Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at $t = \infty$

TL;DR

This paper constructs modules over quantum affine algebras whose graded characters match the specialization at t=∞ of nonsymmetric Macdonald polynomials, revealing new algebraic structures related to these polynomials.

Contribution

It introduces level-zero van der Kallen modules that realize specialized nonsymmetric Macdonald polynomials as graded characters, linking representation theory and special functions.

Findings

Modules' graded characters match polynomial specializations

Explicit module construction via Demazure modules

Connection between algebraic modules and Macdonald polynomials

Abstract

Let $\lambda \in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_{\lambda}$, where $W_{\lambda}$ is the stabilizer of $\lambda$ in a finite Weyl group $W$. In this paper, we give a module $\mathbb{K}_{w}(\lambda)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w \lambda}(q,\,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $\mathbb{K}_{w}(\lambda)$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $V_{w}^{-}(\lambda)$ by the sum of the submodules $V_{z}^{-}(\lambda)$ for all those coset representatives $z$ of minimal length for the cosets in $W/W_{\lambda}$ such that $z > w$ in the Bruhat order $<$ on $W$.