Unipotent Elements in Small Characteristic

0.1. Let k be an algebraically closed field of characteristic exponent p ≥ 1. Let G be a reductive connected algebraic group over k. Let U be the variety of unipotent elements of G. The unipotent classes of G are the orbits of the conjugation action of G on U . The theory of Dynkin and Kostant [Ko] provides a classification of unipotent classes of G assuming that p = 1. It is known that this classification remains valid when p ≥ 2 is assumed to be a good prime for G. But the analogous classification problem in the case where p is a bad prime for G is more complicated. In every case a classification of unipotent classes is known: see [W] for classical groups and [E,Sh,M] for exceptional groups; but from these works it is difficult to see the general features of the classification. One of the aims of this paper is to present a picture of the unipotent elements which should apply for arbitrary p and is as close as possible to the picture for p = 1. In 1.4 we observe that the set of unipotent classes in G can be parametrized by a set S(W) of irreducible representations of the Weyl group W which can be described apriori purely in terms of the root system. This explains clearly why the classification is different for small p. In 1.1 we restate in a more precise form an observation of [L2] according to which U is naturally partitioned into finitely many ”unipotent pieces” which are locally closed subvarieties stable under conjugation by G; the classification of unipotent pieces is independent of p. For p = 1 or a good prime, each unipotent piece is a single conjugacy class. When p is a bad prime a unipotent piece is in general a union of several conjugacy classes but the classes inside a unipotent piece form a very simple partially ordered set under the closure relation. Also each unipotent piece has some topological properties which are independent of p (for example, over a finite field, the number of points of a unipotent piece is given by a formula independent of the characteristic).