{
  "id": "1811.05966",
  "version": "v1",
  "published": "2018-11-14T18:56:38.000Z",
  "updated": "2018-11-14T18:56:38.000Z",
  "title": "Fourier coefficients and small automorphic representations",
  "authors": [
    "Dmitry Gourevitch",
    "Henrik P. A. Gustafsson",
    "Axel Kleinschmidt",
    "Daniel Persson",
    "Siddhartha Sahi"
  ],
  "comment": "52 pages",
  "categories": [
    "math.NT",
    "hep-th",
    "math.RT"
  ],
  "abstract": "In this paper we analyze Fourier coefficients of automorphic forms on adelic reductive groups $G(\\mathbb{A})$. Let $\\pi$ be an automorphic representation of $G(\\mathbb{A})$. It is well-known that Fourier coefficients of automorphic forms can be organized into nilpotent orbits $\\mathcal{O}$ of $G$. We prove that any Fourier coefficient $\\mathcal{F}_\\mathcal{O}$ attached to $\\pi$ is linearly determined by so-called 'Levi-distinguished' coefficients associated with orbits which are equal or larger than $\\mathcal{O}$. When $G$ is split and simply-laced, and $\\pi$ is a minimal or next-to-minimal automorphic representation of $G(\\mathbb{A})$, we prove that any $\\eta \\in \\pi$ is completely determined by its standard Whittaker coefficients with respect to the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $\\mathrm{GL}_n$. In this setting we also derive explicit formulas expressing any maximal parabolic Fourier coefficient in terms of (possibly degenerate) standard Whittaker coefficients for all simply-laced groups. We provide detailed examples for when $G$ is of type $D_5$, $E_6$, $E_7$ or $E_8$ with potential applications to scattering amplitudes in string theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-14T18:56:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11F70",
      "22E55",
      "20G45"
    ],
    "keywords": [
      "small automorphic representations",
      "standard whittaker coefficients",
      "automorphic forms",
      "maximal parabolic fourier coefficient",
      "analyze fourier coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}