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 A α 2 (Ω) on a bounded symmetric domain Ω , of formal dimension d π α > 0 . It is shown that if the Bergman kernel k z (α) is a cyclic vector for the restriction π α | Γ to a lattice Γ ≤ G (resp. (π α (γ) k z (α)) γ ∈ Γ is a frame for A α 2 (Ω) ), then vol (G / Γ) d π α ≤ | Γ z | - 1 . The estimate vol (G / Γ) d π α ≥ | Γ z | - 1 holds for k z (α) being a p z -separating vector (resp. (π α (γ) k z (α)) γ ∈ Γ / Γ z being a Riesz sequence in A α 2 (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) . [ABSTRACT FROM AUTHOR]