ON -DISTINGUISHED REPRESENTATIONS OF THE QUASI-SPLIT UNITARY GROUPS

Abstract

AbstractWe study$\text{Sp}_{2n}(F)$-distinction for representations of the quasi-split unitary group$U_{2n}(E/F)$in$2n$variables with respect to a quadratic extension$E/F$of$p$-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Mœglin–Tadić classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit$L$-packets with no distinguished members that transfer under base change to$\text{Sp}_{2n}(E)$-distinguished representations of$\text{GL}_{2n}(E)$.