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This is a followup to this question about the (left) noetherianity of the group ring of $GL_n(\mathbf{Z})$:

Does GL_n(Z) have a noetherian group ring?

Given that $\mathbf{Z}[GL_n(\mathbf{Z})]$ is not left noetherian, I want to ask if there is a relaxation. What I gather from this fact is that this ring is too large and contains information about all representations of $GL_n(\mathbf{Z})$, while maybe the right thing is to restrict to some class of "reasonable" representations to get more control.

As an analogy, the theory of polynomial representations of $GL_n(K)$ for an algebraically closed field $K$ is "reasonable" and is equivalent to the study of modules over the Schur algebra.

I would like to know if there are similar nice classes of representations for $GL_n(\mathbf{Z})$. One candidate is to look at polynomial representations, but I do not necessarily want to assume that the representations are compatible with base change of the ring $\mathbf{Z}$ to $\mathbf{Q}$, for example, so I am looking for something else. Ideally the class of representations would be equivalent to modules over some nice algebra, where nice might mean left noetherian.

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  • $\begingroup$ 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? $\endgroup$
    – Peter McNamara
    Commented May 4, 2014 at 3:44
  • $\begingroup$ 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 $\endgroup$
    – Steven Sam
    Commented May 4, 2014 at 3:47
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    $\begingroup$ @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 $\endgroup$
    – Andy Putman
    Commented May 4, 2014 at 16:32
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    $\begingroup$ $\text{SL}_n(\mathbb{R})$ 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 $\endgroup$
    – Andy Putman
    Commented May 4, 2014 at 17:24
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    $\begingroup$ 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. $\endgroup$
    – Misha
    Commented May 4, 2014 at 22:18

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