Distinction inside L-packets of SL(n)

Abstract

If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the ${\mathrm{SL}}_n(F)$-distinguished representations inside a distinguished unitary L-packet of ${\mathrm{SL}}_n(E)$ 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 $\pi$ be an ${\mathrm{SL}}_n({\mathbb{A}}_F)$-distinguished square integrable automorphic representation of ${\mathrm{SL}}_n({\mathbb{A}}_E)$. Let $(\sigma ,d)$ be the unique pair associated to $\pi$, where $\sigma$ is a cuspidal representation of ${\mathrm{GL}}_r({\mathbb{A}}_E)$ with $n=dr$. Using an unfolding argument, we prove that an element of the L-packet of $\pi$ is distinguished with respect to ${\mathrm{SL}}_n({\mathbb{A}}_F)$ if and only if it has a degenerate Whittaker model for a degenerate character $\psi$ of type $r^d:=(r,\dots ,r)$ of $N_n({\mathbb{A}}_E)$ which is trivial on $N_n(E+{\mathbb{A}}_F)$, 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 ${\mathrm{SL}}_n({\mathbb{A}}_F)$-distinction inside the L-packet of $\pi$. As a second application we construct examples of distinguished cuspidal automorphic representations $\pi$ of ${\mathrm{SL}}_n({\mathbb{A}}_E)$ such that the period integral vanishes on some canonical copy of $\pi$, and of everywhere locally distinguished representations of ${\mathrm{SL}}_n({\mathbb{A}}_E)$ such that their L-packets do not contain any distinguished representation.

Merged with withdrawn arXiv:1906.11560. We simplified some arguments and removed an unnecessary Grunwald-Wang assumption

References in corpus (1)

• Automorphic period and the central value of Rankin--Selberg L-function

Cited by in corpus (1)

• Distinction and Base Change