Overgroups of subsystem subgroups in exceptional groups: inside a sandwich

TL;DR

This paper explores the relationship between elementary subgroups and stabilizers of Lie subalgebras in Chevalley groups, establishing conditions for normality and analyzing quotient groups.

Contribution

It introduces new connections between elementary subgroups defined by ideals and stabilizers of Lie subalgebras in exceptional groups, extending previous work on overgroup lattices.

Findings

Proves normality of certain elementary subgroups within stabilizers under specific conditions

Analyzes properties of quotient groups formed by these subgroups

Extends understanding of subgroup structure in Chevalley groups

Abstract

The current paper is an addition to the previous paper by author, where the overgroup lattice of the elementary subsystem subgroup $E(\Delta,R)$ of the Chevalley group $G(\Phi,R)$ for a large enough root subsystem $\Delta$ was studied. Now we study the connection between the elementary subgroup $\hat{E}(\sigma)$ given by the net of ideals of the ring $R$ and the stabilizer $S(\sigma)$ of the corresponding Lie subalgebra of the Chevalley algebra. In particular, we prove that under a certain condition the subgroup $\hat{E}(\sigma)$ is normal in $S(\sigma)$, and we also study some properties of the corresponding quotient group.