Symmetric differentials and Jets extension of L^2 holomorphic functions

Let $\Sigma = \mathbb B^n/\Gamma$ be a compact complex hyperbolic space with torsion-free lattice $\Gamma\subset SU(n,1)$ and $\Omega $ a quotient of $\mathbb B^n \times\mathbb B^n$ with respect to the diagonal action of $\Gamma$ which is a holomorphic $\mathbb B^n$-fiber bundle over $\Sigma$. The goal of this article is to investigate the relation between symmetric differentials of $\Sigma$ and the weighted $L^2$ holomorphic functions on the exhaustions $\Omega_\epsilon$ of $\Omega$. If there exists a holomorphic function on $\Omega_\epsilon$ on some $\epsilon$, then there exists a symmetric differential on $\Sigma$. Using this property, we show that $\Sigma$ always has a symmetric differential of degree $N$ for any $N\geq n+1$. Moreover for each symmetric differential over $\Sigma$, we construct a weighted $L^2$ holomorphic function on $\Omega_{1\over \sqrt{n}}$.