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.