Nilpotent pairs in semisimple Lie algebras and their characteristics

Abstract

Recently, V. Ginzburg introduced 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 (e1, e2) consists of two commuting elements in g that can independently be contracted to the origin. We show that any nilpotent pair has a characteristic (h1, h2), which is unique within G-conjugacy. Generalizing Dynkin's approach in case of sl2-triples, we prove that the number of G-orbits of characteristics of nilpotent pairs is finite and provide some estimates for the numerical labels αj(hi), where αj is a suitably chosen set of 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. It is proven that two wonderful pairs having the same characteristic are necessarily 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 a 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.