{
  "id": "1601.07602",
  "version": "v1",
  "published": "2016-01-27T23:56:15.000Z",
  "updated": "2016-01-27T23:56:15.000Z",
  "title": "Remark on representation theory of general linear groups over a non-archimedean local division algebra",
  "authors": [
    "Marko Tadic"
  ],
  "comment": "26 pages",
  "journal": "Rad HAZU, Matematicke Znanosti, vol. 19 / 523, 2015, pages 27-53",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper we give a simple (local) proof of two principal results about irreducible tempered representations of general linear groups over a non-archimedean local division algebra. We give a proof of the parameterization of the irreducible square integrable representations of these groups by segments of cuspidal representations, and a proof of the irreducibility of the tempered parabolic induction. Our proofs are based on Jacquet modules (and the Geometric Lemma, incorporated in the structure of a Hopf algebra). We use only some very basic general facts of the representation theory of reductive p-adic groups (the theory that we use was completed more then three decades ago, mainly in 1970-es). Of the specific results for general linear groups over A, basically we use only a very old result of G.I. Olshanskii, which says that there exist complementary series starting from $Ind(\\rho\\otimes\\rho)$ whenever $\\rho$ is a unitary irreducible cuspidal representation. In appendix of the paper \"On parabolic induction on inner forms of the general linear group over a non-archimedean local field\" of E. Lapid and A. Minguez, there is also a simple local proof of these results, based on a slightly different approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-27T23:56:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50"
    ],
    "keywords": [
      "general linear group",
      "non-archimedean local division algebra",
      "representation theory",
      "parabolic induction",
      "unitary irreducible cuspidal representation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160107602T"
    }
  }
}