Abstract. Let X be an F -rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F . Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F -split. This property has several consequences.

Abstract. Let G be a semisimple group over an algebraically closed field of very good characteristic for G. In the context of geometric invariant theory, G. Kempf and -independently -G. Rousseau have associated optimal cocharacters of G to an unstable vector in a linear G-representation. If the nilpotent element X ∈ Lie(G) lies in the image of the differential of a homomorphism SL 2 → G, we say that homomorphism is optimal for X, or simply optimal, provided that its restriction to a suitable torus of SL 2 is optimal for X in the sense of geometric invariant theory.We show here that any two SL 2 -homomorphisms which are optimal for X are conjugate under the connected centralizer of X. This implies, for example, that there is a unique conjugacy class of principal homomorphisms for G. We show that the image of an optimal SL 2 -homomorphism is a completely reducible subgroup of G; this is a notion defined recently by J.-P. Serre. Finally, if G is defined over the (arbitrary) subfield K of k, and if X ∈ Lie(G)(K) is a K-rational nilpotent element with X [p] = 0, we show that there is an optimal homomorphism for X which is defined over K. Mathematics Subject Classification (2000). 20G15.

Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u in G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the second author done for simple, adjoint G over the complex numbers. When G is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of A(u). Thus we obtain the result, previously known through case-checking, that the structure of the component group A(u) is independent of good characteristic.Comment: 13 pages; AMS LaTeX. This is the final version; it will appear in the Steinberg birthday volume of the Journal of Algebra. This version corrects an oversight pointed out by the referee; see Prop 2

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 G is semisimple and p is sufficiently large, we show that G always has a faithful representation (ρ, V ) with the property that the exponential of dρ(X) lies in ρ(G) for each p-nilpotent X ∈ g. This property permits a simplification of the description given by Suslin, Friedlander, and Bendel of the (even) cohomology ring for the Frobenius kernels G d , d ≥ 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.Theorem. Assume that p is a good prime for the connected reductive group G, and that P is a distinguished parabolic subgroup of G with unipotent radical V. Write n(P ) for the nilpotence class of V (which is the same as the nilpotence class of v), and let the integer m > 0 be minimal with the property that p m ≥ n(P ).1. The p-nilpotence degree of a Richardson element of v = Lie(V) is m; equivalently, the p-exponent of the Lie algebra v is m;

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