Overgroups of subsystem subgroups in exceptional groups: inside a sandwich

Abstract

This paper supplements a previous paper by the author (Overgroups of subsystem subgroups in exceptional groups: a 2 A 1 2A_1 -proof, (2020)), where the overgroup lattice of the elementary subsystem subgroup E ( Δ , R ) E(\Delta ,R) of the Chevalley group G ( Φ , R ) G(\Phi ,R) for a sufficiently large root subsystem Δ \Delta was studied. Currently, the subject-matter is the relationship between the elementary subgroup E ^ ( σ ) \widehat {E}(\sigma ) given by a net of ideals  σ \sigma of the ring  R R , and the stabilizer S ( σ ) S(\sigma ) of the corresponding Lie subalgebra of the Chevalley algebra. In particular, it is proved that under a certain condition the subgroup E ^ ( σ ) \widehat {E}(\sigma ) is normal in S ( σ ) S(\sigma ) , and some properties of the corresponding quotient group are established.