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

Abstract

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 weight module $V(\lambda )$ over a quantum affine algebra $U_{q}$, where $\lambda$ 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 $\mathcal {B}(\lambda )$ of the level-zero extremal weight module $V(\lambda )$ and the crystal ${\mathbb {B}}^{\frac{\infty }{2}}(\lambda )$ of semi-infinite Lakshmibai–Seshadri paths of shape $\lambda$, which is obtained in our previous work. As an application, we obtain a formula expressing the graded character of the Demazure submodule $V_{w_0}^{+}(\lambda )$ in terms of the specialization at $t=0$ of the symmetric Macdonald polynomial $P_{\lambda }(x\,;\,q,\,t)$.