Let [gfr ] be a semisimple Lie algebra over an algebraically closed field [ ] 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
[gfr ]. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting
elements in [gfr ] that can independently be contracted to the origin and such that their
simultaneous centralizer has the minimal possible dimension, that is, rk[gfr ]. 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 [gfr ] 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 [gfr ] = [sfr ][lfr ]([ ]). Trying to achieve a greater
generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and,
again, classified them for [sfr ][lfr ]([ ]). (The precise definitions for all notions related to
nilpotent pairs are found in §1.)