{
  "id": "1504.03458",
  "version": "v1",
  "published": "2015-04-14T08:56:19.000Z",
  "updated": "2015-04-14T08:56:19.000Z",
  "title": "On a uniqueness property of cuspidal unipotent representations",
  "authors": [
    "Yongqi Feng",
    "Eric Opdam"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We prove that for absolutely almost simple groups of classical type, the formal degree of a cuspidal unipotent representation (in the sense of Lusztig) determines its Kazhdan-Lusztig-Langlands parameter, up to twisting by unramified characters. For such groups, a cuspidal unipotent formal degree, normalized appropriately, is the reciprocal of a polynomial whose zeroes are roots of unity of even order. We show that this property is very distinctive among unipotent formal degrees of general discrete series characters, and we show that if a unipotent formal degree has poles or zeroes of odd order, then the root of unity with the largest odd order that occurs in this way is a zero of the formal degree. The main result of this article characterizes unramified Kazhdan-Lusztig-Langlands parameters which support a cuspidal local system in terms of formal degrees. The result implies the uniqueness of so-called cuspidal spectral transfer maps (as introduced in an aritle by Opdam earlier) between unipotent affine Hecke algebras of classical type (up to twisting by unramified characters).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-14T08:56:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cuspidal unipotent representation",
      "uniqueness property",
      "cuspidal unipotent formal degree",
      "unipotent affine hecke algebras",
      "article characterizes unramified kazhdan-lusztig-langlands parameters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150403458F"
    }
  }
}