Symmetric differentials and jets extension of $L^2$ holomorphic functions II: Explicit form

For a symmetric differential on the compact quotient $\Sigma = \mathbb{B}^n / \Gamma$ of the complex unit ball $\mathbb{B}^n \subset \mathbb{C}^n$ by a discrete subgroup $\Gamma \subset \mathrm{Aut}(\mathbb{B}^n)$, there exists a corresponding weighted $L^2$-holomorphic function on $(\mathbb{B}^n \times \mathbb{B}^n)/\Gamma$, where $\Gamma$ acts diagonally on $\mathbb{B}^n \times \mathbb{B}^n$. In this paper, we give an explicit description of this correspondence and derive several applications based on its explicit form.