Extremal Subgroups in Chevalley Groups

Abstract

Throughout this paper G(k) denotes a Chevalley group of rank n defined over the field k, where n⩾3. Let Φ be the root system associated with G(k) and let Π={α1, α2, …, αn} be a set of fundamental roots of Φ, with Φ+ being the set of positive roots of Φ with respect to Π. For α∈Π and γ∈Φ+, let nα(γ) be the coefficient of α in the expression of γ as a sum of fundamental roots; so γ=∑α∈Πnα(γ)α. Also we recall that ht(γ), the height of γ, is given by ht(γ)=∑α∈Πnα(γ). The highest root in Φ+ will be denoted by ρ. We additionally assume that the Dynkin diagram of G(k) is connected.