Asai Gamma Factors and Distinction in families

Let $F$ be a finite extension of $\mathbb{Q}_p$ and let $E$ be a quadratic extension of $F$. A representation $(\pi,V)$ of ${\rm GL}_n(E)$ is said to be ${\rm GL}_n(F)$-distinguished if there exists a non-zero linear functional $\phi$ on $V$ such that $\phi(\pi(h)v) = \phi(v)$ for all $h \in {\rm GL}_n(F)$ and $v \in V$. In this article, we study the notion of ${\rm GL}_n(F)$-distinguished representations for $R[{\rm GL}_n(E)]$ 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 $R[{\rm GL}_n(E)]$ modules of Whittaker type. We then provide a necessary condition for cuspidal $R[{\rm GL}_n(E)]$ modules of Whittaker type to be Whittaker ${\rm GL}_n(F)$-distinguished, expressed in terms of their Asai $\gamma$-factors.