Representation theory over any field I understand that representation theory of complex reductive groups is essentially combinatorial. By general principles, I imagine Galois theory then determines the theory over any field. For example, over a separably closed field, it works exactly the same way as over the complex numbers. Is that an accurate summary? Is there a reference where this exercise is worked out?
 A: As already mentioned, the representation theory of a reductive depends heavily on the characteristic of the ground field. In particular, in positive characteristic it is of a very non-combinatorial nature.
Nevertheless, the determinantion of the irreducible representations is quite uniform over all characteristics. In this case, Galois theory can be used to determine irreducible representations over any field. For connected reductive groups, this "exercise", as you call it, has been worked out by Tits in his paper
Tits, J. Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque. J. Reine Angew. Math. 247 (1971) 196–220.
He even considers irreducible representations over division fields. This exercise is much less trivial than it sounds since it involves, at a minimum, the classification of reductive groups over any field. At the end, the results are very pretty, though. Apart form highest weight theory they involve a homomorphism from the character group of the center of the group $G$ to the Brauer group of the ground field $k$.
A: That's right. There are a few things to check, but the theory that you find in standard sources, such as in Fulton and Harris, do go through for separably closed fields as well. 
