{
  "id": "1811.00803",
  "version": "v1",
  "published": "2018-11-02T10:05:38.000Z",
  "updated": "2018-11-02T10:05:38.000Z",
  "title": "Singularities of Intertwining Operators and Decompositions of Principal Series Representations",
  "authors": [
    "Taeuk Nam",
    "Avner Segal",
    "Lior Silberman"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper, we show that, under certain assumptions, a parabolic induction $Ind_B^G\\lambda$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive group $G$ decomposes into a direct sum of the form: \\[ Ind_B^G\\lambda = \\left(Ind_P^G St_M\\otimes \\chi_0\\right) \\oplus \\left(Ind_P^G \\mathbf{1}_M\\otimes \\chi_0\\right), \\] where $P$ is a parabolic subgroup of $G$ with Levi subgroup $M$ of semi-simple rank $1$, $\\mathbf{1}_M$ is the trivial representation of $M$, $St_M$ is the Steinberg representation of $M$ and $\\chi_0$ is a certain character of $M$. We construct examples of this phenomenon for all simply-connected simple groups of rank at least $2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-02T10:05:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "principal series representations",
      "intertwining operators",
      "decompositions",
      "singularities",
      "construct examples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}