Nilpotent pairs in semisimple Lie algebras and their characteristics

Abstract

Recently, V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra \g. Our aim is to contribute to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pair (e_1,e_2) consists of two commuting elements in \g that can independently be contracted to the origin. A principal nilpotent pair is a double counterpart of a regular nilpotent element. Consequently, the theory of nilpotent pairs should stand out as double counterpart of the theory of nilpotent orbits. We show that any nilpotent pair has a characteristic (h_1,h_2), which is unique within to conjugacy. Generalizing Dynkin's approach to (sl_2)-triples, we prove that the number of G-orbits of characteristics of nilpotent pairs is finite and provide some estimates for the numerical labels α_j(h_i), where {α_j} are suitable simple roots of \g. It was observed by Ginzburg that the number of G-orbits of nilpotent pairs is infinite. This means this class is too wide to have a reasonable theory. To resolve this difficulty, we introduce `wonderful' nilpotent pairs. We prove that if two wonderful pairs have the same characteristic, then these are conjugate. This implies that there are finitely many G-orbits of wonderful pairs. A number of nice properties of wonderful pairs shows that these can be regarded as right double analogue of nilpotent orbits. We also consider several natural classes of wonderful pairs and describe characteristics for principal and almost principal nilpotent pairs.