Abelian Unipotent Subgroups of Reductive Groups

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the “order formula” of D. Testerman for unipotent elements in G; moreover, we show that the same formula determines the p-nilpotence degree of the corresponding nilpotent elements in the Lie algebra g of G. Second, if X is a p-nilpotent element (an element of p-nilpotence degree 1) in g, we show that G always has a representation V for which the exponential homomorphism Ga → GL(V ) determined by X factors through the action of G. This property permits a simplification of the description given by Suslin, Friedlander, and Bendel of the (even) cohomology ring for the higher Frobenius kernels Gd, d ≥ 2.