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

Abstract

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