Let $G$ be a connected semisimple algebraic group defined over $\mathbb Q$. Could some expert give me a complete classification of finite dimensional $\mathbb Q$-irreducible representations of $G$? If convenient please feel free to assume that $G$ is almost simple or simply connected.
Basically what I want to know is the following: given an irreducible representation over $\mathbb C$, is it possible to give a list of all the $\mathbb Q$ irreducible representations whose irreducible component over $\mathbb C$ is the given one. I feel it is possible to do so in the case where $G=SL_n$.
It might be possible find the answer in the Borel-Tits 1965 IHES paper, but it is in French which I can not read.
It seems that the question is difficult to answer, so probably it is helpful to restrict to the cases which I really need. Assume $G$ has no $\mathbb Q$-anisotropic factors and the irreducible component over $\mathbb C$ is Minuscule (maybe I should say the corresponding Lie algebraic representation is Minuscule).