{
  "id": "2109.14498",
  "version": "v2",
  "published": "2021-09-29T15:32:32.000Z",
  "updated": "2022-04-28T08:34:12.000Z",
  "title": "Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains",
  "authors": [
    "Martijn Caspers",
    "Jordy Timo van Velthoven"
  ],
  "categories": [
    "math.FA",
    "math.OA",
    "math.RT"
  ],
  "abstract": "Let $\\pi_{\\alpha}$ be a holomorphic discrete series representation of a connected semi-simple Lie group $G$ with finite center, acting on a weighted Bergman space $A^2_{\\alpha} (\\Omega)$ on a bounded symmetric domain $\\Omega$, of formal dimension $d_{\\pi_{\\alpha}} > 0$. It is shown that if the Bergman kernel $k^{(\\alpha)}_z$ is a cyclic vector for the restriction $\\pi_{\\alpha} |_{\\Gamma}$ to a lattice $\\Gamma \\leq G$ (resp. $(\\pi_{\\alpha} (\\gamma) k^{(\\alpha)}_z)_{\\gamma \\in \\Gamma}$ is a frame for $A^2_{\\alpha}(\\Omega)$), then $\\mathrm{vol}(G/\\Gamma) d_{\\pi_{\\alpha}} \\leq |\\Gamma_z|^{-1}$. The estimate $\\mathrm{vol}(G/\\Gamma) d_{\\pi_{\\alpha}} \\geq |\\Gamma_z|^{-1}$ holds for $k^{(\\alpha)}_z$ being a $p_z$-separating vector (resp. $(\\pi_{\\alpha} (\\gamma) k^{(\\alpha)}_z)_{\\gamma \\in \\Gamma / \\Gamma_z}$ being a Riesz sequence in $A^2_{\\alpha} (\\Omega)$). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for $G = \\mathrm{PSU}(1, 1)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-28T08:34:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bounded symmetric domain",
      "bergman kernel",
      "lattice orbits",
      "density conditions",
      "stabilizers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}