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

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 $W$.