English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let $E/F$ be a quadratic extension of finite fields. By a result of Gow, anirreducible representation $\pi$ of $G = {\rm GL}_n(E)$ has at most onenon-zero $H$-invariant vector, up to multiplication by scalars, when $H$ is${\rm GL}_n(F)$ or ${\rm U}(n,E/F)$. If $\pi$ does have an $H$-invariant vectorit is said to be $H$-distinguished. It is known, from the work of Gow, that$H$-distinction is characterized by base change from ${\rm U}(n,E/F)$, due toKawanaka, when $H$ is ${\rm GL}_n(F)$ (resp. from ${\rm GL}_n(F)$, due toShintani, when $H$ is ${\rm U}(n,E/F)$). Assuming $\pi$ is generic and$H$-distinguished, we give an explicit description of the $H$-invariant vectorin terms of the Bessel function of $\pi$. Let $\psi$ be a non-degeneratecharacter of $N_G/N_H$ and let $B_{\pi,\psi}$ be the (normalized) Besselfunction of $\pi$ on the $\psi$-Whittaker model. For the $H$-average\[W_{\pi,\psi} = \frac{1}{|H|} \sum_{h\in H} \pi(h) B_{\pi,\psi}\] of theBessel function, we prove that \[W_{\pi,\psi}(I_n) = \frac{{\rm dim}\rho}{{\rmdim} \pi} \cdot \frac{|{\rm GL}_n(E)|}{|{\rm GL}_n(F)| |{\rm U}(n,E/F)|},\]where $\rho$ is the representation of ${\rm U}(n,E/F)$ (resp. ${\rm GL}_n(F)$)that base changes to $\pi$ when $H$ is ${\rm GL}_n(F)$ (resp. ${\rmU}(n,E/F)$). As an application we classify the members of a generic $L$-packetof ${\rm SL}_n(E)$ that admit invariant vectors for ${\rm SL}_n(F)$. Finally weprove a $p$-adic analogue of our result for square-integrable representationsin terms of formal degrees by employing the formal degree conjecture ofHiraga-Ichino-Ikeda \cite{hii08}.

Test vectors for finite periods and base change