Let $k$ be a field and $G/k$ a connected reductive group. Fix a maximal torus $T$, and let $X$ denote the group of characters of $T_{\overline k}$, where $\overline k$ is a separable closure of $k$. Let $R\subset X$ denote the set of roots of $(G_{\overline k},T_{\overline k})$ and fix an ordering, with positive roots $R^+$ and dominant weights $X^+$. Everything has an action $\operatorname{Gal}(\overline k/k)$.

Then any $\lambda\in X^+$ gives an irreducible representation $V_{\lambda}$ of $G_{\overline k}$ with highest weight $\lambda$.

The question: if $k\subset F\subset\overline k$ is such that $\operatorname{Gal}(\overline k/F)$ stabilizes $\lambda$, is it true that $V_{\lambda}$ is actually the extension of scalars of a representation of $G_F$?

According to Théorème 3.3 of [Tits, *Représentations linéaires irréductibles d'un groupe réductif*], there exists a representation of $G_F$ with values in $GL_{m,D}$, where $D$ is a central division algebra over $F$, whose base change to $\overline k$ gives indeed $V_{\lambda}$. But $D$ is not necessarily equal to $F$.

`$\overline{k}$`

for an algebraic closure and`$k_s$`

for a separable closure. $\endgroup$