Overgroups of subsystem subgroups in exceptional groups: nonideal levels

Abstract

In the present paper, a description of overgroups for the subsystem subgroups E ( Δ , R ) E(\Delta ,R) of the Chevalley groups G ( Φ , R ) G(\Phi ,R) over the ring R R , where Φ \Phi is a simply laced root system and Δ \Delta is its sufficiently large subsystem, is almost entirely finished. Namely, objects called levels are defined and it is shown that for any such overgroup H H there exists a unique level σ \sigma with E ( σ ) ≤ H ≤ Stab G ( Φ , R ) ⁡ ( L max ( σ ) ) E(\sigma )\le H\le \operatorname {Stab}_{G(\Phi ,R)}(L_{\max }(\sigma )) , where E ( σ ) E(\sigma ) is an elementary subgroup associated with the level σ \sigma and L max ( σ ) L_{\max }(\sigma ) is the corresponding subalgebra of the Chevalley algebra. Unlike the previous papers, here levels can be more complicated than nets of ideals.