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

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 $π$ be an irreducible generic representation of $G(E)$. When $π$ 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 $π$. This shows particularly that $π$ 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.

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