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arXiv · 2412.10003

Parabolic restrictions and double deformations of weight multiplicities

Cédric Lecouvey
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Abstract

We introduce some (p,q)-deformations of the weight multiplicities for the representations of any simple Lie algebra g over the complex numbers. This is done by associating the indeterminate q to the positive roots of a parabolic subsystem of g and the indeterminate p to the remaining positive roots. When p=q, we just recover the usual Lusztig analogues of weight multiplicities. We then study the positivity of the coefficients in these double deformations. In particular, the positivity holds when p=1 in which case the polynomials have a natural algebraic interpretation in terms of a parabolic Brylinski filtration. For the parabolic restriction from type C to type A, this positivity result was conjectured by Lee. We also establish this positivity, in any finite type and for any p, for a stabilized version of our double deformation. In addition, we study the double deformation obtained by replacing the pair (p,q) by (p+1,q+1), show it has nonnegative coefficients and admits a combinatorial description in terms of crystals.

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Cédric Lecouvey. 2024-12-13. Parabolic restrictions and double deformations of weight multiplicities. https://arxiv.org/abs/2412.10003

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