有限周期和基变化的测试向量
Test vectors for finite periods and base change
摘要：
设$E/F$是有限域的二次扩张.根据Gow的一个结果，当$H$是${\rm GL}_n(F)$或${\rm U}(n,E/F)$时，$G = {\rm GL}_n(E)$的不可约表示$π$至多有一个非零的$H$不变向量，直到与标量相乘.如果$π$确实有一个$H$不变向量，则称其为$H$可区分的。从Gow的工作中已知，当$H$是${\rm GL}_n(F)$时，由于Kawanaka，$H$ -区别的特征在于从${\rm U}(n,E/F)$的碱基变化（分别为：当$H$是${\rm U}(n,E/F)$时，由于Shintani，来自${\rm GL}_n(F)$）。假设$π$是通用的和$H$区别，我们给出了一个明确的描述$H$不变向量的贝塞尔函数的$π$。设$ψ$是$N_G/N_H$的非退化特征标，$B_{π,ψ}$是$ψ$ -Whittaker模型上$π$的（归一化）贝塞尔函数.对于Bessel函数的$H$ -平均\[W_{π,ψ} = \frac{1}{|H|} \sum_{h\in H} π(h) B_{π,ψ}\]，我们证明了\[W_{π,ψ}(I_n) = \frac{{\rm dim}ρ}{{\rm dim} π} \cdot \frac{|{\rm GL}_n(E)|}{|{\rm GL}_n(F)| |{\rm U}(n,E/F)|},\]，其中$ρ$是${\rm U}(n,E/F)$的表示. ${\rm GL}_n(F)$），当$H$为${\rm GL}_n(F)$时，其基础变为$π$（分别为${\rm U}(n,E/F)$）。作为一个应用程序，我们分类的一个通用的$L$ -分组的${\rm SL}_n(E)$，承认不变向量${\rm SL}_n(F)$的成员。最后，我们证明了一个$p$ -adic模拟我们的结果平方可积表示的形式程度，采用形式程度猜想的Hiraga-Ichino-Ikeda \cite{hii08}。
英文摘要：
Let $E/F$ be a quadratic extension of finite fields. By a result of Gow, an irreducible representation $π$ of $G = {\rm GL}_n(E)$ has at most one non-zero $H$-invariant vector, up to multiplication by scalars, when $H$ is ${\rm GL}_n(F)$ or ${\rm U}(n,E/F)$. 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 ${\rm U}(n,E/F)$, due to Kawanaka, when $H$ is ${\rm GL}_n(F)$ (resp. from ${\rm GL}_n(F)$, due to Shintani, when $H$ is ${\rm U}(n,E/F)$). 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 \[W_{π,ψ}(I_n) = \frac{{\rm dim}ρ}{{\rm dim} π} \cdot \frac{|{\rm GL}_n(E)|}{|{\rm GL}_n(F)| |{\rm U}(n,E/F)|},\] where $ρ$ is the representation of ${\rm U}(n,E/F)$ (resp. ${\rm GL}_n(F)$) that base changes to $π$ when $H$ is ${\rm GL}_n(F)$ (resp. ${\rm U}(n,E/F)$). As an application we classify the members of a generic $L$-packet of ${\rm SL}_n(E)$ that admit invariant vectors for ${\rm SL}_n(F)$. 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}.
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