Let $K$ be a finite field of characteristic $p$, $G/K$ be a connected, reductive, split algebraic group. Fix some maximal split torus $T$ and a system of positive roots $\Phi^+ \subset \Phi (G,T)$.


Consider a $p$-restricted weight $\lambda \in X\_\*(T)$: this is a (dominant) weight such that $0 \le \langle \lambda, \alpha \rangle < p$ for all simple roots $\alpha$.


Just like for any other dominant weight $\lambda$, we have an associated highest weight module $L(\lambda)$: this is a rational, finite-dimensional $K$-representation of $G$ characterized by having highest weight $\lambda$.


Question: if $\lambda$ is $p$-restricted, is it true that $L(\lambda)$ is absolutely irreducible? If so, I'd be grateful to anyone who provides a reference.



Absolute irreducibility of p-restricted highest weight modules
Yes, it's absolutely irreducible. The standard reference is R. Steinberg's 1963 paper [*here*](http://www.ams.org/mathscinet-getitem?mr=0155937) (which is freely available online). His 1967-68 Yale lectures (see $\S13$), now published by AMS in a LaTeX version, may be a good alternative source. 


It should be emphasized, however, that our knowledge of these irreducibles is indirect and incomplete. This is somewhat parallel to the infinite dimensional theory in characteristic 0, but far more complicated to resolve. 


ADDED: For a treatment in somewhat more modern language, see also my LMS Lecture Note volume 326 (2006), section 5.2. As in Steinberg's early work, the finite groups of Lie type are the main theme; but I've tried to make the splitting field notion more explicit. (Revisions are posted on my webpage [*here*](http://people.math.umass.edu/~jeh/pub/book.html).)


One other comment is that for the algebraic groups (or the finite groups over arbitrary finite fields), Steinberg's tensor product theorem shows that *all* irreducibles are then absolutely irreducible when constructed over appropriate finite fields.


