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In this paper, we give a characterization of the crystal bases$\mathcal{B}_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of Demazure submodules$V_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of a level-zero extremal weightmodule $V(\lambda)$ over a quantum affine algebra $U_{q}$, where $\lambda$ isan arbitrary level-zero dominant integral weight, and $W_{\mathrm{af}}$ denotesthe affine Weyl group. This characterization is given in terms of the initialdirection of a semi-infinite Lakshmibai-Seshadri path, and is established undera suitably normalized isomorphism between the crystal basis$\mathcal{B}(\lambda)$ of the level-zero extremal weight module $V(\lambda)$and the crystal $\mathbb{B}^{\frac{\infty}{2}}(\lambda)$ of semi-infiniteLakshmibai-Seshadri paths of shape $\lambda$, which is obtained in our previouswork. As an application, we obtain a formula expressing the graded character ofthe Demazure submodule $V_{w_0}^{+}(\lambda)$ in terms of the specialization at$t=0$ of the symmetric Macdonald polynomial $P_{\lambda}(x\,;\,q,\,t)$.

Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials