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

Abstract

In this paper, we give a representation-theoretic interpretation of the specialization E w ∘ λ ( q , ∞ ) E_{w_\circ \lambda } (q,\infty ) of the nonsymmetric Macdonald polynomial E w ∘ λ ( q , t ) E_{w_\circ \lambda }(q,t) at t = ∞ t=\infty in terms of the Demazure submodule V w ∘ − ( λ ) V_{w_\circ }^- (\lambda ) of the level-zero extremal weight module V ( λ ) V(\lambda ) over a quantum affine algebra of an arbitrary untwisted type. Here, λ \lambda is a dominant integral weight, and w ∘ w_\circ denotes the longest element in the finite Weyl group W W . Also, for each x ∈ W x \in W , we obtain a combinatorial formula for the specialization E x λ ( q , ∞ ) E_{x \lambda } (q, \infty ) at t = ∞ t=\infty of the nonsymmetric Macdonald polynomial E x λ ( q , t ) E_{x \lambda } (q,t) and also a combinatorical formula for the graded character gch ⁡ V x − ( λ ) \operatorname {gch} V_{x}^- (\lambda ) of the Demazure submodule V x − ( λ ) V_{x}^- (\lambda ) of V ( λ ) V(\lambda ) . Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape λ \lambda .