{
  "id": "2201.11293",
  "version": "v1",
  "published": "2022-01-27T03:07:07.000Z",
  "updated": "2022-01-27T03:07:07.000Z",
  "title": "On the asymptotic support of Plancherel measures for homogeneous spaces",
  "authors": [
    "Benjamin Harris",
    "Yoshiki Oshima"
  ],
  "comment": "57 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a real linear reductive group and let $H$ be a unimodular, locally algebraic subgroup. Let $\\operatorname{supp} L^2(G/H)$ be the set of irreducible unitary representations of $G$ contributing to the decomposition of $L^2(G/H)$, namely the support of the Plancherel measure. In this paper, we will relate $\\operatorname{supp} L^2(G/H)$ with the image of moment map from the cotangent bundle $T^*(G/H)\\to \\mathfrak{g}^*$. For the homogeneous space $X=G/H$, we attach a complex Levi subgroup $L_X$ of the complexification of $G$ and we show that in some sense \"most\" of representations in $\\operatorname{supp} L^2(G/H)$ are obtained as quantizations of coadjoint orbits $\\mathcal{O}$ such that $\\mathcal{O}\\simeq G/L$ and that the complexification of $L$ is conjugate to $L_X$. Moreover, the union of such coadjoint orbits $\\mathcal{O}$ coincides asymptotically with the moment map image. As a corollary, we show that $L^2(G/H)$ has a discrete series if the moment map image contains a nonempty subset of elliptic elements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-27T03:07:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "plancherel measure",
      "homogeneous space",
      "asymptotic support",
      "coadjoint orbits",
      "moment map image contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}