0 M ar 1 99 9 PRINCIPAL NILPOTENT PAIRS IN A SEMISIMPLE LIE ALGEBRA

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 sln, the conjugacy classes of principal nilpotent pairs and the irreducible representations of the Symmetric group, Sn, 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 Sn-modules in terms of Young’s symmetrisers. First results towards a complete classification of all principal nilpotent pairs in a simple Lie algebra are presented at the end of this paper in an Appendix, written by A. Elashvili and D. Panyushev.