TITLE: Passing from T-equivariant to G-equivariant cohomology
QUESTION [12 upvotes]: Let G=GLn(ℂ) and let T be a maximal torus. Let X be a topological space with a G-action. My question is: when is the canonical map $$H^*_G(X;\mathbb{Z})\to H^*_T(X;\mathbb{Z})$$ injective?
Some remarks: I am trying to understand Torsten's answer here, which claims injectivity for the case where X is the space of length m quotients of $\mathcal{O}_C^m$ for a smooth proper curve C over ℂ. (there is a remark "G being special" that is especially cryptic to me and is probably where the answer lies). I am familiar with some arguments to prove injectivity when the ring of coefficients has n! invertible, but here I specifically want to consider integral coefficients.

REPLY [12 votes]: This isn't a complete answer, but for example, if you know that $H_T^*(X;\mathbb{Z})$ injects into the cohomology of the fixed point set $X^T$, then for $G=GL_n(\mathbb{C})$, the canonical map $H_G^*(X;\mathbb{Z})\to H_T^*(X;\mathbb{X})$ is injective.  See Theorem 2.10 and Corollary 2.11 in T. Holm and R. Sjamaar, "Torsion and abelianization in equivariant cohomology." Transform. Groups 13 (2008), no. 3-4, 585–615.