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

Let $\lambda \in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W\mathrm{/}{W}_{\lambda }$, 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 $t=\mathrm{\infty }$ 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 $W\mathrm{/}{W}_{\lambda }$ such that $z>w$ in the Bruhat order $<$ on $W$.