Normalizer of the Chevalley group of type ${\operatorname E}_7$

Abstract

The simply connected Chevalley group $G(\mathrm {E}_7,R)$ of type $\mathrm {E}_7$ is considered in the 56-dimensional representation. The main objective is to prove that the following four groups coincide: the normalizer of the elementary Chevalley group $E(\mathrm {E}_7,R)$, the normalizer of the Chevalley group $G(\mathrm {E}_7,R)$ itself, the transporter of $E(\mathrm {E}_7,R)$ into $G(\mathrm {E}_7,R)$, and the extended Chevalley group $\bar {G}(\mathrm {E}_7,R)$. This holds over an arbitrary commutative ring $R$, with all normalizers and transporters being calculated in $\mathrm {GL}(56,R)$. Moreover, $\bar {G}(\mathrm {E}_7,R)$ is characterized as the stabilizer of a system of quadrics. This last result is classically known over algebraically closed fields, here it is proved that the corresponding group scheme is smooth over $\mathbb Z$, which implies that it holds over arbitrary commutative rings. These results are a key step in a subsequent paper, devoted to overgroups of exceptional groups in minimal representations.