Test vectors for finite periods and base change

Let $E\mathrm{/}F$ be a quadratic extension of finite fields. By a result of Gow, an irreducible representation $\pi$ of $G={\mathrm{G}\mathrm{L}}_n(E)$ has at most one non-zero $H$-invariant vector, up to multiplication by scalars, when $H$ is ${\mathrm{G}\mathrm{L}}_n(F)$ or $\mathrm{U}(n,E\mathrm{/}F)$. If $\pi$ does have an $H$-invariant vector it is said to be $H$-distinguished. It is known, from the work of Gow, that $H$-distinction is characterized by base change from $\mathrm{U}(n,E\mathrm{/}F)$, due to Kawanaka, when $H$ is ${\mathrm{G}\mathrm{L}}_n(F)$ (resp. from ${\mathrm{G}\mathrm{L}}_n(F)$, due to Shintani, when $H$ is $\mathrm{U}(n,E\mathrm{/}F)$). Assuming $\pi$ is generic and $H$-distinguished, we give an explicit description of the $H$-invariant vector in terms of the Bessel function of $\pi$. Let $\psi$ be a non-degenerate character of $N_G\mathrm{/}{N}_H$ and let $B_{\pi ,\psi }$ be the (normalized) Bessel function of $\pi$ on the $\psi$-Whittaker model. For the $H$-average $$W_{\pi ,\psi }=\frac{1}{\mathrm{\mid }H\mathrm{\mid }}\sum _{h\in H}\pi (h)B_{\pi ,\psi }$$ of the Bessel function, we prove that $$W_{\pi ,\psi }(I_n)=\frac{\mathrm{d}\mathrm{i}\mathrm{m}\rho }{\mathrm{d}\mathrm{i}\mathrm{m}\pi }\cdot \frac{\mathrm{\mid }{\mathrm{G}\mathrm{L}}_n(E)\mathrm{\mid }}{\mathrm{\mid }{\mathrm{G}\mathrm{L}}_n(F)\mathrm{\mid }\mathrm{\mid }\mathrm{U}(n,E\mathrm{/}F)\mathrm{\mid }},$$ where $\rho$ is the representation of $\mathrm{U}(n,E\mathrm{/}F)$ (resp. ${\mathrm{G}\mathrm{L}}_n(F)$) that base changes to $\pi$ when $H$ is ${\mathrm{G}\mathrm{L}}_n(F)$ (resp. ${\rm U}(n,E/F)$)\mathrm{.}Asanapplicationweclassifythemembersofageneric$L$-packet of ${\mathrm{S}\mathrm{L}}_n(E)$ that admit invariant vectors for ${\mathrm{S}\mathrm{L}}_n(F)$. Finally we prove a $p$-adic analogue of our result for square-integrable representations in terms of formal degrees by employing the formal degree conjecture of Hiraga-Ichino-Ikeda \cite{hii08}.