Given a prime $p$ and an integer $d \geq 1$, we may consider the adjoint representation of $\operatorname{SL}_d(\mathbb{F}_p)$ on $\mathfrak{sl}_d(\mathbb{F}_p)$ given by $g \cdot A=gAg^{-1}$.
It is stated in this thread that for $d=2$ and $p>2$ this representation is irreducible. What about for $d>2$? Is it true for all primes? What is a good reference?
No. For each prime $p$ there are arbitrarily large $d$ so that the adjoint representation of $\mathrm{SL}_d$ is not irreducible: if $d$ is a multiple of $p$ then the scalar matrices are in $\mathfrak{sl}_d$ and give a non-trivial subrepresentation of the adjoint representation.