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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
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