Weighted L² Holomorphic Functions on Ball-Fiber Bundles Over Compact Kähler Manifolds

Let M~ be a complex manifold, Γ be a torsion-free cocompact lattice of Aut (M~) and ρ: Γ → SU(N, 1) be a representation. Suppose that there exists a ρ -equivariant totally geodesic isometric holomorphic embedding (Formula presented.). In this paper, we investigate a relation between weighted L² holomorphic functions on the fiber bundle Ω : = M× _ρB^N and the holomorphic sections of the pull-back bundle ı∗(SmTΣ∗) over M. In particular, Aα2(Ω) has infinite dimension for any α> - 1 and if n< N , then A-12(Ω) also has the same property. As an application, if Γ is a torsion-free cocompact lattice in (Formula presented.), and (Formula presented.) is a maximal representation, then for any α> - 1 , Aα2(Bn×ρBN) has infinite dimension. If n< N , then A-12(Bn×ρBN) also has the same property.