Elements in Small Characteristic

Let k be an algebraically closed field of characteristic exponent p ≥ 1. Let G be a connected reductive algebraic group over k and let g be the Lie algebra of G. Note that G acts on G and on g by the adjoint action. Let UG be the variety of unipotent elements of G. Let Ng be the variety of nilpotent elements of g. In [L2], we have proposed a definition of a partition of UG into smooth locally closed G-stable pieces which are indexed by the unipotent classes in the group over C of the same type as G and which in many ways seem to depend very smoothly on p; moreover we studied in detail the pieces of UG for types A and C. In [L3] we have studied in detail the pieces of UG for types B and D; however the definition in [L3] was not based on the proposal of [L2] (which involved the partial order of unipotent classes). In this paper we propose another general definition of the pieces of UG which is not based on the partial order of unipotent classes and we show that this new definition unifies the definitions in [L2], [L3] in the sense that for types A,C it can be identified with the definition in [L2] while for types B,D it can be identified with the definition in [L3]. The idea of the new definition is as follows. One needs to consider a grading g = ⊕ngn analogous to the one associated to a nilpotent element over C by the Morozov-Jacobson theorem. Let G0 be the closed connected subgroup of G corresponding to g0. Now G0 acts naturally on g2. The main ingredient in the definition of a piece is the definition of a suitable open G0invariant subset g2 of g2. When p = 1 or p≫ 0, the subset g ! 2 is by definition the unique open G0-orbit in g2. But this definition is not correct for general p. Now note that when p = 1 the centralizer in G of any element of the open G0-orbit in g2 is contained in G≥0, the parabolic subgroup of G whose Lie algebra is ∑