Let $G$ be a simple simply connected algebraic group and $T$ be a maximal torus in $G$. Let $B$ be a Borel containing $T$ and $N(T)/T$ be the Weyl group. We have nice actions of $T$ and $W$ on the Flag variety $G/B$. For the torus action the $T$-equivariant cohomology and the structure of the cohomology modules have been studied explicitly. Has any one studied the same for the Weyl group action on $G/B$ ? Any reference or results in this direction is highly appreciated.
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$\begingroup$ What's your action of W on the flag variety? I don't know of any natural ones (because I don't know of any natural embeddings of W into G.) $\endgroup$– dhyCommented Dec 29, 2015 at 13:40
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3$\begingroup$ The natural action is of $W$ on the right of $G/T$, not $G/B$, but they're homotopic and so have the same cohomology. If that's what you want, then as a $W$-module it's the regular representation. This is proved in e.g. Humphreys' gray book on Coxeter groups, by a Galois theory argument. $\endgroup$– Allen KnutsonCommented Dec 29, 2015 at 18:43
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