{
  "id": "2407.17366",
  "version": "v1",
  "published": "2024-07-24T15:43:30.000Z",
  "updated": "2024-07-24T15:43:30.000Z",
  "title": "Automorphisms of the DAHA of type $\\check{C_1}C_1$ and their action on Askey-Wilson polynomials and functions. I. The flip $(a,b,c,d)\\mapsto(a,b,qd^{-1},qc^{-1})$",
  "authors": [
    "Tom H. Koornwinder",
    "Marta Mazzocco"
  ],
  "comment": "34 pages",
  "categories": [
    "math.CA",
    "math.RT"
  ],
  "abstract": "In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type $\\check{C_1}C_1$ which have a relatively simple action on the generators and on the parameters, notably a symmetry $t_4$ which sends the Askey-Wilson parameters $(a,b,c,d)$ to $(a,b,qd^{-1},qc^{-1})$. We study how these symmetries act on the basic representation and on the symmetric and non-symmetric Askey-Wilson (AW) polynomials and functions. Interestingly $t_4$ maps AW polynomials to functions. In the second part of the paper we look for a version of non-symmetric AW functions that behave well under these symmetries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-24T15:43:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S99",
      "20C08",
      "33D45"
    ],
    "keywords": [
      "askey-wilson polynomials",
      "automorphisms",
      "non-symmetric aw functions",
      "maps aw polynomials",
      "double affine hecke algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}