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

摘要

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 $ mathop {m gch}olimits 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 $. P