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Hecke modules from metaplectic ice

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Abstract

We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of p-adic groups and R-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on p-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of R-matrices of quantum groups depending on the cover degree and associated root system.

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Acknowledgements

This work was supported by NSF grants DMS-1406238 (Brubaker), DMS-1601026 (Bump), and DMS-1500977 (Friedberg) and by the Max Planck Institute for Mathematics (Buciumas). We would like to thank Sergey Lysenko and Anna Puskás for useful conversations, and the referee for helpful comments.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Ben Brubaker

  2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, 91904, Jerusalem, Israel

    Valentin Buciumas

  3. Department of Mathematics, Stanford University, Stanford, CA, 94305-2125, USA

    Daniel Bump

  4. Department of Mathematics, Boston College, Chestnut Hill, MA, 02467-3806, USA

    Solomon Friedberg

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  1. Ben Brubaker
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  2. Valentin Buciumas
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  4. Solomon Friedberg
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Correspondence to Solomon Friedberg.

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Brubaker, B., Buciumas, V., Bump, D. et al. Hecke modules from metaplectic ice. Sel. Math. New Ser. 24, 2523–2570 (2018). https://doi.org/10.1007/s00029-017-0372-0

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