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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
##### Abstract

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

##### Keywords
Galois distinction, Galois periods, $\mathrm{SL}(n)$, unitary representations, automorphic representations
Primary: 11F70
Secondary: 22E50