The following is a weaker version of what is called splitting principle in  

 [Appendix C, page 12](http://www.ma.huji.ac.il/~karshon/monograph), see also for a lighter version [Brions Eq cohom and eq intersection theory, page 6](http://www-fourier.ujf-grenoble.fr/~mbrion/notes.html):  

Let $G$ be a compact (complex) connected Lie group with torus $T\subset G$ and $N$ its normalizer, $W=N/T$ its Weyl group. Let $X$ be any $G$-variety. 
Then, there is an isomorphism 


$ H\_G^\* (X) \cong (H\_T^\*(X))^W $  

of graded algebras. For $X=pt$ this is known to be Chevalley's restriction theorem. 


My question is, can one drop the assumption on $G$ and $T$ to be compact? I came across an article [VarVas, page 12](http://arxiv.org/abs/0901.3992), where that has been claimed (with G=Gl\_n, X quasi-projective) to be a standard result and no reference is provided.



Splitting principle in equivariant cohomology
The embedding of the unitary group $U\_n$ into $GL\_n(\mathbb C)$ is a homotopy equivalence; this is easily seen to imply that $H^\*\_{U\_n}(X)$ is isomorphic to $H^\*\_{GL\_n}(X)$. So the result for compact groups implies that for $GL\_n$.


The same idea works for any reductive complex algebraic group $G$, since the embedding of a maximal compact subgroup into $G$ is a homotopy equivalence.


