Parabolic restrictions and double deformations of weight multiplicities

TL;DR

This paper introduces a new two-parameter deformation of weight multiplicities in Lie algebra representations, exploring their positivity, algebraic interpretation, and combinatorial structure, extending known results and conjectures.

Contribution

It develops a novel (p,q)-deformation framework for weight multiplicities, proving positivity and combinatorial properties, and generalizing previous Lusztig analogues and conjectures.

Findings

Positivity of coefficients when p=1, linked to parabolic Brylinski filtration.

Positivity in finite types for any p, using stabilized double deformation.

Nonnegative coefficients and crystal combinatorics for the (p+1,q+1) deformation.

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.