I need a reasonably detailed reference for the proof of the fact that, in characteristic 0, any linear representation of a reductive algebric group is completely reducible. I looked in Humphries and Borel, and what I found there was cursory at best.
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2$\begingroup$ You have misspelled the name of James Humphreys. $\endgroup$– Jason StarrCommented Jan 27, 2016 at 15:20
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4$\begingroup$ @Jason: Presumably my British ancesters were mostly illiterate, so I'm used to all the variant spellings. Anyway, the question being asked is probably too elementary for this site. The concept of "reductive" algebraic group originates in the work of Borel-Tits, but related ideas in characteristic 0 are older in the theory of Lie groups. There is an identification (from the Chevalley classification) of semisimple Lie groups and semisimple algebraic groups over $\mathbb{C}$. Then the question reduces to "reductive" Lie algebras and Weyl's complete reducibility theorem. $\endgroup$– Jim HumphreysCommented Jan 27, 2016 at 18:33
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$\begingroup$ jmilne.org/math/CourseNotes/iAG200.pdf Chapter 22, Section o. $\endgroup$– zenoCommented Jan 27, 2016 at 19:20
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$\begingroup$ As zeno points out, there are explicit statements. Actually, my early Chapter V on characteristic 0 theory does treat the case of a semisimple group with few preliminaries other than Weyl's complete reducibility theorem: see 13.2 and 14.3. [to be continued] $\endgroup$– Jim HumphreysCommented Jan 28, 2016 at 14:07
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$\begingroup$ The more difficult question is to get complete reducibility in char 0 for (say connected) reductive groups from the Borel-Tits definition. It seems to take a lot of work to show that such a group is the almost-direct product of a torus (for which all rational representations are completely reducible in any characteristic) and a semisimple group (possibly trivial). The root structure gets one close to the full classification. Maybe there's a shortcut, but in prime characteristic complete reducibility usually fails for semisimple groups. $\endgroup$– Jim HumphreysCommented Jan 28, 2016 at 14:09
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