{
  "id": "1212.6004",
  "version": "v1",
  "published": "2012-12-25T09:05:08.000Z",
  "updated": "2012-12-25T09:05:08.000Z",
  "title": "Irreducible representations of product of real reductive groups",
  "authors": [
    "Dmitry Gourevitch",
    "Alexander Kemarsky"
  ],
  "comment": "The authors were surprised not to find this result in the literature. Comments are welcome",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G_1,G_2$ be real reductive groups and $(\\pi,V)$ a smooth, irreducible, admissible representation of $G_1 \\times G_2$. We prove that $(\\pi,V)$ is the completed tensor product of $(\\pi_i,V_i)$, $i=1,2$, where $(\\pi_i,V_i)$ is a smooth,irreducible,admissible representation of $G_i$, $i=1,2$. We deduce this from the analogous theorem for Harish-Chandra modules, for which one direction was proven in [AG] and the other direction we prove here. As a corollary, we deduce that strong Gelfand property for a pair $H \\subset G$ of real reductive groups is equivalent to the usual Gelfand property of the pair $\\Delta H \\subset G \\times H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-25T09:05:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05",
      "22E46"
    ],
    "keywords": [
      "real reductive groups",
      "irreducible representations",
      "usual gelfand property",
      "strong gelfand property",
      "admissible representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.6004G"
    }
  }
}