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The question itself and some of the comments seem out of focus to me, so let me add to what Richard and George write the following summary version of an answer. I'd stress that nothing here is really complicated or subtle to prove apart from the basic Cartan-Weyl classification and (in characteristic 0) complete reducibility for finite dimensional representations.

First, the group itself is defined and split over the prime field (here $\mathbb{Q}$), hence over any larger field. Chevalley's theory implies that the representations discussed here are absolutely irreducible over $K$. (For a semisimple group defined but not split over a field, more analysis is needed of representations which require a field extension to become absolutely irreducdible.)

Anyway, for a connected semisimple group the "rational" and "polynomial" representations are the same, unlike the reductive group GL$(n,K)$. The group also being simply connected in this case, the rational/polynomial representations are essentially those of the Lie algebra and are more easily classified by dominant integral highest weights in that setting. So each irreducible representation or simple module in question has a unique highest weight $\lambda$. The easy textbook criterion for such a module to be self-dual is just that $\lambda = -w_0 \lambda$ where $w_0$ is the longest element of the Weyl group. As Richard Borcherds points out, this is -1 just for simple types listed, including type $A_1$.

So far nothing really depends on characteristic 0. But as George McNinch observes, there are plenty of cases where nonsimple modules in prime characteristic fail to be completely reducible and are typically not self-dual. So you do need to invoke complete reducibility (and non-canonical direct sum decompositions) to dispose of the characteristic 0 question.

P.S. It's certainly possible to treat the rank 1 case here by direct ad hoc methods in characteristic 0, including the needed proof of complete reducibility (using the easily computed Casimir operator). For irreducible representations, self-duality is a trivial consequence of the fact that these representations are uniquely classified (up to isomorphism) by their dimensions 1, 2, 3, .... But such an ad hoc argument fails to provide much enlightenment. And the general theory allows one to see that the group representations and Lie algebra representations are essentially the same, whether the groups are regarded as Lie groups or algebraic groups (or just as abstract groups). Of course, finite dimensionality is a key point throughout, since the infinite dimensional representation theory involves harder questions.

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The question itself and some of the comments seem out of focus to me, so let me add to what Richard and George write the following summary version of an answer. I'd stress that nothing here is really complicated or subtle to prove apart from the basic Cartan-Weyl classification and (in characteristic 0) complete reducibility for finite dimensional representations.

First, the group itself is defined and split over the prime field (here $\mathbb{Q}$), hence over any larger field. But it should also be specified Chevalley's theory implies that the representations discussed here are absolutely irreducible over $K$ to avoid possible extra analysis. (That is For a semisimple group defined but not so difficult in characteristic 0 and has also been worked out by Tits and others split over any field.a field, more analysis is needed of representations which require a field extension to become absolutely irreducdible.)

Anyway, for a connected semisimple group the "rational" and "polynomial" representations are the same, unlike the reductive group GL$(n,K)$. The group also being simply connected in this case, the rational/polynomial representations are essentially those of the Lie algebra and are more easily classified by dominant integral highest weights in that setting. So each irreducible representation or simple module in question has a unique highest weight $\lambda$. The easy textbook criterion for such a module to be self-dual is just that $\lambda = -w_0 \lambda$ where $w_0$ is the longest element of the Weyl group. As Richard Borcherds points out, this is -1 just for simple types listed, including type $A_1$.

So far nothing really depends on characteristic 0. But as George McNinch observes, there are plenty of cases where nonsimple modules in prime characteristic fail to be completely reducible and are typically not self-dual. So you do need to invoke complete reducibility (and non-canonical direct sum decompositions) to dispose of the characteristic 0 question.

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The question itself and some of the comments seem out of focus to me, so let me add to what Richard and George write the following summary version of an answer. I'd stress that nothing here is really complicated or subtle to prove apart from the basic Cartan-Weyl classification and (in characteristic 0) complete reducibility for finite dimensional representations.

First, the group itself is defined and split over the prime field (here $\mathbb{Q}$), hence over any larger field. But it should also be specified that the representations discussed are absolutely irreducible over $K$ to avoid possible extra analysis. (That is not so difficult in characteristic 0 and has also been worked out by Tits and others over any field.)

Anyway, for a connected semisimple group the "rational" and "polynomial" representations are the same. The group being simply connected in this case, the rational/polynomial representations are essentially those of the Lie algebra and are more easily classified by dominant integral highest weights in that setting. So each irreducible representation or simple module in question has a unique highest weight $\lambda$. The easy textbook criterion for such a module to be self-dual is just that $\lambda = -w_0 \lambda$ where $w_0$ is the longest element of the Weyl group. As Richard Borcherds points out, this is -1 just for simple types listed, including type $A_1$.

So far nothing really depends on characteristic 0. But as George McNinch observes, there are plenty of cases where nonsimple modules in prime characteristic fail to be completely reducible and are typically not self-dual. So you do need to invoke complete reducibility (and non-canonical direct sum decompositions) to dispose of the characteristic 0 question.