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Parabolic restrictions and double deformations of weight multiplicities

Cédric Lecouvey

TL;DR

This work introduces a two-parameter deformation $K_{\nu,\mu}(p,q)$ of weight multiplicities for simple Lie algebras using a parabolic root subsystem, unifying Lusztig’s $q$-analogues (at $p=q$) and weight-space dimensions (at $p=q=1$). A fundamental decomposition expresses $K_{\nu,\mu}(p,q)$ in terms of parabolic branching coefficients and Levi-subalgebra Lusztig analogues, linking positivity to parabolic Brylinski–Kostant filtrations for $p=1$ and to stabilized variants for broader positivity results. The paper proves a nonnegative $K_{\nu,\mu}(p+1,q+1)$ for all dominant weights, and a stabilization phenomenon for $K_{\nu+k\rho^{\diamondsuit},\mu+k\rho^{\diamondsuit}}(p,q)$ with explicit positive expansions, generalizing classical stabilization in types $B$, $C$, $D$. Combinatorial descriptions via crystals yield concrete models: a $p=1$ description in terms of branching within crystal graphs and parabolic Kostka polynomials, and a $p+1,q+1$ description using a new $\mu$-charge statistic on crystal vertices. The results connect to Brylinski–Kostant filtrations, Hall–Littlewood polynomials, and potential unequal-parameter affine Hecke algebra interpretations, offering a robust framework for studying parabolic weight multiplicities with two deformation parameters.

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.

Parabolic restrictions and double deformations of weight multiplicities

TL;DR

This work introduces a two-parameter deformation of weight multiplicities for simple Lie algebras using a parabolic root subsystem, unifying Lusztig’s -analogues (at ) and weight-space dimensions (at ). A fundamental decomposition expresses in terms of parabolic branching coefficients and Levi-subalgebra Lusztig analogues, linking positivity to parabolic Brylinski–Kostant filtrations for and to stabilized variants for broader positivity results. The paper proves a nonnegative for all dominant weights, and a stabilization phenomenon for with explicit positive expansions, generalizing classical stabilization in types , , . Combinatorial descriptions via crystals yield concrete models: a description in terms of branching within crystal graphs and parabolic Kostka polynomials, and a description using a new -charge statistic on crystal vertices. The results connect to Brylinski–Kostant filtrations, Hall–Littlewood polynomials, and potential unequal-parameter affine Hecke algebra interpretations, offering a robust framework for studying parabolic weight multiplicities with two deformation parameters.

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.

Paper Structure

This paper contains 14 sections, 15 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Double deformation of weight multiplicities
  3. Decomposition of the double deformation
  4. The general case
  5. The case $p=1$ and the parabolic Brylinski-Kostant filtration
  6. Other positivity results
  7. The double deformation $K_{\nu,\mu}(p+1,q+1)$
  8. Stabilized version of the double deformation $K_{\nu,\mu}(p,q)$
  9. The stabilization phenomenon
  10. Positivity of the stabilized forms
  11. Combinatorial descriptions
  12. Combinatorial description of the double deformation $K_{\nu,\mu }(p,q)$
  13. Combinatorial description of the double deformation $K_{\nu,\mu }(p+1,q+1)$
  14. Some analogues of the Hall-Littlewood polynomials

Key Result

Lemma 2.1

For any $\overline{w}$ in $\overline{W}$, we have $\overline{w}(R_{+}\setminus\overline{R}_{+})=R_{+}\setminus\overline{R}_{+}$, that is the set $R_{+}\setminus\overline{R}_{+}$ is invariant under the action of $\overline{W}$.

Theorems & Definitions (37)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Remark 2.3
  • Conjecture 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 27 more