{
  "id": "2202.02119",
  "version": "v1",
  "published": "2022-02-04T13:09:25.000Z",
  "updated": "2022-02-04T13:09:25.000Z",
  "title": "The most continuous part of the Plancherel decomposition for a real spherical space",
  "authors": [
    "Job J. Kuit",
    "Eitan Sayag"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "In this article we give a precise description of the Plancherel decomposition of the most continuous part of $L^{2}(Z)$ for a real spherical homogeneous space $Z$. Our starting point is the recent construction of Bernstein morphisms by Delorme, Knop, Kr\\\"otz and Schlichtkrull. The most continuous part decomposes into a direct integral of unitary principal series representations. We give an explicit construction of the $H$-invariant functionals on these principal series. We show that for generic induction data the multiplicity space equals the full space of $H$-invariant functionals. Finally, we determine the inner products on the multiplicity spaces by refining the Maass-Selberg relations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-04T13:09:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G20",
      "22E46",
      "22F30",
      "43A85"
    ],
    "keywords": [
      "real spherical space",
      "plancherel decomposition",
      "invariant functionals",
      "unitary principal series representations",
      "generic induction data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}