# Cohomology Finite covering and Weyl group

Let $G$ a compact Lie group, $T$ a maximal torus in $G$ and $W=N(T)/T$ its Weyl group. Then we have a finite covering (why is a covering?) $W \rightarrow G/T \rightarrow G/N(T)$ Has $G/N(T)$ a manifold structure? How can I prove (rigorously) that there is an isomorfism $H^{*}(G/T)^{W} \simeq H^{*}(G/N(T))$? where $H^{*}(G/T)^{W}$ denotes the $W$-invariant elements of $H^{*}(G/T)$

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can you just prove everything by writing out the details? it should not that hard. – Bombyx mori Jan 17 '13 at 22:19
I think that concern with invariant forms... – ArthurStuart Jan 17 '13 at 22:48
cohomology is just a disguise; this also holds for representation rings in general. Can you make a functor rhetoric proof? – Bombyx mori Jan 17 '13 at 22:53
No... but I am intrested on it – ArthurStuart Jan 17 '13 at 22:57