Unipotent Elements in Small Characteristic, Iv

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 and on g by the coadjoint action. (For any k-vector space V we denote by V ∗ the dual vector space.) Let GC be the reductive group over C of the same type as G. Let UG be the variety of unipotent elements of G. Let Ng be the variety of nilpotent elements of g. Let Ng∗ be the variety of nilpotent elements of g (following [KW] we say that a linear form ξ : g −→ k is nilpotent if its kernel contains some Borel subalgebra of g). In [L1, L2, L3] we have proposed a definition of a partition of UG and of Ng into smooth locally closed G-stable pieces which are indexed by the unipotent classes in GC and which in many ways depend very smoothly on p. In this paper we propose a definition of an analogous partition of Ng∗ into pieces which are indexed by the unipotent classes in GC. (This definition is only of interest for p > 1, small; for p = 1 or p large we can identify Ng with Ng∗ and the partition of Ng∗ is deduced from the partition of Ng.) We will illustrate this in the case where G is of type A, C or D and p is arbitrary. (We do not treat the case where G is of type B which seems to be more complicated.) Notation. If f is a permutation of a set X we denote X = {x ∈ X; f(x) = x}. The cardinal of a finite set X is denoted by |X|. For any subspace U of g let Ann(U) = {ξ ∈ g; ξ|U = 0}.