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

For a symmetric differential on the compact quotient $\mathrm{\Sigma }={\mathbb{B}}^n\mathrm{/}\mathrm{\Gamma }$ of the complex unit ball ${\mathbb{B}}^n\subset {\mathbb{C}}^n$ by a discrete subgroup $\mathrm{\Gamma }\subset \mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{B}}^n)$, there exists a corresponding weighted $L^2$-holomorphic function on $({\mathbb{B}}^n\times {\mathbb{B}}^n)\mathrm{/}\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ 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.