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 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 G is semisimple and p is sufficiently large, we show that G always has a faithful representation (p, V) with the property that the exponential of dp(X) lies in p(G) for each p-nilpotent X is an element of g. This property permits a simplification of the description given by Suslin et al. of the (even) cohomology ring for the Frobenius kernels G(d), d greater than or equal to 2. The previous authors already observed that the natural representation of a classical group has the above property (with no restriction on p). Our methods apply to any Chevalley group and hence give the result also for quasisimple groups with "exceptional type" root systems. The methods give explicit sufficient conditions on p; for an adjoint semisimple G with Coxeter number h, the condition p > 2h - 2 is always good enough. (C) 2002 Elsevier Science B.V. All rights reserved.