  This is the first of a series of papers devoted to certain pairs of commuting
nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable
properties and which are expected to play a major role in Representation
theory. The properties of these pairs and their role is similar to those of the
principal nilpotents. To any principal nilpotent pair we associate a
two-parameter analogue of the Kostant partition function, and propose the
corresponding two-parameter analogue of the weight multiplicity formula.
  In a different direction, each principal nilpotent pair gives rise to a
harmonic polynomial on the Cartesian square of the Cartan subalgebra, that
transforms under an irreducible representation of the Weyl group. In the
special case of sl_n, the conjugacy classes of principal nilpotent pairs and
the irreducible representations of the Symmetric group, S_n, are both
parametrised (in a compatible way) by Young diagrams. In general, our theory
provides a natural generalization to arbitrary Weyl groups of the classical
construction of simple S_n-modules in terms of Young's symmetrisers.
