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](https://mathoverflow.net/a/18181/425), 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.



Passing from T-equivariant to G-equivariant cohomology
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.


