A Classification of the Principal Nilpotent Pairs in Simple Lie Algebras and Related Problems

Abstract

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in g. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in g that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rkg. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well-known results by B. Kostant on regular nilpotent elements in g and has interesting applications to representation theory. In particular, he proved that the number of G-orbits of principal nilpotent pairs is finite and gave a classification for g = sl(V). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for sl(V). (The precise definitions for all notions related to nilpotent pairs are found in §1.)