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 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 p_z-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).