Test vectors for finite periods and base change

Abstract

Let $E/F$ be a quadratic extension of finite fields. By a result of Gow, an irreducible representation $\pi$ of $G={\mathrm{GL}}_n(E)$ has at most one non-zero $H$-invariant vector, up to multiplication by scalars, when $H$ is ${\mathrm{GL}}_n(F)$ or $\mathrm{U}(n,E/F)$. 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 $\mathrm{U}(n,E/F)$, due to Kawanaka, when $H$ is ${\mathrm{GL}}_n(F)$ (resp. from ${\mathrm{GL}}_n(F)$, due to Shintani, when $H$ is $\mathrm{U}(n,E/F)$). 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 $N_G/N_H$ and let $B_{\pi ,\psi }$ be the (normalized) Bessel function of $\pi$ on the $\psi$-Whittaker model. For the $H$-average \[W_{π,ψ} = \frac{1}{|H|} \sum_{h\in H} π(h) B_{π,ψ}\] of the Bessel function, we prove that \[W_{π,ψ}(I_n) = \frac{{\rm dim}ρ}{{\rm dim} π} \cdot \frac{|{\rm GL}_n(E)|}{|{\rm GL}_n(F)| |{\rm U}(n,E/F)|},\] where $\rho$ is the representation of $\mathrm{U}(n,E/F)$ (resp. ${\mathrm{GL}}_n(F)$) that base changes to $\pi$ when $H$ is ${\mathrm{GL}}_n(F)$ (resp. $\mathrm{U}(n,E/F)$). As an application we classify the members of a generic $L$-packet of ${\mathrm{SL}}_n(E)$ that admit invariant vectors for ${\mathrm{SL}}_n(F)$. 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}.

Final version to appear in Adv. Math

References in corpus (3)

• Galois self-dual cuspidal types and Asai local factors
• Local Langlands correspondence for Asai L-functions and epsilon factors
• Distinguished representations for SL(n)

Cited by in corpus (3)

• Supercuspidal representations of ${\mathrm{GL}}_n(F)$ distinguished by a unitary involution
• Distinguished representations, Shintani base change and a finite field analogue of a conjecture of Prasad
• Characterisation of the poles of the $\ell$-modular Asai $L$-factor