Abelian unipotent subgroups of reductive groups

✍ Scribed by George J. McNinch

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 264 KB
• Volume: 167
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(01)00038-x

✦ Synopsis

Let G be a connected reductive group deÿned over an algebraically closed ÿeld 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 su ciently large, we show that G always has a faithful representation ( ; V ) with the property that the exponential of d (X ) lies in (G) for each p-nilpotent X ∈ g. This property permits a simpliÿcation of the description given by Suslin et al. 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 su cient conditions on p; for an adjoint semisimple G with Coxeter number h, the condition p ¿ 2h -2 is always good enough.