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Can anyone provide a reference to proofs of statements of the following type: The higher algebric group cohomology of a reductive group $G$ over $\mathbb{C}$ vanishes.

I am interested not just in finite dimensional modules but also "rational representations" for instance the functions on a vector space $\mathbb{C}^{n}$ on which $G$ acts.

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  • $\begingroup$ I'd second BCnrd's request for a clearer formulation. As far as I know, the deep cohomology questions center either on real/complex Lie groups and infinite dimensional representations, or on reductive algebraic groups in prime characteristic (where interesting representations are usually finite dimensional but complete reducibility usually fails). $\endgroup$
    – Jim Humphreys
    Jul 19, 2010 at 14:07
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    $\begingroup$ @Jim: Maybe it wasn't meant as a deep question; maybe Brian simply answered it? $\endgroup$
    – Greg Kuperberg
    Jul 19, 2010 at 14:30
  • $\begingroup$ Thanks for the responses. The question may indeed have just been trivial in light of BCnrd's observation. $\endgroup$
    – Oren Ben-Bassat
    Jul 19, 2010 at 18:21
  • $\begingroup$ @BCnrd: I suggest reposting your comment as the answer to the question. $\endgroup$
    – Greg Kuperberg
    Jul 19, 2010 at 20:43

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Rational representations are directed unions of finite-dimensional ones, on which all linear representations of $G$ are completely reducible (either by an ad hoc definition of "reductive group" or a theorem applied to a good definition). So the functor of $G$-invariants on the category of rational representations is exact, hence one gets the desired higher vanishing (by whatever reasonable method one chooses to define the higher cohomologies).

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    $\begingroup$ It's probably helpful to add a reference, though people studying infinite dimensional representations of reductive groups in characteristic 0 tend to focus on Lie groups. So I'm not sure what a "standard" reference would be, though Hochschild's books get into the interface with algebraic groups. Even though Jantzen's book Representations of Algebraic Groups is aimed mainly at prime characteristic, much of the foundational material he covers at first is in the spirit of this question. $\endgroup$
    – Jim Humphreys
    Jul 24, 2010 at 15:44

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