In this question, all representations are finite-dimensional representations over $\mathbb{C}$.


Fix some $n \geq 3$. Assume that $V$ is a representation of $\text{SL}(n,\mathbb{Z})$. Also, assume that $W$ is a subrepresentation of $V$ and set $V' = V/W$, so we have a short exact sequence


$$0 \longrightarrow W \longrightarrow V \longrightarrow V' \longrightarrow 0$$


of $\text{SL}(n,\mathbb{Z})$ representations. Assume that the actions of $\text{SL}(n,\mathbb{Z})$ on $W$ and $V'$ extend to actions of $\text{SL}(n,\mathbb{C})$. Question : Must the action of $\text{SL}(n,\mathbb{Z})$ on $V$ extend to an action of $\text{SL}(n,\mathbb{C})$?


Margulis superrigidity says that the action virtually extends, but I can't construct representations where it doesn't extend on the nose. Of course, if it extends then $V = V' \oplus W$ as $\text{SL}(n,\mathbb{C})$-representations.



Question about the representation theory of SL(n,Z)
Consider the surjective map of $SL(n,\Bbb Z)$-modules $Hom\_{\Bbb C}(V',V)\to Hom\_{\Bbb C}(V',V')$.
Tim tells us that the identity map from $V'$ to $V'$ lifts to an $f:V'\to V$ which is invariant under a finite index subgroup $\Gamma $ of $SL(n,\Bbb Z)$. Then by averageing one can make it invariant under $SL(n,\Bbb Z)$.


