Overgroups of subsystem subgroups in exceptional groups: A 2𝐴₁-proof

Abstract

In the present paper, a weak form of sandwich classification for the overgroups of the subsystem subgroup E ( Δ , R ) E(\Delta ,R) of the Chevalley group G ( Φ , R ) G(\Phi ,R) is proved in the case where Φ \Phi is a simply laced root system and Δ \Delta is its sufficiently large subsystem. Namely, it is shown that, for such an overgroup H H , there exists a unique net of ideals σ \sigma of the ring R R such that E ( Φ , Δ , R , σ ) ≤ H ≤ Stab G ( Φ , R ) ⁡ ( L ( σ ) ) E(\Phi ,\Delta ,R,\sigma )\le H\le \operatorname {Stab}_{G(\Phi ,R)}(L(\sigma )) , where E ( Φ , Δ , R , σ ) E(\Phi ,\Delta ,R,\sigma ) is an elementary subgroup associated with the net and L ( σ ) L(\sigma ) is the corresponding subalgebra of the Chevalley Lie algebra.