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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 $\mathcal A_g(\mathbb C)$.

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  • $\begingroup$ It helps to specify $g$ 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 book-in-progress 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 lie-groups to your long list of tags.) $\endgroup$
    – Jim Humphreys
    Commented Dec 6, 2012 at 20:50
  • $\begingroup$ P.S. It may be worth looking at the 1967 IHES Publ. Math. 33 paper by Bass-Milnor-Serre (available at numdam.org), which treats the congruence subgroup problem here and considers representations (final section). For symplectic groups of rank at least 2, there is extra complication if you work over imaginary number fields. But in rank 1 (the classical modular group) it's far worse. For general discrete subgroups you ask which lattices are arithmetic subgroups (as in Mostow, Margulis) and then which arithmetic groups are congruence groups. The story is rather long. $\endgroup$
    – Jim Humphreys
    Commented Dec 6, 2012 at 23:42

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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.

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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 super-rigidity, 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 (Bass-Milnor-Serre paper)

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  • $\begingroup$ Apologies: in this case, superrigidity and complete reducibility follow from the congruence subgroup property. Complete reducibility does not follow directly from super-rigidity 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 semi-simple Lie groups. $\endgroup$
    – Venkataramana
    Commented Jan 13, 2013 at 12:31

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