{
  "id": "1806.08248",
  "version": "v1",
  "published": "2018-06-21T13:53:33.000Z",
  "updated": "2018-06-21T13:53:33.000Z",
  "title": "$K$-invariant cusp forms for reductive symmetric spaces of split rank one",
  "authors": [
    "Erik P. van den Ban",
    "Job J. Kuit",
    "Henrik Schlichtkrull"
  ],
  "comment": "12 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G/H$ be a reductive symmetric space of split rank $1$ and let $K$ be a maximal compact subgroup of $G$. In a previous article the first two authors introduced a notion of cusp forms for $G/H$. We show that the space of cusp forms coincides with the closure of the $K$-finite generalized matrix coefficients of discrete series representations if and only if there exist no $K$-spherical discrete series representations. Moreover, we prove that every $K$-spherical discrete series representation occurs with multiplicity $1$ in the Plancherel decomposition of $G/H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-21T13:53:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E30",
      "22E45"
    ],
    "keywords": [
      "reductive symmetric space",
      "invariant cusp forms",
      "split rank",
      "spherical discrete series representation occurs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}