{
  "id": "2006.08126",
  "version": "v1",
  "published": "2020-06-15T04:41:40.000Z",
  "updated": "2020-06-15T04:41:40.000Z",
  "title": "Harmonic Analysis and Gamma Functions on Symplectic Groups",
  "authors": [
    "Dihua Jiang",
    "Zhilin Luo",
    "Lei Zhang"
  ],
  "comment": "99 pages",
  "categories": [
    "math.NT",
    "math.FA",
    "math.OA",
    "math.RT"
  ],
  "abstract": "Over a $p$-adic local field $F$ of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group $G={\\mathbb G}_m\\times{\\mathrm Sp}_{2n}$. It is associated to the Langlands $\\gamma$-functions attached to any irreducible admissible representations $\\chi\\otimes\\pi$ of $G(F)$ and the standard representation $\\rho$ of the dual group $G^\\vee({\\mathbb C})$, and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on ${\\rm GL}_1(F)$, which is associated to a $\\gamma$-function $\\beta_\\psi(\\chi_s)$ (a product of $n+1$ certain abelian $\\gamma$-functions). Our work on ${\\rm GL}_1(F)$ plays an indispensable role in the development of our work on $G(F)$. These two types of harmonic analyses both specialize to the well-known local theory developed in Tate's thesis when $n=0$. The approach is to use the compactification of ${\\rm Sp}_{2n}$ in the Grassmannian variety of ${\\rm Sp}_{4n}$, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis and many other works) on the doubling local zeta integrals for the standard $L$-functions of ${\\rm Sp}_{2n}$. The method can be viewed as an extension of the work of Godement-Jacquet for the standard $L$-function of ${\\rm GL}_n$ and is expected to work for all classical groups. We will consider the archimedean local theory and the global theory in our future work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-15T04:41:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F66",
      "43A32",
      "46S10",
      "11F70",
      "22E50",
      "43A80"
    ],
    "keywords": [
      "harmonic analysis",
      "symplectic group",
      "gamma functions",
      "archimedean local theory",
      "well-known local theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 99,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}