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.