$E_7$型Chevalley群的正规化子
Normalizer of the Chevalley group of type $E_7$
摘要：
我们考虑56维表示中$E_7$型的单连通Chevalley群$G(E_7,R)$.本文的主要目的是证明以下四个群是一致的：初等Chevalley群$E(E_7,R)$的正规化子，Chevalley群$G(E_7,R)$自身的正规化子，$E(E_7,R)$到$G(\mathrm E_7,R)$的转运子，以及扩展Chevalley群$\overline G(E_7,R)$ .这在任意交换环$R$上成立，所有的正规化子和转运子都在$GL(56,R)$中计算。此外，我们刻画了$\overline G(E_7,R)$作为一个系统的二次稳定器。最后一个结果在代数闭域上是经典已知的，这里我们证明了相应的群概型在$\mathbb Z$上是光滑的，这意味着它在任意交换环上成立。这些结果是我们随后的论文中的关键步骤之一，致力于极小表示中例外群的扩群。
英文摘要：
We consider the simply connected Chevalley group $G(E_7,R)$ of type $E_7$ in the 56-dimensional representation. The main objective of the paper is to prove that the following four groups coincide: the normalizer of the elementary Chevalley group $E(E_7,R)$, the normalizer of the Chevalley group $G(E_7,R)$ itself, the transporter of $E(E_7,R)$ into $G(\mathrm E_7,R)$, and the extended Chevalley group $\overline G(E_7,R)$. This holds over an arbitrary commutative ring $R$, with all normalizers and transporters being calculated in $GL(56,R)$. Moreover, we characterize $\overline G(E_7,R)$ as the stabilizer of a system of quadrics. This last result is classically known over algebraically closed fields, here we prove that the corresponding group scheme is smooth over $\mathbb Z$, which implies that it holds over arbitrary commutative rings. These results are one of the key steps in our subsequent paper, dedicated to the overgroups of exceptional groups in minimal representations.
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