Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

• Jasper V. Stokman ^[1] ; Siddhartha Sahi ^[2] ; Vidya Venkateswaran ^[3]
  1. [1] University of Amsterdam

     University of Amsterdam

     Países Bajos

     
  2. [2] Rutgers University

     Rutgers University

     City of New Brunswick, Estados Unidos

     
  3. [3] Center for Communications Research, USA

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 3, 2021
• Idioma: inglés
• DOI: 10.1007/s00029-021-00654-1
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• Resumen

  • We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters gi, and which we refer to as metaplectic representations. We realize these representations as quotients of certain parabolically induced modules, and we apply the method of Baxterization (localization) to obtain actions of corresponding Weyl groups on rational functions on the torus. Our construction both generalizes and provides a conceptual proof of earlier results of Chinta, Gunnells, and Puskas, which had depended on a crucial computer verification. A key motivation is that when the parameters gi are specialized to certain Gauss sums, the resulting representation and its localization arise naturally in the consideration of p-parts of Weyl group multiple Dirichlet series. In this special case, similar results have been previously obtained in the literature by the study of Iwahori Whittaker functions for principal series of metaplectic covers of reductive p-adic groups. However this technique is not available for generic parameters gi. It turns out that the metaplectic representations can be extended to the double affine Hecke algebra, where they share many important properties with Cherednik’s basic polynomial representation, which they generalize. This allows us to introduce families of metaplectic polynomials, which depend on the gi, and which generalize Macdonald polynomials. In this paper we discuss in some detail the situation for type A, which is of considerable interest in algebraic combinatorics. We postpone some of the proofs, as well as a discussion of other types, to the sequel.