Timeline for Is there a notion of "tame" representations of $GL_n(Z)$?
Current License: CC BY-SA 3.0
9 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| May 4, 2014 at 22:18 | comment | added | Misha | Sam: In addition to Margulis you also need congruence subgroup property (to reduce the problem to congruence quotients) and a tiny argument that $Sl(n,Z)$ does not have finite dimensional unitary representations with infinite image. Needless to say, none of this works for $n=2$. Infinite dimensional unitary theory for groups like $Sl(n,Z)$ is very poorly understood, I think nothing beyond what the property T gives you. | |
| May 4, 2014 at 17:24 | comment | added | Andy Putman | $\text{SL}_n(\mathbb{R})$ and the representation theory of finite groups like $\text{SL}_n(\mathbb{Z}/\ell)$. A good place to learn about this kind of stuff is Dave Witte Morris's book people.uleth.ca/~dave.morris/books/IntroArithGroups.html | |
| May 4, 2014 at 16:32 | comment | added | Andy Putman | @StevenSam : The key result for finite-dimensional representations is the Margulis Superrigidity Theorem, which for $\text{SL}_n(\mathbb{Z})$ says that as long as $n \geq 3$, any finite-dimensional representation "virtually extends" to $\text{SL}_n(\mathbb{R})$, i.e. there exists a finite-index subgroup $\Gamma$ of $\text{SL}_n(\mathbb{Z})$ such that the restriction of the representation to $\Gamma$ extends. This means essentially that the finite-dimentional representation theory of $\text{SL}_n(\mathbb{Z})$ is a sort of mixture of the representation theory of | |
| May 4, 2014 at 16:09 | comment | added | Steven Sam | Okay, could you give me a reference for finite-dimensional representations? | |
| May 4, 2014 at 15:58 | comment | added | Misha | Do you mean a finite dimensional or infinite dimensional representation? Finite dimensional ones are reasonably well understood. | |
| May 4, 2014 at 3:47 | comment | added | Steven Sam | I don't explicitly have an example. What I meant is just that for the representations I want to consider, it will not be known a priori that they are compatible with base change. I was motivated by the discussion in Section 6.2 of arxiv.org/abs/1008.1368 | |
| May 4, 2014 at 3:44 | comment | added | Peter McNamara | Could you give an example of a representation of GL_n(Z) that you'd like to include that is not compatible with base change to Q? | |
| May 4, 2014 at 0:46 | history | asked | Steven Sam | CC BY-SA 3.0 |