$\mathfrak{K}$ -分裂代数群局部Zeta函数的一致性与函数方程
Uniformity and Functional Equations for Local Zeta Functions of $\mathfrak{K}$-Split Algebraic Groups
摘要：
本文研究了定义在$\mathfrak{K}$上的代数群$\mathcal{G}$的局部zeta函数，以及$\mathbb{Q}$的有限扩张$\mathfrak{K}$的忠实$\mathfrak{K}$ -有理表示$ρ$.这些由$\mathfrak{p}$上的积分给出-$\mathcal{G}$的adic点由$ρ$确定，对于$\mathfrak{K}$的素数$\mathfrak{p}$。证明了对于所有幂幺根满足某种提升性质的$\mathfrak{K}$ -分裂群，局部zeta函数几乎是一致的.如果$\mathcal{G}$是还原的，则自动满足此属性。我们提供了一个进一步的标准方面的不变量的$\mathcal{G}$和$ρ$，保证当地zeta函数满足函数方程的几乎所有素数的$\mathfrak{K}$。我们通过使用Iwahori和Matsumoto [IM]的$\mathfrak{p}$ -adic Bruhat分解将zeta函数表示为与多面体锥的格点上的生成函数的$\mathcal{G}$相关联的Weyl群$W$上的加权和来获得这些结果。函数方程反映了外尔群的对称性和组合对象的相互作用之间的相互作用。我们构建家庭的群体表示违反我们的第二个结构标准，其局部zeta函数不满足函数方程。我们的工作推广了Igusa [Igu]和du Sautoy和Lubotzky [dSL]的结果，并对生成的无挠幂零群的zeta函数有影响.
英文摘要：
We study the local zeta functions of an algebraic group $\mathcal{G}$ defined over $\mathfrak{K}$ together with a faithful $\mathfrak{K}$-rational representation $ρ$ for a finite extension $\mathfrak{K}$ of $\mathbb{Q}$. These are given by integrals over $\mathfrak{p}$-adic points of $\mathcal{G}$ determined by $ρ$ for a prime $\mathfrak{p}$ of $\mathfrak{K}$. We prove that the local zeta functions are almost uniform for all $\mathfrak{K}$-split groups whose unipotent radical satisfies a certain lifting property. This property is automatically satisfied if $\mathcal{G}$ is reductive. We provide a further criterion in terms of invariants of $\mathcal{G}$ and $ρ$ which guarantees that the local zeta functions satisfy functional equations for almost all primes of $\mathfrak{K}$. We obtain these results by using a $\mathfrak{p}$-adic Bruhat decomposition of Iwahori and Matsumoto [IM] to express the zeta function as a weighted sum over the Weyl group $W$ associated to $\mathcal{G}$ of generating functions over lattice points of a polyhedral cone. The functional equation reflects an interplay between symmetries of the Weyl group and reciprocities of the combinatorial object. We construct families of groups with representations violating our second structural criterion whose local zeta functions do not satisfy functional equations. Our work generalizes results of Igusa [Igu] and du Sautoy and Lubotzky [dSL] and has implications for zeta functions of finitely generated torsion-free nilpotent groups.
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