Specialization of nonsymmetric Macdonald polynomials at $t=\infty$ and Demazure submodules of level-zero extremal weight modules

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 $\lambda$.