LEVEL-ZERO VAN DER KALLEN MODULES AND SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT t = ∞

Abstract

Let λ ∈ P+ be a level-zero dominant integral weight, and w the coset representative of minimal length for a coset in W/Wλ, where Wλ is the stabilizer of λ in a finite Weyl group W. In this paper, we give a module $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at t = ∞ of the nonsymmetric Macdonald polynomial Ewλ(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 $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $$ {V}_w^{-}\left(\uplambda \right) $$ by the sum of the submodules $$ {V}_z^{-}\left(\uplambda \right) $$ for all those coset representatives z of minimal length for cosets in W/Wλ such that z > w in the Bruhat order < on W.