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 one for the graded character $\mathrm{gch}{V}_x^{-}(\lambda )$ 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 $\lambda$.

arXiv comment45 pages