To fix notation, by $SU(n,p)$, I mean the subgroup of $SL_n(\mathbb F_{p^2})$ consisting of matrices $A$ which satisfy $\overline A^t A = 1$, where $\overline A$ is the matrix given by raising all the entries of $A$ to the power of $p$.

So my question is what are the irreducible algebraic representations of this group defined over $\overline {\mathbb{F_p}}$? In particular, are they the same as the representations of $SL(n,p)$? Mucking around with the MeatAxe in GAP, I've checked this is true for a few small primes and $n\leqslant 3$.

Is it true in general?