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

TL;DR

This paper establishes new density conditions for lattice orbits of Bergman kernels on bounded symmetric domains, linking the volume of the quotient space, the formal dimension, and stabilizer sizes, thus refining previous density theorems.

Contribution

It introduces improved density estimates for lattice orbits of Bergman kernels, incorporating stabilizer subgroup sizes, and extends sharp results to specific cases like rac{1}{PSU}(1,1).

Findings

Derived inequalities relating volume, dimension, and stabilizer size.

Improved density bounds for lattice orbits of Bergman kernels.

Partial recovery of sharp results for rac{1}{PSU}(1,1).

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 $G = \mathrm{PSU}(1, 1)$.