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

Let ${\pi }_{\alpha }$ be a holomorphic discrete series representation of a connected semi-simple Lie group $G$ with finite center, acting on a weighted Bergman space $A_{\alpha }^2(\mathrm{\Omega })$ on a bounded symmetric domain $\mathrm{\Omega }$, of formal dimension $d_{{\pi }_{\alpha }}>0$. It is shown that if the Bergman kernel $k_z^{(\alpha )}$ is a cyclic vector for the restriction $\pi_{\alpha} |{\Gamma}$toalattice$\Gamma \leq G$(resp\mathrm{.}$(\pi{\alpha} (\gamma) k^{(\alpha)}z){\gamma \in \Gamma}$isaframefor$A^2_{\alpha}(\Omega)$), then $\mathrm{v}\mathrm{o}\mathrm{l}(G\mathrm{/}\mathrm{\Gamma })d_{{\pi }_{\alpha }}\le \mathrm{\mid }{\mathrm{\Gamma }}_z{\mathrm{\mid }}^{-1}$. The estimate $\mathrm{v}\mathrm{o}\mathrm{l}(G\mathrm{/}\mathrm{\Gamma })d_{{\pi }_{\alpha }}\ge \mathrm{\mid }{\mathrm{\Gamma }}_z{\mathrm{\mid }}^{-1}$ holds for $k_z^{(\alpha )}$ being a $p_z$-separating vector (resp. $(\pi_{\alpha} (\gamma) k^{(\alpha)}z){\gamma \in \Gamma / \Gamma_z}$ being a Riesz sequence in $A_{\alpha }^2(\mathrm{\Omega })$). 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 = $\mathrm{P}\mathrm{S}\mathrm{U}$(1, 1)$.