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.