  In this paper, we give a representation-theoretic interpretation of the
specialization $E_{w_{\circ} \lambda} (q,\infty)$ of the nonsymmetric Macdonald
polynomial $E_{w_{\circ} \lambda}(q,t)$ at $t=\infty$ in terms of the Demazure
submodule $V_{w_\circ}^{-} (\lambda)$ of the level-zero extremal weight module
$V(\lambda)$ over a quantum affine algebra of an arbitrary untwisted type,
here, $\lambda$ is a dominant integral weight, and $w_{\circ}$ denotes the
longest element in the finite Weyl group $W$. Also, for each $x \in W$, we
obtain a combinatorial formula for the specialization $E_{x \lambda} (q,
\infty)$ at $t=\infty$ of the nonsymmetric Macdonald polynomial $E_{x \lambda}
(q,t)$, and also one for the graded character $\mathrm{gch} V_{x}^- (\lambda)$
of the Demazure submodule $V_{x}^- (\lambda)$ of $V(\lambda)$, both of these
formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape
$\lambda$.
