Rigidity of the period map up to finite covers

The period map is an important holomorphic map from the moduli space $M_g$ of genus g curves to the moduli space $A_g$ of g-dimensional principally polarized abelian varieties, sending a curve to its Jacobian. It was shown by Farb that the period map is the unique nonconstant holomorphic map from $M_g$ to $A_h$ for h \le g. In my work, we study holomorphic maps from a certain finite cover $R_g$ of $M_g$ to $A_h$ for h \le g, and prove that the unique  nonconstant holomorphic map from $R_g$ to $A_h$ is the composition of the covering map $R_g$ to $M_g$ and the period map from $M_g$ to $A_g$.
The proof is based on classifying linear representations of certain finite-index subgroups of the mapping class group.