Let $G = G(K)$ be a Chevalley group over an algebraically closed field $K$ of characteristic $p > 0$. Consider the finite group $G(q) = G(\mathbb{F}_q)$. (For example, if $G = \operatorname{SL}_n(K)$ then $G(q) = \operatorname{SL}_n(\mathbb{F}_q)$).

It was proven by Steinberg that every irreducible $\mathbb{F}_q[G(q)]$-module is absolutely irreducible, and correspond to $q$-restricted dominant weights (i.e. weights $\lambda$ such that $0 \leq \langle \lambda, \alpha \rangle < q$ for all simple roots $\alpha$).

Consider an indecomposable $\mathbb{F}_q[G(q)]$-module $V$. Is $V \otimes_{\mathbb{F}_q} K$ an indecomposable $K[G(q)]$-module?

EDIT: In case this is not true, I also would be interested if what I am asking for is true under some mild restrictions on $q$ and/or $G$.

simplegroups here. There are also some close relatives such as $\mathrm{SL}(2, p)$ for $p>3$ having finite representation type. $\endgroup$ – Jim Humphreys Mar 26 '18 at 17:55