MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 5 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.