Weighted $L^2$ Holomorphic Functions on Ball-Fiber Bundles Over Compact Kähler Manifolds

Abstract

Let ${\widetilde{M}}$ be a complex manifold, $\Gamma$ be a torsion-free cocompact lattice of $\text {Aut}({\widetilde{M}})$ and $\rho :\Gamma \rightarrow SU(N,1)$ be a representation. Suppose that there exists a $\rho$-equivariant totally geodesic isometric holomorphic embedding $\imath :{{\widetilde{M}}}\rightarrow {\mathbb {B}}^N$. Let $M:={{\widetilde{M}}}/\Gamma$ and $\Sigma :={\mathbb {B}}^N/\rho (\Gamma )$. 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 $\imath ^*(S^mT^*_\Sigma )$ 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), $n\ge 2$, and $\rho :\Gamma \rightarrow SU(N,1)$ 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.