{
  "id": "1008.1592",
  "version": "v2",
  "published": "2010-08-09T20:22:00.000Z",
  "updated": "2010-08-12T04:39:26.000Z",
  "title": "Fourier transforms of orbital integrals on the Lie algebra of $\\operatorname{SL}_2$",
  "authors": [
    "Loren Spice"
  ],
  "comment": "35 pages; v2: updated introduction to refer to work of Langlands",
  "categories": [
    "math.RT"
  ],
  "abstract": "The Harish-Chandra--Howe local character expansion expresses the characters of reductive, $p$-adic groups in terms of Fourier transforms of nilpotent orbital integrals on their Lie algebras, and Murnaghan--Kirillov theory expresses many characters of reductive, $p$-adic groups in terms of Fourier transforms of semisimple orbital integrals (also on their Lie algebras). In many cases, the evaluation of these Fourier transforms seems intractable; but, for $\\operatorname{SL}_2$, the nilpotent orbital integrals have already been computed. In this paper, we use a variant of Huntsinger's integral formula, and the theory of $p$-adic special functions, to compute semisimple orbital integrals.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-08-12T04:39:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "22E35"
    ],
    "keywords": [
      "fourier transforms",
      "lie algebra",
      "semisimple orbital integrals",
      "nilpotent orbital integrals",
      "harish-chandra-howe local character expansion expresses"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.1592S"
    }
  }
}