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.

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.