[Paper Review] A Kato-Lusztig formula for nonsymmetric Macdonald polynomials

Bogdan Ion|arXiv (Cornell University)|Jun 3, 2004

Algebraic structures and combinatorial models7 references8 citations

TL;DR

This paper establishes a nonsymmetric analogue of the Kato-Lusztig formula, linking irreducible Weyl character coefficients to degenerate symmetric Macdonald polynomials via Kazhdan–Lusztig polynomials. It proves polynomiality of Macdonald polynomial coefficients and extends Demazure's character formula to p-adic zonal spherical functions.

ABSTRACT

We prove a nonsymmetric analogue of a formula of Kato and Lusztig which describes the coefficients of the expansion of irreducible Weyl characters in terms of (degenerate) symmetric Macdonald polynomials as certain Kazhdan–Lusztig polynomials. We also establish precise polynomiality results for coefficients of symmetric and nonsymmetric Macdonald polynomials and a version of Demazure’s character formula for p–adic zonal spherical functions.

Motivation & Objective

• To extend the Kato-Lusztig formula, originally for symmetric Macdonald polynomials, to the nonsymmetric setting.
• To clarify the polynomiality properties of coefficients in expansions of symmetric and nonsymmetric Macdonald polynomials.
• To provide a Demazure-type character formula for p-adic zonal spherical functions using nonsymmetric Macdonald polynomials.
• To establish a precise connection between irreducible Weyl characters and degenerate symmetric Macdonald polynomials through Kazhdan–Lusztig polynomials in the nonsymmetric framework.

Proposed method

• Adapting the Kato-Lusztig framework to the nonsymmetric setting of Macdonald polynomials.
• Using Kazhdan–Lusztig polynomials to express coefficients in the expansion of irreducible Weyl characters in terms of degenerate symmetric Macdonald polynomials.
• Analyzing the polynomial structure of coefficients in symmetric and nonsymmetric Macdonald polynomial expansions.
• Applying representation-theoretic techniques to derive a Demazure-type character formula for p-adic zonal spherical functions.
• Leveraging the structure of affine Weyl groups and nonsymmetric Macdonald polynomials to generalize known character formulas.
• Establishing precise polynomiality results for coefficients via algebraic and combinatorial methods in the context of Hecke algebras and p-adic groups.

Experimental results

Research questions

• RQ1 How can the Kato-Lusztig formula be generalized to the nonsymmetric case of Macdonald polynomials?
• RQ2 What are the precise polynomiality properties of coefficients in expansions of symmetric and nonsymmetric Macdonald polynomials?
• RQ3 Can a Demazure-type character formula be formulated for p-adic zonal spherical functions using nonsymmetric Macdonald polynomials?
• RQ4 What is the role of Kazhdan–Lusztig polynomials in expressing irreducible Weyl character coefficients in terms of degenerate symmetric Macdonald polynomials in the nonsymmetric setting?
• RQ5 How do the structural properties of nonsymmetric Macdonald polynomials enable the extension of classical character formulas to p-adic settings?

Key findings

• A nonsymmetric Kato-Lusztig formula is established, expressing coefficients of irreducible Weyl characters in terms of degenerate symmetric Macdonald polynomials via Kazhdan–Lusztig polynomials.
• The coefficients in the expansions of symmetric and nonsymmetric Macdonald polynomials are shown to satisfy precise polynomiality conditions.
• A version of Demazure’s character formula is proven for p-adic zonal spherical functions using nonsymmetric Macdonald polynomials.
• The framework successfully generalizes classical results from symmetric to nonsymmetric Macdonald polynomials in the context of Weyl character theory.
• The connection between irreducible characters and Macdonald polynomials is formalized through Kazhdan–Lusztig polynomials in the nonsymmetric setting.
• The results provide a unified algebraic and combinatorial structure linking representation theory, Hecke algebras, and p-adic harmonic analysis.