Some questions on global distinction for $\mathrm{SL}(n)$

Abstract

Let $E/F$ be a quadratic extension of number fields and let $\pi$ be an ${\mathrm{SL}}_n({\mathbb{A}}_F)$-distinguished cuspidal automorphic representation of ${\mathrm{SL}}_n({\mathbb{A}}_E)$. Using an unfolding argument, we prove that an element of the $\mathrm{L}$-packet of $\pi$ is distinguished if and only if it is $\psi$-generic for a non-degenerate character $\psi$ of $N_n({\mathbb{A}}_E)$ trivial on $N_n(E+{\mathbb{A}}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of ${\mathrm{SL}}_n$. We then use this result to analyze the non-vanishing of the period integral on different realizations of a distinguished cuspidal automorphic representation of ${\mathrm{SL}}_n({\mathbb{A}}_E)$ with multiplicity $>1$, and show that in general some canonical copies of a distinguished representation inside different $\mathrm{L}$-packets can have vanishing period. We also construct examples of everywhere locally distinguished representations of ${\mathrm{SL}}_n({\mathbb{A}}_E)$ the $\mathrm{L}$-packets of which do not contain any distinguished representation.

To be merged with arXiv:2010.05678 and completed and simplified there, in particular we get rid of the Grunwald-Wang assumption

References in corpus (1)

• A local global question in automorphic forms