Weighted $L^2$ Holomorphic functions on ball-fiber bundles over compact Kähler manifolds

Abstract: Let $\widetilde{M}$ be a complex manifold and $\Gamma$ be a torsion-free cocompact lattice of $\text{Aut}(\widetilde{M})$. Let $\rho\colon\Gamma\to SU(N,1)$ be a representation and $M:=\widetilde M/\Gamma$ be an $n$-dimensional compact complex manifold which admits a holomorphic embedding $\imath$ into $\Sigma:=\mathbb B^N/\rho(\Gamma)$. In this paper, we investigate a relation between weighted $L^2$ holomorphic functions on the fiber bundle $\Omega:=M\times_\rho\mathbb B^N$ and the holomorphic sections of the pull-back bundle $\imath^{-1}(S^mT^*_\Sigma)$ over $M$. In particular, $A^2_\alpha(\Omega)$ has infinite dimension for any $\alpha>-1$ and if $n<N$, then $A^2_{-1}(\Omega)$ also has the same property. As an application, if $\Gamma$ is a torsion-free cocompact lattice in $SU(n,1)$, $n\geq 2$, and $\rho\colon \Gamma\to SU(N,1)$ is a maximal representation, then for any $\alpha>-1$, $A^2_\alpha(\mathbb B^n\times_{\rho} \mathbb B^N)$ has infinite dimension. If $n<N$, then $A_{-1}^2(\mathbb B^n\times_{\rho} \mathbb B^N)$ also has the same property.