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 $\mathcal A_g(\mathbb C)$.
2 Answers
The result you are looking for is the Margulis superrigidity theorem. See Chapter 13 of the book "Introduction to Arithmetic Groups" by Dave Witte Morris for more details.
Perhaps I can take the liberty of amplifying on Andy Putman's answer. We are looking for, say,(finite dimensional complex) representations of $\Gamma = Sp_{2g}({\mathbb Z})$ for $g\geq 2$, so we are in "higher rank". Using superrigidity, it can be proved that all representations of $\Gamma$ are completely reducible. We may therefore, need only describe the irreducible complex representations $\rho $ of $\Gamma$.
Any such $\rho$ is a tensor product of the form $\tau \otimes \sigma$ where $\tau$ is an irreducible {\bf algebraic} representation of $Sp_{2g}({\mathbb C})$ (hence given by highest weight theory) and $\sigma$ is an irreducible representation of $Sp_{2g}({\mathbb Z}/m{\mathbb Z})$ (a finite congruence quotient of $\Gamma $). This can be easily deduced from the reference of Prof. Jim Humphreys (BassMilnorSerre paper)

$\begingroup$ Apologies: in this case, superrigidity and complete reducibility follow from the congruence subgroup property. Complete reducibility does not follow directly from superrigidity but one has to have an extra argument using vanishing of the first cohomology of $\Gamma $ with values in finite dimensional representations. This is also done in Margulis' book on discrete subgroups of semisimple Lie groups. $\endgroup$ Jan 13, 2013 at 12:31
$g$
here, since the case of a rank one simple Lie group is far more problematic than higher ranks (beginning with the congruence subgroup problem). The literature is formidable, so the bookinprogress suggested by Andy is a good place to start. Though the question in your header looks straightforward, it conceals a lot of heavy mathematics. (By the way, you might add liegroups to your long list of tags.) $\endgroup$