My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good beginning reference for these type of questions. Some Representations of $Sp(2g, \mathbb Z)$ dont seem to extend such as coset spaces associated to quotients with respect to congruence subgroups. The question is are these the only ones in some sense. I am a novice in this branch of mathematics so any references would be appreciated. The question arose because I want to understand complex vector bundles admitting a flat Connection on $\mathcal A_g(\mathbb C)$.