Let $G$ be a compact connected Lie group, $T \subset G$ its maximal torus and $W=N(T)/T$ its Weyl group. The formula 2.11 of Atiyah, Bott The Moment Map and Equivariant Cohomology states that for any topological space $X$ with an action of $G$ the $G$- and $T$-equivariant cohomology are connected by $$H_G^*(X) \simeq H_T^*(X)^W.$$
The proof is only briefly sketched, so I wonder whether somesome could help me to decypher it. Or, maybe, there is a reference with complete exposition?
Equivariant cohomology (from MO:131270). Given an action of a group $G$ on a topological space $X$, the associated homotopy quotient is $$X_G := (EG \times X)/G,$$ where $EG$ is the total space of a universal principal $G$-bundle $EG \to BG$ and the quotient is by the (free) diagonal action of $G$ on $EG \times X$. The $G$-equivariant cohomology of $X$ is defined to be $$H^\ast_G(X) := H^\ast(X_G),$$ the cohomology of $X_G$ (say singular over $\mathbb{Q}$).
Sketch of the proof (from Atiyah, Bott). If one divides $EG \times X$ by $T$ first, one obtains a fibering $$G/T \to X_T \to X_G$$ with fiber $G/T$. Now the fact that the Euler class of $G/T$ is non-zero already implies that, over $\mathbb Q$, $H_G^*(X)$ imbeds into $H_T^*(X)$. Finally the identification of the image follows from the well-known fact that $W$ acts on $H^*(G/T)$ as the regular representation.
Details of the proof.
- The Euler class is non-zero because it is equal to $$\chi(G/T)=\chi((G/T)^T)=\chi(N(T)/T)=|W|.$$
- The fact that the cogomology $H^*(X_G)$ of the base imbeds into the cohomology $H^*(X_T)$ of the total space is equivalent to the degeneration of the Serre spectral sequence $$E_2^{pq}=H^p(X_G) \otimes H^q(G/T) \Rightarrow H^{p+q}(X_T)$$ at $E_2$-page. The Euler class is used in one particular case, when the fiber is a sphere, called Gysin long exact sequence, but what should one apply here?
- Why is the cohomology space $H^*(G/T)$ isomorphic to the regular representation $\mathbb Q[W]$? In particular, the invariant subspace is one dimensional.
- The last step is to take $W$-invariants of $$H^*(X_G) \otimes H^*(G/T) \simeq \operatorname{gr_*} H^*(X_T):$$ and use that the Weyl group acts trivially on $H^*(X_G)$ and trivially only on $H^0(G/T) \subset H^*(G/T)$, and also that $H^*(X_T)$ is the smallest grading of $\operatorname{gr_*} H^*(X^T)$.
