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Generalized Weyl modules, alcove paths and Macdonald polynomials

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Abstract

Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr–Shimozono formula in order to prove that the \(t=\infty \) specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.

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Authors and Affiliations

  1. Department of Mathematics, National Research University Higher School of Economics, Usacheva str. 6, Moscow, Russia, 119048

    Evgeny Feigin & Ievgen Makedonskyi

  2. Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia, 143026

    Evgeny Feigin

  3. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111, Bonn, Germany

    Ievgen Makedonskyi

Authors
  1. Evgeny Feigin
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  2. Ievgen Makedonskyi
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Correspondence to Evgeny Feigin.

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Feigin, E., Makedonskyi, I. Generalized Weyl modules, alcove paths and Macdonald polynomials. Sel. Math. New Ser. 23, 2863–2897 (2017). https://doi.org/10.1007/s00029-017-0346-2

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  • DOI: https://doi.org/10.1007/s00029-017-0346-2

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