Given a complex reductive group, with Weyl group $W$, one can associate to it lots of "algebras of size $|W|$". For example $B$ equivariant functions on $G/B$ with convolution, grothendieck groups of categories associated with $G$, the group ring $\mathbb C[W]$...

So far they are all more or less specializations of the Iwahori-Hecke algebra! But there is another natural algebra of size $W$, namely the cohomology ring of the flag variety $H^*(G/B)$. So my question is, how is this ring related to the Hecke algebra?