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 $\mathcal A\_g(\mathbb C)$. 



Representation theory of Discrete Subgroups of Lie groups 
The result you are looking for is the Margulis superrigidity theorem. See Chapter 13 of the [book](http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html) "Introduction to Arithmetic Groups" by Dave Witte Morris for more details.


