关于偶的极大单型与极大因子的实现
On the realisation of maximal simple types and epsilon factors of pairs
摘要：
设$G$是非阿基米德局部域$F$上一般线性群的有理点群。我们证明了$G$（Bushnell和Kutzko的最大简单类型）的开紧模中心子群的某些表示可以被实现为具体空间。在零水平的情况下，我们的结果基本上是由于Gelfand。这允许我们，对于$G$的超尖点表示$π$，计算$π$的区别矩阵系数。通过积分，我们得到了$π$的显式Whittaker函数.我们用它来计算对的双因子，对于$G$的超尖点表示$π_1$，$π_2$，当$π_1$和$π_2$的contradient仅在“驯服水平”上不同时（更确切地说，$π_1$和$\checkπ_2$包含相同的简单字符）。我们使用Jacquet、Piatetskiii-Shapiro和Shalika的定义，通过计算定义因子的函数方程的两边来实现这一点。我们还研究了在$π_1$的扭曲下，由驯服分歧准特征标引起的扭曲因子的行为。我们的结果推广了Bushnell和Henniart提出的$π_1=\checkπ_2$的特殊情况。
英文摘要：
Let $G$ be the group of rational points of a general linear group over a non-archimedean local field $F$. We show that certain representations of open, compact-mod-centre subgroups of $G$, (the maximal simple types of Bushnell and Kutzko) can be realized as concrete spaces. In the level zero case our result is essentially due to Gelfand. This allows us, for a supercuspidal representation $π$ of $G$, to compute a distinguished matrix coefficient of $π$. By integrating, we obtain an explicit Whittaker function for $π$. We use this to compute the epsilon factor of pairs, for supercuspidal representations $π_1$, $π_2$ of $G$, when $π_1$ and the contragredient of $π_2$ differ only at the `tame level' (more precisely, $π_1$ and $\checkπ_2$ contain the same simple character). We do this by computing both sides of the functional equation defining the epsilon factor, using the definition of Jacquet, Piatetskii-Shapiro, Shalika. We also investigate the behaviour of the epsilon factor under twisting of $π_1$ by tamely ramified quasi-characters. Our results generalise the special case $π_1=\checkπ_2$ totally wildly ramified, due to Bushnell and Henniart.
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