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

2021/09/29 by Martijn Caspers, Caspers, Martijn, Jordy Timo van Velthoven +1 · 1 citation
Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Geometry and complex manifolds #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2109.14498

openalex publication_date 2021/09/29 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Let π_α be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space A²_α (Ω) on a bounded symmetric domain Ω, of formal dimension d_πα > 0. It is shown that if the Bergman kernel k^(α)_z is a cyclic vector for the restriction π_α |_Γ to a lattice Γ≤ G (resp. (π_α (γ) k^(α)_z)_{γ∈ Γ} is a frame for A²_α(Ω)), then vol(G/Γ) d_πα ≤ |Γ_z|^-1. The estimate vol(G/Γ) d_πα ≥ |Γ_z|^-1 holds for k^(α)_z being a p_z-separating vector (resp. (π_α (γ) k^(α)_z)_{γ∈ Γ/ Γz} being a Riesz sequence in A²_α (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).

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• Density conditions with stabilisers for lattice orbits of discrete series representations

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