完备离散赋值环上典型群的正则特征标
Regular characters of classical groups over complete discrete valuation rings
摘要：
设$\mathfrak{o}$是一个完备的离散赋值环，其有限剩余域$\mathsf{k}$为奇特征，$\mathbf{G}$是$\mathfrak{o}$上的一个辛或特殊正交群概型.对于任何$\ell\in\mathbb{N}$，令$G^\ell$表示$\mathbf{G}(\mathfrak{o})$的第$\ell$个主同余子群。群$\mathbf{G}(\mathfrak{o})$的一个不可约特征标称为正则的，如果它对于某个$\ell$在子群$G^{\ell+1}$上是平凡的，并且如果它对$G^\ell/G^{\ell+1}\simeq \mathrm{Lie}(\mathbf{G})(\mathsf{k})$的限制由最小$\mathbf{G}(\mathsf{k}^{\rm alg})$稳定子维数的特征标组成。本文研究了$\mathfrak{o}$上这类典型群的正则特征标，当$\mathsf{k}$的特征标大于2时，构造并枚举了$\mathbf{G}(\mathfrak{o})$的所有正则特征标.因此，我们计算其表示zeta函数的正则部分。
英文摘要：
Let $\mathfrak{o}$ be a complete discrete valuation ring with finide residue field $\mathsf{k}$ of odd characteristic, and let $\mathbf{G}$ be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $\ell\in\mathbb{N}$ let $G^\ell$ denote the $\ell$-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{\ell+1}$ for some $\ell$, and if its restriction to $G^\ell/G^{\ell+1}\simeq \mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}(\mathsf{k}^{\rm alg})$ stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.
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