Paper detail

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 $A^2_α (Ω)$ on a bounded symmetric domain $Ω$, of formal dimension $d_{π_α} > 0$. It is shown that if the Bergman kernel $k^{(α)}_z$ is a cyclic vector for the restriction $π_α |_Γ$ to a lattice $Γ\leq G$ (resp. $(π_α (γ) k^{(α)}_z)_{γ\in Γ}$ is a frame for $A^2_α(Ω)$), then $\mathrm{vol}(G/Γ) d_{π_α} \leq |Γ_z|^{-1}$. The estimate $\mathrm{vol}(G/Γ) d_{π_α} \geq |Γ_z|^{-1}$ holds for $k^{(α)}_z$ being a $p_z$-separating vector (resp. $(π_α (γ) k^{(α)}_z)_{γ\in Γ/ Γ_z}$ being a Riesz sequence in $A^2_α (Ω)$). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for $G = \mathrm{PSU}(1, 1)$.