TITLE: Hecke algebra and $H^*(G/B)$
QUESTION [7 upvotes]: 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?

REPLY [6 votes]: The group algebra $\mathbb{C}[W]$ is the Borel-Moore homology of the union of conormals to the Schubert cells (the preimage of $\mathfrak{n}$ under the Springer resolution) endowed with a convolution structure; the coinvariant ring is its cohomology.  These have the same dimension, since this union of conormals is a union of complex affine spaces, but there's no canonical isomorphism or duality between them, since this variety is horribly non-singular and non-compact.