Let $V$ be a finite-dimensional irreducible representation of the Lie group $\text{SL}\_n(\mathbb{R})$. Must $V$ remain irreducible when you restrict the action to $\text{SL}\_n(\mathbb{Z})$? More generally, when you restrict it to other lattices in $\text{SL}\_n(\mathbb{R})$?



Restricting representations to lattices
The result is essentially the statement of Borel's stability theorem for $\mathrm{SL}\_n(\mathbf{R})$, see for example
Theorem 4.39 of the following:


<http://people.uleth.ca/~dave.morris/books/IntroArithGroups.pdf>


Sometimes Borel's stability theorem is phrased in terms of one of the corollaries, which in this case would be that any lattice in $\mathrm{SL}\_n(\mathbf{R})$ is Zariski dense. Then to deduce the result one would also have to note that all finite dimensional representations of $\mathrm{SL}\_n(\mathbf{R})$
are algebraic.


