Nilpotent pairs in semisimple Lie algebras and their characteristics

In a recent article [Gi99], V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra g. He also obtained several results for more general pairs. As a next step, we considered in [Pa99] almost principal nilpotent pairs. The aim of this paper is to make a contribution to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pair e = (e1, e2) consists of two commuting elements in g that can independently be contracted to the origin (see precise definition in sect. 1). A principal nilpotent pair is a double counterpart of a regular (=principal) nilpotent element. Consequently, the theory of nilpotent pairs should stand out as double counterpart of the theory of nilpotent orbits. As the cornerstone of the latter is the Morozov–Jacobson theorem and the concept of a characteristic, the primary goal is to realize to which extent these can be generalized to the double setting.