Abstract.
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. 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 ?? 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. 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.
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Oblatum 30-III-1999 & 16-XII-1999¶Published online: 29 March 2000
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Ginzburg, V. Principal nilpotent pairs in a semisimple Lie algebra 1. Invent. math. 140, 511–561 (2000). https://doi.org/10.1007/s002220050371
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DOI: https://doi.org/10.1007/s002220050371