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.

中文翻译：

---

特异表示、Shintani 基数变化和 Prasad 猜想的有限域模拟

令$E/F$ 是域的二次扩展，$G$ 是$F$ 上的连通准分裂归约群。令$G^{op}$为Prasad考虑的对偶对合对$G$进行扭曲得到的对立群。假设字段 $F$ 是有限的。令 $\pi$ 是 $G(E)$ 的不可约泛型表示。当 $\pi$ 是 $G^{op}(F)$ 的某些表示的 Shintani 基变化提升时，我们根据 $\pi 的 Whittaker 向量给出一个显式的非零 $G(F)$-不变向量$. 这特别表明 $\pi$ 是 $G(F)$-distinguished。当$F$域为$p$-adic 时，本文还证明了对偶对合取$G(F)$ 的不可约可容许泛型表示为其逆参。作为此结果的一个特例，$G_2、\ F_4$ 或 $E_8$ 的所有泛型表示都是自对偶的。