{
  "id": "2308.10844",
  "version": "v1",
  "published": "2023-08-21T16:44:03.000Z",
  "updated": "2023-08-21T16:44:03.000Z",
  "title": "Affine Hecke algebras and symmetric quasi-polynomial duality",
  "authors": [
    "Vidya Venkateswaran"
  ],
  "comment": "34 pages",
  "categories": [
    "math.RT",
    "math.NT",
    "math.QA"
  ],
  "abstract": "In a recent paper with Sahi and Stokman, we introduced quasi-polynomial generalizations of Macdonald polynomials for arbitrary root systems via a new class of representations of the double affine Hecke algebra. These objects depend on a deformation parameter $q$, Hecke parameters, and an additional torus parameter. In this paper, we study $\\textit{antisymmetric}$ and $\\textit{symmetric}$ quasi-polynomial analogs of Macdonald polynomials in the $q \\rightarrow \\infty$ limit. We provide explicit decomposition formulas for these objects in terms of classical Demazure-Lusztig operators and partial symmetrizers, and relate them to Macdonald polynomials with prescribed symmetry in the same limit. We also provide a complete characterization of (anti-)symmetric quasi-polynomials in terms of partially (anti-)symmetric polynomials. As an application, we obtain formulas for metaplectic spherical Whittaker functions associated to arbitrary root systems. For $GL_{r}$, this recovers some recent results of Brubaker, Buciumas, Bump, and Gustafsson, and proves a precise statement of their conjecture about a ``parahoric-metaplectic\" duality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-21T16:44:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "33D52",
      "11F68",
      "22E50",
      "05E05"
    ],
    "keywords": [
      "symmetric quasi-polynomial duality",
      "macdonald polynomials",
      "arbitrary root systems",
      "double affine hecke algebra",
      "metaplectic spherical whittaker functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}