Let G=GL_{n}(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.