TITLE: Cohomology ring of a flag variety and representation theory
QUESTION [12 upvotes]: I'm interested in the cohomology ring $H^*(G/B)$ of a flag variety $G/B$, where $G$ is a complex semi-simple Lie group and $B$ the Borel subgroup. Borel (1953) showed that this ring is isomorphic to the coinvariants algebra of the associated root system. $H^*(G/B)$ also has a distinguished basis given by Schubert cells. Demazure (1974) and Bernstein, Gelfand, Gelfand (1973) identified elements of the coinvariants algebra corresponding to the Schubert cells. 
I know that flag varieties play an important role in representation theory (Borel-Weil-Bott theorem, for instance). I'd like to know whether the cohomology ring $H^*(G/B)$ carries any useful representation-theoretic information. References to literature would also be very appreciated.

REPLY [6 votes]: Both the study of ordinary Schubert calculus (the ordinary, e.g. Borel-Moore, cohomology of Schubert varieties of the flag variety) and the study of Kazhdan-Lusztig theory (the intersection cohomology of Schubert varieties of the flag variety) are directly related to the representation theory of the Hecke algebra of the associated Weyl group and its specializations.
The literature is vast, but a couple of places to start would be Stephen Griffeth and Arun Ram's article Affine Hecke Algebras and the Schubert Calculus and the article by David Kazhdan and George Lusztig Schubert Varieties and Poincare Duality (the second link is the AMS review on mathscinet, available by subscription only).