$\DeclareMathOperator\SL{SL}$Let $ G $ be a noncompact simple Lie group. For example $ \SL\_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by conjugation. Is this representation of $ \Gamma $ always irreducible?


For example, I think it is true that all lattices in $ \SL\_2(\mathbb{R}) $ and $ \SL\_2(\mathbb{C}) $ are Ad-irreducible.



A lattice in $ \operatorname{SL}_n $ is Ad-irreducible
Per the request to post it as an answer.


Notice that the Ad representation is a polynomial representation into $\operatorname{GL}(\operatorname{Lie}(G))$.
We do know that $\operatorname{Ad}(G)$ acts irreducibly, and $\Gamma$ is Zariski dense by Borel's density theorem. Hence $\operatorname{Ad}\rvert\_{\Gamma}$ is also irreducible.


