The following is a weaker version of what is called splitting principle in
Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6:
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, where that has been claimed (with G=Gl_n, X quasi-projective) to be a standard result and no reference is provided.