{
  "id": "2305.16537",
  "version": "v1",
  "published": "2023-05-25T23:41:15.000Z",
  "updated": "2023-05-25T23:41:15.000Z",
  "title": "Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL_2",
  "authors": [
    "Kazuki Oshita",
    "Masao Tsuzuki"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We introduce a local zeta-function for an irreducible admissible supercuspidal representation $\\pi$ of the metaplectic double cover of $\\SL_2$ over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of $\\pi$. We obtain an expression of the gamma factor, which shows its entireness on $\\C$. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on ${\\rm PGL}_2\\cong {\\rm SO}_3$ associated with the prehomogenous vector space of binary symmetric matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-25T23:41:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gamma factor",
      "local zeta functions",
      "functional equation",
      "metaplectic cover",
      "local zeta-function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}