The Isomorphism and Isogeny Theorems for Reductive Algebraic Groups

Abstract

Let G be a (connected) reductive algebraic group defined over an algebraically closed field K and let T be a maximal torus of G.LetX be the character group ofTand$R \subset X$ the root system of Grelative toT.ThusX is isomorphic to ${\mathbb{Z}^r}$ with r = dim T, the rank of G, and for each $\alpha \in R$ there exists a unipotent subgroup ${U_\alpha }$ which is isomorphic to the additive group of K and is normalized by T according to the character $\alpha$:1.1 If ${u_\alpha }:K \to {U_\alpha }$ is an isomorphism then $t{u_\alpha }\left( a \right){t^{ - 1}} = {u_\alpha }\left( {\alpha \left( t \right)a} \right)$for all$t \in T$and $a \in K.$