[Paper Review] Test vectors for finite periods and base change

U. K. Anandavardhanan, Nadir Matringe|arXiv (Cornell University)|May 10, 2018

Advanced Algebra and Geometry22 references3 citations

TL;DR

This paper provides an explicit formula for the unique $H$-invariant vector in the Whittaker model of a generic, $H$-distinguished representation $\pi$ of $\mathrm{GL}_n(E)$, where $H$ is either $\mathrm{GL}_n(F)$ or $\mathrm{U}(n,E/F)$, using the Bessel function of $\pi$. The key result expresses the value of the $H$-average of the Bessel function at the identity as a ratio of dimensions of base-changed representations, linking distinction to base change in finite and $p$-adic settings.

ABSTRACT

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

Motivation & Objective

• To explicitly describe the unique $H$-invariant vector in the Whittaker model of a generic, $H$-distinguished representation $\pi$ of $\mathrm{GL}_n(E)$, where $H$ is $\mathrm{GL}_n(F)$ or $\mathrm{U}(n,E/F)$.
• To establish a precise formula for the $H$-average of the Bessel function of $\pi$ at the identity matrix.
• To connect $H$-distinction to base change from $\mathrm{U}(n,E/F)$ or $\mathrm{GL}_n(F)$ via explicit formulas involving dimensions and formal degrees.
• To extend the finite field result to the $p$-adic setting using the formal degree conjecture of Hiraga-Ichino-Ikeda.
• To classify which members of a generic $L$-packet of $\mathrm{SL}_n(E)$ admit $\mathrm{SL}_n(F)$-invariant vectors.

Proposed method

• The authors use the $\psi$-Whittaker model of a generic representation $\pi$ of $\mathrm{GL}_n(E)$, where $\psi$ is a non-degenerate character trivial on $N_H$, and define the normalized Bessel function $B_{\pi,\psi}$.
• They compute the $H$-average of the Bessel function: $W_{\pi,\psi}(I_n) = \frac{1}{|H|} \sum_{h \in H} \pi(h) B_{\pi,\psi}$, and derive its value at the identity.
• The key identity is $W_{\pi,\psi}(I_n) = \frac{\dim \rho}{\dim \pi} \cdot \frac{|\mathrm{GL}_n(E)|}{|\mathrm{GL}_n(F)| \cdot |\mathrm{U}(n,E/F)|}$, where $\rho$ is the base-changed representation from $H$.
• For the $p$-adic case, they employ the formal degree conjecture of Hiraga-Ichino-Ikeda to relate the formal degree of $\pi$ to that of its base-changed representation $\rho$, leading to a $p$-adic analogue of the finite field result.
• They use the functional equation and epsilon factors of $L$-functions, particularly $\gamma(s, \pi, r, \psi)$, to relate the $L$-value ratio to the formal degree ratio.
• They apply known results on the epsilon factor $\epsilon(1/2, \pi, r, \psi) = 1$ for square-integrable, distinguished representations to simplify the final expression.

Experimental results

Research questions

• RQ1 What is the explicit value of the $H$-average of the Bessel function of a generic, $H$-distinguished representation $\pi$ of $\mathrm{GL}_n(E)$ at the identity matrix?
• RQ2 How does the $H$-invariant vector in the Whittaker model relate to the base-changed representation $\rho$ from $H$?
• RQ3 Can the finite field result on test vectors be extended to the $p$-adic setting using the formal degree conjecture?
• RQ4 What is the precise relationship between the formal degree of $\pi$ and the dimension of the base-changed representation $\rho$ in the $p$-adic case?
• RQ5 Which representations in a generic $L$-packet of $\mathrm{SL}_n(E)$ admit $\mathrm{SL}_n(F)$-invariant vectors?

Key findings

• The $H$-average of the Bessel function at the identity is given by $W_{\pi,\psi}(I_n) = \frac{\dim \rho}{\dim \pi} \cdot \frac{|\mathrm{GL}_n(E)|}{|\mathrm{GL}_n(F)| \cdot |\mathrm{U}(n,E/F)|}$, where $\rho$ is the base-changed representation from $H$.
• In the $p$-adic setting, for square-integrable representations, the linear functional $\lambda$ on the Whittaker model satisfies $\lambda \sim \frac{d(\rho)}{d(\pi)} \cdot \ell$, where $d(\cdot)$ denotes the formal degree.
• The epsilon factor $\epsilon(1/2, \pi, r, \psi) = 1$ for square-integrable, $H$-distinguished representations, simplifying the functional equation analysis.
• The result implies that the $H$-invariant vector is explicitly determined by the Bessel function and the base-changed representation, with the constant of proportionality made explicit via the formal degree conjecture.
• The paper classifies $\mathrm{SL}_n(F)$-invariant vectors in generic $L$-packets of $\mathrm{SL}_n(E)$ using the test vector formula.
• The finite field result is shown to be the finite analogue of the $p$-adic result, with the Bessel function satisfying $\ell(B_{\pi}) = 1$ under suitable normalization.