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I understand the finite-dimensional representations of $\text{SL}(2,\mathbb C)$ as a Lie group and their correspondence with Lie algebra representations of $\mathfrak{sl}(2,\mathbb C)$.

Does anyone know a good reference that explains the analogous theory for the finite dimensional representations of $\text{SL}(2)$ as an algebraic group (over an arbitrary field)? Under what conditions does a similar correspondence hold with the Lie algebra representations?

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You could try looking in Jantzen's book "Representations of algebraic groups". This assumes the ground field is algebraically closed. For the full story, see Tits' paper "Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque" (Crelle, 1971).

The basic story is this:

Over any field of char 0, the situation is not too surprising. For SL(2), the representations are exactly the ones you'd expect -- every representation is a direct sum of irreducibles and the irreducibles are exactly the symmetric powers of the standard representation. This goes over to a general split connected reductive group in the way you'd expect, with irreducibles being parametrised by the dominant integral weights of the Lie algebra. If the group is non-split, then not all of the weights correspond to representations defined over the ground field, but there is still a relatively nice description (cf. Tits' paper, or the summary in Gross's "Algebraic modular forms").

For char p the story is much more subtle, even for SL(2). The Frobenius map now gives you unexpected morphisms between the symmetric powers of the standard rep, sending $Sym^k$ to $Sym^{pk}$. The images of these maps in general don't have invariant complements, so the representations are not semisimple, and the situation becomes much more complicated and interesting. Jantzen has much more information about this.

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Thanks David. That's very helpful. –  John M Sep 8 '11 at 20:08
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