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.



Cohomology ring of a flag variety and representation theory
Let $\mathfrak{g}$ be the Lie algebra of the group $G$. You might consider reading about the Springer resolution $$\mu:T^\*(G/B)\rightarrow\mathcal{N},$$ where $\mathcal{N}$ is the nilpotent cone of $\mathfrak{g}$. The fibres of this map are isomorphic over individual adjoint orbits of $G$. These are called Springer fibres. For each such fibre, there is a representation of the Weyl group on its Borel-Moore homology (or dually, its cohomology). The fibre above $0$ is the zero-section of $T^\*(G/B)$, giving us a representation of $W$ on $H^\*(G/B)$.


A reference would be Representation Theory and Complex Geometry, by Chriss and Ginzburg.


