Nilpotent pairs, dual pairs, and sheets

Recently, V.Ginzburg introduced the notion of a principal nilpotent pair (= pn-pair) in a semisimple Lie algebra {\frak g}. It is a double counterpart of the notion of a regular nilpotent element. A pair (e_1,e_2) of commuting nilpotent elements is called a pn-pair, if the dimension of their simultaneous centralizer is equal to the rank of {\frak g} and some bi-homogeneity condition is satisfied. Ginzburg proved that many familiar results of the `ordinary' theory have analogues for pn-pairs. The aim of this article is to develop the theory of nilpotent pairs a bit further and to present some applications of it to dual pairs and sheets. It is shown that a large portion of Ginzburg's theory can be extended to the pairs whose simultaneous centraliser is of dimension rk{\frak g}+1. Such pairs are called almost pn-pairs. It is worth noting that the very existence of almost pn-pairs is a purely "double" phenomenon, because the dimension of "ordinary" orbits is always even. We prove that to any principal or almost nilpotent pair one naturally associates a dual pair. Moreover, this dual pair is reductive if and only if e_1 and e_2 can be included in commuting sl_2-triples. We also study sheets containing members of pn-pairs. Some cases are described, where these sheets are smooth and admit a section.