Hecke algebra and $H^*(G/B)$

Question

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?

Answer 1

The early work of Borel showed in effect how to interpret the cohomology algebra of a flag variety as the coinvariant algebra associated to the Weyl group, which affords the regular representation of $W$.(A useful exposition, if you can locate it, is the 1982 Pitman Research Notes Geometry of Coxeter Groups by Howard Hiller, though the account of Coxeter groups in general is too sketchy.) This has been understood from various viewpoints such as compact groups and complex semisimple Lie groups, with further combinatorial development by Bernstein-Gelfand-Gelfand and Demazure. By now there is a lot of related literature, including connections with the Springer Correspondence.

On the other hand, the Iwahori-Hecke algebra associated to the Coxeter group $W$ is a deformation of the integral group ring, therefore also close to the group algebra and representation theory of $W$. A fundamental source is the 1979 Invent. Math. paper by Kazhdan and Lusztig here. But as far as I know the answer to the original question (how is the cohomology ring related to the Hecke algebra) is that the two are only indirectly related.

P.S. Bischof has pointed to some of the rich developments following the papers by BGG and Demazure, but I'm not enough of a specialist to know which references are most relevant here. One important direction I should mention involves work by Kostant and Kumar on what they call the "nil Hecke ring", motivated partly by wanting to work with analogues of Schubert varieties in the infinite dimensional Kac-Moody framework. See in particular: Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Adv. in Math. 62 (1986), no. 3, 187–237.

Answer 2

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.