ScalingStacks

Since we are only considering rational coefficients, the $T$- and $G$-equivariant Chow rings agree with equivariant cohomology rings. In particular, the Chern class map induces isomorphisms of graded algebras

(3.7)

$\displaystyle S\cong\operatorname{CH}^{\bullet}_{T}(k)_{\mathbb{Q}}\text{ and }S^{W}\cong\operatorname{CH}^{\bullet}_{T}(k)_{\mathbb{Q}}^{W}\cong\operatorname{CH}^{\bullet}_{G}(k)_{\mathbb{Q}},$

see [Tot99] or [EG98, Section 3.2]. Similarly, for higher Chow groups

(3.8)

$\displaystyle\operatorname{CH}^{\bullet}_{G}(k,i)_{\mathbb{Q}}\cong(\operatorname{CH}^{\bullet}_{T}(k,i)_{\mathbb{Q}})^{W}\cong(S\otimes\operatorname{CH}^{\bullet}(k,i)_{\mathbb{Q}})^{W}$

using [Kri17, Theorem 1.5] and [Kri13, Theorem 5.7].