@pschwahn@AxelBoldt - right, that's one way to proceed. I've been doing a lot of work lately with representations of GL(n,𝔽) for 𝔽 an arbitrary field of characteristic zero. For subfields of ℂ this trick of complexifying and reducing to the case 𝔽 = ℂ works fine. But in fact the representation theory works exactly the same way even for fields of characteristic zero that aren't subfields of ℂ!
It's not that I really care about such fields. I just find it esthetically annoying to work only with subfields of ℂ when dealing with something that's purely algebraic and shouldn't really involve the complex numbers. So I had to learn a bit about how we can develop the representation theory of GL(n,𝔽) for an arbitrary field of characteristic zero. Milne's book 𝐴𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐 𝐺𝑟𝑜𝑢𝑝𝑠 does this, and a preliminary version is free:
but unfortunately it's quite elaborate if all you want is the basics of the representation theory of GL(n,𝔽).
(For 𝔽 not of characteristic zero everything changes dramatically, since you can't symmetrize by dividing by n!. Nobody even knows all the irreps of the symmetric groups.)