{
  "id": "1311.2434",
  "version": "v4",
  "published": "2013-11-11T13:14:17.000Z",
  "updated": "2014-02-24T13:01:30.000Z",
  "title": "Basic functions and unramified local L-factors for split groups",
  "authors": [
    "Wen-Wei Li"
  ],
  "comment": "42 pages, largely revised",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "According to a program of Braverman, Kazhdan and Ng\\^o Bao Ch\\^au, for a large class of split unramified reductive groups $G$ and representations $\\rho$ of the dual group $\\hat{G}$, the unramified local $L$-factor $L(s,\\pi,\\rho)$ can be expressed as the trace of $\\pi(f_{\\rho,s})$ for a suitable function $f_{\\rho,s}$ with non-compact support whenever $\\mathrm{Re}(s) \\gg 0$. Such functions can be plugged into the trace formula to study certain sums of automorphic $L$-functions. It also fits into the conjectural framework of Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term basic functions; this is supposed to lead to a generalized Tamagawa-Godement-Jacquet theory for $(G,\\rho)$. In this article, we derive some basic properties for the basic functions $f_{\\rho,s}$ and interpret them via invariant theory. In particular, their coefficients are interpreted as certain generalized Kostka-Foulkes polynomials defined by Panyushev. These coefficients can be encoded into a rational generating function.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-02-24T13:01:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "11F70"
    ],
    "keywords": [
      "unramified local l-factors",
      "split groups",
      "term basic functions",
      "large class",
      "non-compact support"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.2434L"
    }
  }
}