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

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 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 π_α| _Γ...

The density theorem for discrete series representations restricted to lattices

This article considers the relation between the spanning properties of lattice orbits of discrete series representations and the associated lattice co-volume. The focus is on the density theorem, which provides a trichotomy characterizing the existence of cyclic vectors and separating vectors, and frames and Riesz sequences. We provide an...

On sufficient density conditions for lattice orbits of relative discrete series

This note provides new criteria on a unimodular group G and a discrete series representation (π, H_π) of formal degree d_π> 0 under which any lattice Γ ≤ G with vol(G/Γ)dπ≤1 (resp. vol(G/Γ)dπ≥1) admits g∈ H_π such that π(Γ) g is a frame (resp. Riesz sequence). The results apply to all projective discrete...