{
  "id": "2204.13729",
  "version": "v1",
  "published": "2022-04-28T18:18:16.000Z",
  "updated": "2022-04-28T18:18:16.000Z",
  "title": "Quasi-polynomial representations of double affine Hecke algebras",
  "authors": [
    "Siddhartha Sahi",
    "Jasper Stokman",
    "Vidya Venkateswaran"
  ],
  "comment": "136 pages",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "We introduce an explicit family of representations of the double affine Hecke algebra $\\mathbb{H}$ acting on spaces of quasi-polynomials, defined in terms of truncated Demazure-Lusztig type operators. We show that these quasi-polynomial representations provide concrete realizations of a natural family of cyclic $Y$-parabolically induced $\\mathbb{H}$-representations. We recover Cherednik's well-known polynomial representation as a special case. The quasi-polynomial representation gives rise to a family of commuting operators acting on spaces of quasi-polynomials. These generalize the Cherednik operators, which are fundamental in the study of Macdonald polynomials. We provide a detailed study of their joint eigenfunctions, which may be regarded as quasi-polynomial, multi-parametric generalizations of nonsymmetric Macdonald polynomials. We also introduce generalizations of symmetric Macdonald polynomials, which are invariant under a multi-parametric generalization of the standard Weyl group action. We connect our results to the representation theory of metaplectic covers of reductive groups over non-archimedean local fields. We introduce root system generalizations of the metaplectic polynomials from our previous work by taking a suitable restriction and reparametrization of the quasi-polynomial generalizations of Macdonald polynomials. We show that metaplectic Iwahori-Whittaker functions can be recovered by taking the Whittaker limit of these metaplectic polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-28T18:18:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "33D52",
      "11F68"
    ],
    "keywords": [
      "double affine hecke algebra",
      "quasi-polynomial representation",
      "cheredniks well-known polynomial representation",
      "multi-parametric generalization",
      "standard weyl group action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 136,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}