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##
Victor Ginzburg - *Principal nilpotent pairs in a semisimple Lie algebra*

VICTOR GINZBURG, University of Chicago |

Principal nilpotent pairs in a semisimple Lie algebra |

We introduce and study a new class of pairs of commuting nilpotent
elements in a semisimple Lie algebra. These pairs enjoy quite
remarkable properties and 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
*GL*_{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
symmetrisers.

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