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

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

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