  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
$W$.
