Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains

Martijn Caspers, Jordy Timo van Velthoven*

*Corresponding author for this work

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Abstract

Let πα be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space Aα2(Ω) on a bounded symmetric domain Ω , of formal dimension dπα>0. It is shown that if the Bergman kernel kz(α) is a cyclic vector for the restriction πα| Γ to a lattice Γ ≤ G (resp. (πα(γ)kz(α))γ∈Γ is a frame for Aα2(Ω)), then vol(G/Γ)dπα≤|Γz|-1. The estimate vol(G/Γ)dπα≥|Γz|-1 holds for kz(α) being a pz-separating vector (resp. (πα(γ)kz(α))γ∈Γ/Γz being a Riesz sequence in Aα2(Ω)). 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= PSU (1 , 1).

Original languageEnglish
Pages (from-to)609-628
Number of pages20
JournalMathematische Zeitschrift
Volume302
Issue number1
DOIs
Publication statusPublished - 2022

Keywords

  • Bergman space
  • Cyclic vector
  • Discrete series representation
  • Frame
  • Lattice
  • Riesz sequence
  • Separating vector
  • Twisted von Neumann algebra

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