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 $W$.
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