Extremal Subgroups in Chevalley Groups

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={α 1 , α 2 , …, α n } 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 n α (γ) be the coefficient of α in the expression of γ as a sum of fundamental roots; so γ=∑ α∈Π n α (γ)α. Also we recall that ht(γ), the height of γ, is given by ht(γ)=∑ α∈Π n α (γ). The highest root in Φ + will be denoted by ρ. We additionally assume that the Dynkin diagram of G ( k ) is connected.