Asai Gamma Factors and Distinction in families

Abstract

Let $F$ be a finite extension of ${\mathbb{Q}}_p$ and let $E$ be a quadratic extension of $F$. A representation $(\pi ,V)$ of ${\mathrm{GL}}_n(E)$ is said to be ${\mathrm{GL}}_n(F)$-distinguished if there exists a non-zero linear functional $\varphi$ on $V$ such that $\varphi (\pi (h)v)=\varphi (v)$ for all $h\in {\mathrm{GL}}_n(F)$ and $v\in V$. In this article, we study the notion of ${\mathrm{GL}}_n(F)$-distinguished representations for $R[{\mathrm{GL}}_n(E)]$ modules of Whittaker type, where $R$ is a Noetherian algebra over the ring of Witt vectors of ${\overline{\mathbb{F}}}_{\ell }$ with $\ell \ne p$. We first derive a functional equation, which gives the existence of the Asai $\gamma$-factors associated with $R[{\mathrm{GL}}_n(E)]$ modules of Whittaker type. We then provide a necessary condition for cuspidal $R[{\mathrm{GL}}_n(E)]$ modules of Whittaker type to be Whittaker ${\mathrm{GL}}_n(F)$-distinguished, expressed in terms of their Asai $\gamma$-factors.

15 pages