Let $p$ be a prime, $\mathbb{F}_q$ a finite field of order $q$ of characteristic $p$, and $\overline{\mathbb{F}_q}$ a fixed algebraic closure of $\mathbb{F}_q$. Let $\rho:G\to {\rm GL}_n(\mathbb{F}_q)$ be a representation of a group $G$. My question is the following:
Question: Suppose that $n\geq 2$ and $p\geq 5$. Suppose also that the image of $\rho$ is ${\rm SL}_n(\mathbb{F}_q)$. Then is it true that $\rho$ is absolutely irreducible? (i.e. irreducible over $\overline{\mathbb{F}_q}$.)
It is true for $n=2$: If $\rho$ is not absolutely irreducible, then it is conjugate over $\overline{\mathbb{F}_q}$ to a representation that is upper triangular which implies the image of $\rho$ is solvable. But ${\rm SL}_2(\mathbb{F}_q)$ is not solvable if $p\geq 5$.
I think the answer to the question is yes for any $n\geq 2$. It would be great if someone could point out a reference.