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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 GLn, 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 symmetrisers.


next up previous
Next: Mark Haiman - The Up: Algebraic Combinatorics, Group Representations Previous: Tudose Geanina - Littlewood-Richardson