. A G ] 3 A pr 1 99 9 Nilpotent pairs , dual pairs , and sheets

Recently, V.Ginzburg introduced the notion of a principal nilpotent pair (= pn-pair) in a semisimple Lie algebra g [Gi99]. It is a double counterpart of the notion of a regular nilpotent element in g. A pair e = (e1, e2) ∈ g × g is called nilpotent, if [e1, e2] = 0 and there exists a pair h = (h1, h2) of semisimple elements such that [h1, h2] = 0, [hi, ej ] = δijej (i, j ∈ {1, 2}). A pn-pair e is a nilpotent pair such that the simultaneous centralizer zg(e) is of dimension rk g. Note that, by a theorem of R.Richardson, rk g is the least possible value for this dimension. Evident similarity between the “double” and “ordinary” theory is manifestly seen in the following results of [Gi99]: zg(h) is a Cartan subalgebra; the eigenvalues of adh1 and adh2 are integral; both e1 and e2 are Richardson elements; ZG(e) is a connected Abelian unipotent group; ZG(e) acts transitively on the set of semisimple pairs satisfying the above commutator relations. Excerpts from Ginzburg’s theory, which by no means exhaust [Gi99], are presented in section 1.