Q:

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.

A:

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). 

