Representation theory of Discrete Subgroups of Lie groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T04:24:14Z http://mathoverflow.net/feeds/question/115632 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115632/representation-theory-of-discrete-subgroups-of-lie-groups Representation theory of Discrete Subgroups of Lie groups Anant Atyam 2012-12-06T17:49:18Z 2013-01-13T12:22:06Z <p>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 $\mathcal A_g(\mathbb C)$. </p> http://mathoverflow.net/questions/115632/representation-theory-of-discrete-subgroups-of-lie-groups/115635#115635 Answer by Andy Putman for Representation theory of Discrete Subgroups of Lie groups Andy Putman 2012-12-06T19:05:03Z 2012-12-06T19:05:03Z <p>The result you are looking for is the Margulis superrigidity theorem. See Chapter 13 of the <a href="http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html" rel="nofollow">book</a> "Introduction to Arithmetic Groups" by Dave Witte Morris for more details.</p> http://mathoverflow.net/questions/115632/representation-theory-of-discrete-subgroups-of-lie-groups/118803#118803 Answer by Aakumadula for Representation theory of Discrete Subgroups of Lie groups Aakumadula 2013-01-13T12:22:06Z 2013-01-13T12:22:06Z <p>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 $\Gamma$.</p> <p>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) </p>