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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).

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    $\begingroup$ This might be a bit ambitious for arbitrary $G$. Also, even over $\mathbb{C}$, the answer of "by dominant integral weights" may not be a satisfactory classification. It perhaps does not portend well for a classification over $\mathbb{Q}$. $\endgroup$ – Peter Crooks Feb 8 '15 at 19:13
  • $\begingroup$ @ Peter Crooks: over $\mathbb C$, the classification by dominant integral weights is satisfied. $\endgroup$ – ronggang Feb 8 '15 at 19:54
  • $\begingroup$ I have a couple of lists: Restriction of scalars from an absolutely (over $\mathbb C$) irreducible representation over a number field, representations in $GL_n( D)$ where $D$ is a central division algebra over $\mathbb Q$. $\endgroup$ – ronggang Feb 8 '15 at 19:54
  • $\begingroup$ 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 $G=SL_n$. $\endgroup$ – ronggang Feb 8 '15 at 19:55
  • $\begingroup$ Right, over $\mathbb{C}$, this is the answer. I was unsure about whether such an answer was a satisfactory classification for your purposes. Anyway, I would suggest editing your post to take into account the comment you have just made. I think it will help yield the sort of answer you seek. $\endgroup$ – Peter Crooks Feb 8 '15 at 20:02

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