Passing from T-equivariant to G-equivariant cohomology

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.

• Two things. The relevant map of spaces here is $X_T := (X \times EG)/T \to X_G := (X\times EG)/G$ with fiber $G/T$ or homotopically $G/B$. So $H^*(X_T)$ can be computed from a spectral sequence from $H^*(X_G)$. If for some reason you knew already the latter had even-degree cohomology... The other thing is that your map factors through $H^*_{N(T)}(X;\mathbb Z)$, and I suspect any kernel has to happen already there. Commented Apr 16, 2016 at 7:21
• 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 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. Commented Apr 17, 2016 at 1:14
• Thanks Tara for this useful response. If you make it an answer, I'll accept it. Commented Apr 18, 2016 at 1:46

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.