Extending representation of $\operatorname{GL}(n,\mathbb{Z})$ to $\operatorname{GL}(n,\mathbb{Q})$ Given a representation $\rho\colon\operatorname{GL}(n;\mathbb{Z}) \rightarrow\operatorname{GL}(V)$ on a finite-dimensional, complex vector space $V$, I am trying to understand a condition that would let me find a representation $$\tilde \rho\colon\operatorname{GL}(n;\mathbb{Q}) \rightarrow \operatorname{GL}(V)$$ with $\tilde \rho|_{\operatorname{GL}(n;\mathbb{Z})} = \rho.$
This is easy enough when $n=1$. Any representation $\rho$ can be extended to $\mathbb{Q}^{\times}$ by defining $$\rho(x) := \begin{cases} \rho(1): & x > 0; \\ \rho(-1): & x < 0. \end{cases}$$ In other words, pulling back along the sign homomorphism $$\mathrm{sgn} : \mathbb{Q}^{\times} \rightarrow \mathbb{Z}^{\times}, \; x \mapsto \begin{cases} 1: & x > 0; \\ -1: & x < 0. \end{cases}$$
I can't think of a surjective homomorphism $\operatorname{GL}(n;\mathbb{Q}) \rightarrow\operatorname{GL}(n;\mathbb{Z})$ for $n > 1$ so I can't use this argument anymore.
 A: In general this is impossible (I assume that $n\ge 2$). The simplest counter-example is the homomorphism $\operatorname{GL}(n, {\mathbb Z})\to F=\operatorname{GL}(n, {\mathbb Z}/p)$. The image is a finite nonabelian group and admits a (finite-dimensional) faithful (real) linear representation $F\to\operatorname{GL}(V)$, while the group $\operatorname{GL}(n, {\mathbb Q})$ is essentially simple (its derived subgroup $\operatorname{SL}(n,{\mathbb Q})$ is simple). Thus, the homomorphism
$$
\operatorname{GL}(n,{\mathbb Z})\to\operatorname{GL}(n, {\mathbb Z}/p)\to\operatorname{GL}(V)
$$
does not extend to $\operatorname{GL}(n,{\mathbb Q})$. There are variations on this construction obtained by taking finite index subgroups in $\operatorname{GL}(n,{\mathbb Z})$ and using their (finite-dimensional) linear representations and combining them with the induced representation construction. In the case $n=2$ there are even more constructions.
Nevertheless, remarkably, there is an extension theorem of sorts in this situation:
Theorem. (G.Margulis) For $n\ge 3$, given any finite-dimensional linear representation $f\colon\operatorname{SL}(n,{\mathbb Z})\to\operatorname{GL}(V)$, either the image of $f$ is bounded or there exists a finite index subgroup $\Gamma<\operatorname{SL}(n,{\mathbb Z})$ such that $f|\Gamma$ extends to a homomorphism $\operatorname{SL}(n,{\mathbb Q})\to \operatorname{GL}(V)$.
Actually, Margulis proves much more than this... See this wikipeadia article as well as
"Ergodic theory and semisimple groups" by Robert Zimmer. (Margulis also has a book with more detailed proofs but it is much harder to read.)
