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Weight polytopes and saturation of Demazure characters

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For G a reductive group and \(T\subset B\) a maximal torus and Borel subgroup, Demazure modules are certain B-submodules, indexed by elements of the Weyl group, of the finite irreducible representations of G. In order to describe the T-weight spaces that appear in a Demazure module, we study the convex hull of these weights—the Demazure polytope. We characterize these polytopes both by vertices and by inequalities, and we use these results to prove that Demazure characters are saturated, in the case that G is simple of classical Lie type. Specializing to \(G=GL_n\), we recover results of Fink, Mészáros, and St. Dizier, and separately Fan and Guo, on key polynomials, originally conjectured by Monical, Tokcan, and Yong.

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Acknowledgements

The authors are very grateful to D. Anderson and A. Yong for their feedback on an earlier version of the manuscript. The authors also wish to thank J. Kamnitzer for questions and comments on the first draft. The first author wishes to thank J. Hong for discussions on this problem during the course of their studies of affine Schubert varieties [5]. Finally, we thank the anonymous referee for many helpful suggestions, comments, and corrections.

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Authors and Affiliations

  1. BICMR, Peking University, Beijing, 100871, People’s Republic of China

    Marc Besson

  2. University of Queensland, St. Lucia, QLD, 4067, Australia

    Sam Jeralds

  3. Marian University, 3200 Cold Spring Rd, Indianapolis, IN, 46222, USA

    Joshua Kiers

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Correspondence to Marc Besson.

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Besson, M., Jeralds, S. & Kiers, J. Weight polytopes and saturation of Demazure characters. Math. Ann. 388, 4449–4486 (2024). https://doi.org/10.1007/s00208-023-02617-7

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  • DOI: https://doi.org/10.1007/s00208-023-02617-7

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