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 $W$.