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LEVEL-ZERO VAN DER KALLEN MODULES AND SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT t = ∞

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Abstract

Let λ  ∈ P+ be a level-zero dominant integral weight, and w the coset representative of minimal length for a coset in W/Wλ, where Wλ is the stabilizer of λ in a finite Weyl group W. In this paper, we give a module \( {\mathbbm{K}}_w^{-}\left(\uplambda \right) \) over the negative part of a quantum affine algebra whose graded character is identical to the specialization at t = ∞ of the nonsymmetric Macdonald polynomial Ewλ(q, t) multiplied by a certain explicit finite product of rational functions of q of the form (1 − q−r)−1 for a positive integer r. This module \( {\mathbbm{K}}_w^{-}\left(\uplambda \right) \) (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module \( {V}_w^{-}\left(\uplambda \right) \) by the sum of the submodules \( {V}_z^{-}\left(\uplambda \right) \) for all those coset representatives z of minimal length for cosets in W/Wλ such that z > w in the Bruhat order < on W.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo, 152-8551, Japan

    SATOSHI NAITO

  2. Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, Japan

    DAISUKE SAGAKI

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  1. SATOSHI NAITO
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  2. DAISUKE SAGAKI
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Correspondence to DAISUKE SAGAKI.

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Satoshi Naito is partially supported by JSPS Grant-in-Aid for Scientific Research (B) 16H03920.

Daisuke Sagaki is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 15K04803.

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NAITO, S., SAGAKI, D. LEVEL-ZERO VAN DER KALLEN MODULES AND SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT t = ∞. Transformation Groups 26, 1077–1111 (2021). https://doi.org/10.1007/s00031-020-09586-0

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