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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 $E_{w_{\circ} λ} (q,\infty)$ of the nonsymmetric Macdonald polynomial $E_{w_{\circ} λ}(q,t)$ at $t=\infty$ in terms of the Demazure submodule $V_{w_\circ}^{-} (λ)$ of the level-zero extremal weight module $V(λ)$ over a quantum affine algebra of an arbitrary untwisted type, here, $λ$ 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 $E_{x λ} (q, \infty)$ at $t=\infty$ of the nonsymmetric Macdonald polynomial $E_{x λ} (q,t)$, and also one for the graded character $\mathrm{gch} V_{x}^- (λ)$ of the Demazure submodule $V_{x}^- (λ)$ of $V(λ)$, both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape $λ$.