Splitting principle in equivariant cohomology

Question

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.

Answer 1

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.