An A2-proof of Structure Theorems for Chevalley Groups of Types

In the present paper we prove the main structure theorem for Chevalley groups G = G(Φ, R) of types Φ = E6,E7 over a commutative ring R. More precisely, we describe subgroups in G normalized by the elementary subgroup E(Φ, R). This result is not new, since structure theorems are known for all Chevalley groups [25, 27, 28, 30], [38]–[40], and [58, 61] (see [42, 65, 34, 56] for further references). The gist of the present paper resides not in the results themselves, but rather in the method of their proof based on the geometry of exceptional groups. We believe that this method is novel and of significant interest. Actually the Schwerpunkt of the present paper abides in a new descent procedure, which enables reduction to groups of smaller rank. This procedure is both simpler and more powerful than any other method known today. Our results on the geometry of the 27-dimensional module for Chevalley groups of type E6 and of the 56dimensional module for Chevalley groups of type E7 pave the way to much more general results such as description of subgroups normalized by some of elementary matrices. Groups of types E8 and F4 can be handled in essentially the same style, and we intend to return to these cases in our subsequent publications. However, from a technical viewpoint the proofs in these cases are noticeably more involved because these groups do not have microweight representations.