TITLE: Representation theory of inner forms
QUESTION [5 upvotes]: I once heard something like "inner forms of reductive groups have the same representation theory". 
Is this assertion misguided? 
If this assertion is not misguided, then is there a precise statement to this effect (perhaps in Tannakian terms)?

REPLY [4 votes]: Well, it is not misguided if by "representation" you mean "algebraic representation". More generally, if $E$ is a right $G$-torsor over $Spec F$ and $X$ is a $G$-variety you can form a ``twisted form'' $E\wedge_G X=E\times X/(e,x)\sim (eg,gx)$ which is ${}_EG$-variety, where ${}_EG$ is the inner twisted form of $G$ corresponding to $E$. This gives an equivalence between the category of $G$-varieties and the category of ${}_EG$-varieties. The same construction applies to algebraic representations (and the equivalence is additive and preserves tensor products).