{
  "id": "2211.03724",
  "version": "v1",
  "published": "2022-11-07T17:51:40.000Z",
  "updated": "2022-11-07T17:51:40.000Z",
  "title": "Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity",
  "authors": [
    "Valentin Buciumas",
    "Manish M. Patnaik"
  ],
  "comment": "Comments welcome!",
  "categories": [
    "math.RT",
    "math.NT",
    "math.QA"
  ],
  "abstract": "We describe the structure of the Whittaker or Gelfand-Graev module on a $n$-fold metaplectic cover of a $p$-adic group $G$ at both the Iwahori and spherical level. We express our answer in terms of the representation theory of a quantum group at a root of unity attached to the Langlands dual group of $G$. To do so, we introduce an algebro-combinatorial model for these modules and develop for them a Kazhdan-Lusztig theory involving new generic parameters. These parameters can either be specialized to Gauss sums to recover the $p$-adic theory or to the natural grading parameter in the representation theory of quantum groups. As an application of our results, we deduce geometric Casselman-Shalika type results for metaplectic covers, conjectured in a slightly different form by S. Lysenko, as well as prove a variant of G. Savin's local Shimura type correspondences at the Whittaker level.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-07T17:51:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "metaplectic cover",
      "quantum group",
      "adic group",
      "savins local shimura type correspondences",
      "deduce geometric casselman-shalika type results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}