenerate Whittaker model for a degenerate character $ψ$ of type $r^{d} : = (r, \dots, r)$ of $N_{n} (𝔸_{E})$ which is trivial on $N_{n} (E + 𝔸_{F})$ , where $N_{n}$ is the group of unipotent upper triangular matrices of ${SL}_{n}$ . As a first application, under the assumptions that $E ∕ F$ splits at infinity and $r$ is odd, we establish a local–global principle for ${SL}_{n} (𝔸_{F})$ -distinction inside the $L$ -packet of $π$ . As a second application we construct examples of distinguished cuspidal automorphic representations $π$ of ${SL}_{n} (𝔸_{E})$ such that the period integral vanishes on some canonical realization of $π$ , and of everywhere locally distinguished representations of ${SL}_{n} (𝔸_{E})$ such that their $L$ -packets do not contain any distinguished representation.

Keywords

Galois distinction, Galois periods, $\mathrm{SL}(n)$, unitary representations, automorphic representations

Mathematical Subject Classification

Primary: 11F70

Secondary: 22E50

Milestones

Received: 2 December 2020

Revised: 2 August 2021

Accepted: 3 January 2022

Published: 24 March 2023

Authors

U. K. Anandavardhanan
Department of Mathematics Indian Institute of Technology Bombay Mumbai, Maharashtra India

Nadir Matringe
Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Paris Cité - Campus des Grands Moulins Paris France	Laboratoire Mathématiques et Applications Université de Poitiers Poitiers France

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Vol. 17, No. 1, 2023