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

2015/11/22 by Satoshi Naito, Naito, Satoshi, Fumihiko Nomoto +3 · 1 citation
Mathematics · #17B37 #20G42 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Primary 05E05 #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Secondary 33D52

paper · pdf · doi:10.48550/arxiv.1511.07005

openalex publication_date 2015/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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• Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials
• Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at t = ∞
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• Nonsymmetric Macdonald polynomials and Demazure characters
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