Let $C$ be a complex curve with universal covering $\tilde{C}$ (which in my case is the upper half plane). Any group-cocylce $e \in H^1(\pi_1(C^n),H^0(\tilde{C}{}^n,\mathcal{O}^{\times}))$ defines a line bundle $L_e$ on $C^n$.
My question is the following: suppose for any $\gamma \in \pi_1(C^n)$ the cocycle $e_\gamma$ is symmetric function in its $n$ variables in $\tilde{C}$, does the line bundle $L_e$ descend to quotient given by the symmetric product $C^{(n)} = C^n / \mathrm{S}_n$ where $\mathrm{S}_n$ is the symmetric group?
If the answer is no in general, is there an additional condition on $e$ such that the statement holds?
