{
  "id": "1602.01610",
  "version": "v1",
  "published": "2016-02-04T09:58:15.000Z",
  "updated": "2016-02-04T09:58:15.000Z",
  "title": "The Degenerate Eisenstein Series Attached to the Heisenberg Parabolic Subgroups of Quasi-Split Forms of $Spin_8$",
  "authors": [
    "Avner Segal"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In previews works, joint with N. Gurevitch, a family of Rankin-Selberg integrals were shown to represent the twisted standard $\\mathcal{L}$-function $\\mathcal{L}\\left(s,\\pi,\\chi,\\mathfrak{st}\\right)$ of a cuspidal representation $ \\pi$ of the exceptional group of type $G_2$. This integral representation binds the analytic behavior of this $\\mathcal{L}$-functions with that of a degenerate Eisenstein series defined over the family of quasi-split forms of $Spin_8$ associated to an induction from a character on the Heisenberg parabolic subgroup. This paper is divided into two parts. In part 1 we study the poles of this degenerate Eisenstein series in the right half plane $\\mathfrak{Re}(s)>0$. In part 2 we use the results of part 1 to give a criterion for $\\pi$ to be a {\\bf CAP} representation with respect to the Borel subgroup in terms of poles of $\\mathcal{L}\\left(s,\\pi,\\chi,\\mathfrak{st}\\right)$. We also settle a conjecture of J. Hundley and D. Ginzburg and prove a few results relating the analytic behavior of $\\mathcal{L}\\left(s,\\pi,\\chi,\\mathfrak{st}\\right)$ and the set of Fourier coefficients supported by $\\pi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-04T09:58:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "degenerate eisenstein series",
      "heisenberg parabolic subgroup",
      "quasi-split forms",
      "analytic behavior",
      "right half plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160201610S"
    }
  }
}