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

Abstract:

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 a combinatorical formula for the graded character $\operatorname {gch} V_{x}^- (\lambda )$ 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 $\lambda$.