Normalizer of the Chevalley group of type $E_7$

Abstract

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.