{
  "id": "2106.13562",
  "version": "v1",
  "published": "2021-06-25T11:11:02.000Z",
  "updated": "2021-06-25T11:11:02.000Z",
  "title": "Branching of unitary $\\operatorname{O}(1,n+1)$-representations with non-trivial $(\\mathfrak{g},K)$-cohomology",
  "authors": [
    "Clemens Weiske"
  ],
  "comment": "34 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G=\\operatorname{O}(1,n+1)$ with maximal compact subgroup $K$ and let $\\Pi$ be a unitary irreducible representation of $G$ with non-trivial $(\\mathfrak{g},K)$-cohomology. Then $\\Pi$ occurs inside a principal series representation of $G$, induced from the $\\operatorname{O}(n)$-representation $\\bigwedge\\nolimits^p(\\mathbb{C}^n)$ and characters of a minimal parabolic subgroup of $G$ at the limit of the complementary series. Considering the subgroup $G'=\\operatorname{O}(1,n)$ of $G$ with maximal compact subgroup $K'$, we prove branching laws and explicit Plancherel formulas for the restrictions to $G'$ of all unitary representations occurring in such principal series, including the complementary series, all unitary $G$-representations with non-trivial $(\\mathfrak{g},K)$-cohomology and further relative discrete series representations in the cases $p=0,n$. Discrete spectra are constructed explicitly as residues of $G'$-intertwining operators which resemble the Fourier transforms on vector bundles over the Riemannian symmetric space $G'/K'$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-25T11:11:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E45",
      "22E46"
    ],
    "keywords": [
      "maximal compact subgroup",
      "non-trivial",
      "cohomology",
      "complementary series",
      "principal series representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}