# Group Cohomology for Reductive Groups

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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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). –  Jim Humphreys Jul 19 '10 at 14:07
@Jim: Maybe it wasn't meant as a deep question; maybe Brian simply answered it? –  Greg Kuperberg Jul 19 '10 at 14:30
Thanks for the responses. The question may indeed have just been trivial in light of BCnrd's observation. –  Oren Ben-Bassat Jul 19 '10 at 18:21
@BCnrd: I suggest reposting your comment as the answer to the question. –  Greg Kuperberg Jul 19 '10 at 20:43

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).