Hecke modules from metaplectic ice

Ben Brubaker; Valentin Buciumas; Daniel Bump; Solomon Friedberg

doi:10.1007/s00029-017-0372-0

Hecke modules from metaplectic ice

Ben Brubaker, Valentin Buciumas, Daniel Bump, Solomon Friedberg

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

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.

Original language	English (US)
Pages (from-to)	2523-2570
Number of pages	48
Journal	Selecta Mathematica, New Series
Volume	24
Issue number	3
DOIs	https://doi.org/10.1007/s00029-017-0372-0
State	Published - Jul 1 2018

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2017, Springer International Publishing AG, part of Springer Nature.

Keywords

Hecke algebra
Metaplectic group
Quantum group
R-matrix
Whittaker function

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title = "Hecke modules from metaplectic ice",

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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AU - Brubaker, Ben

AU - Buciumas, Valentin

AU - Bump, Daniel

AU - Friedberg, Solomon

N1 - Funding Information: 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. Publisher Copyright: © 2017, Springer International Publishing AG, part of Springer Nature.

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N2 - 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.

AB - 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.

KW - Hecke algebra

KW - Metaplectic group

KW - Quantum group

KW - R-matrix

KW - Whittaker function

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Hecke modules from metaplectic ice

Abstract

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Keywords

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