Distinguished representations, Shintani base change and a finite field analogue of a conjecture of Prasad

Abstract: Let $E/F$ be a quadratic extension of fields, and $G$ a connected quasi-split reductive group over $F$. Let $G^{op}$ be the opposition group obtained by twisting $G$ by the duality involution considered by Prasad. Assume that the field $F$ is finite. Let $\pi$ be an irreducible generic representation of $G(E)$. When $\pi$ is a Shintani base change lift of some representation of $G^{op}(F)$, we give an explicit nonzero $G(F)$-invariant vector in terms of the Whittaker vector of $\pi$. This shows particularly that $\pi$ is $G(F)$-distinguished. When the field $F$ is $p$-adic, the paper also proves that the duality involution takes an irreducible admissible generic representation of $G(F)$ to its contragredient. As a special case of this result, all generic representations of $G_2,\ F_4$ or $E_8$ are self-dual.