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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

If EF 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 EF be a quadratic extension of number fields, and let π be an SL n(𝔸F)-distinguished square-integrable automorphic representation of SL n(𝔸E). Let (σ,d) be the unique pair associated to π, where σ is a cuspidal representation of GL r(𝔸E) with n = dr. Using an unfolding argument, we prove that an element of the L -packet of π is distinguished with respect to SL n(𝔸F) if and only if it has a degenerate Whittaker model for a degenerate character ψ of type rd := (r,,r) of Nn(𝔸E) which is trivial on Nn(E + 𝔸F), where Nn is the group of unipotent upper triangular matrices of SL n. As a first application, under the assumptions that EF splits at infinity and r is odd, we establish a local–global principle for SL n(𝔸F)-distinction inside the L -packet of π. As a second application we construct examples of distinguished cuspidal automorphic representations π of SL n(𝔸E) such that the period integral vanishes on some canonical realization of π, and of everywhere locally distinguished representations of SL n(𝔸E) such that their L -packets do not contain any distinguished representation.

Galois distinction, Galois periods, $\mathrm{SL}(n)$, unitary representations, automorphic representations
Mathematical Subject Classification
Primary: 11F70
Secondary: 22E50
Received: 2 December 2020
Revised: 2 August 2021
Accepted: 3 January 2022
Published: 18 February 2023
U. K. Anandavardhanan
Department of Mathematics
Indian Institute of Technology Bombay
Mumbai, Maharashtra
Nadir Matringe
Institut de Mathématiques de Jussieu-Paris Rive Gauche
Université Paris Cité - Campus des Grands Moulins
Laboratoire Mathématiques et Applications
Université de Poitiers