Given a (smooth but not holomorphic) surface bundle over surface $S_g\to M\to S_h$ (where $g,h>1$, there is an induced surface group representation $$\pi_1(S_h)\to \text{SP}(2g,\mathbb R).$$
I was told that there is an associated equivariant holomorphic map $$\tilde{S_h}=\mathbb H^2\to \text{SP}(2g,\mathbb R)/U(g).$$
I wonder why there is such an equivariant holomorphic map and is it generally true that if we have a homomorphism between lattices, then there is an associated equivariant map between their symmetric spaces? Here I see ${S_h}$ as a Fuchsian group and $\tilde{S_h}$ is its symmetric space, and $\text{SP}(2g,\mathbb R)/U(g)$ is the symmetric space of $\text{SP}(2g,\mathbb R)$.