# Cohomology ring of a flag variety and representation theory

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.

• You should follow up the tag 'flag-verieties' here, for instance the related earlier question: mathoverflow.net/questions/120984 – Jim Humphreys Dec 2 '13 at 19:52
• P.S. One clarification: the coinvariant algebra belongs to the Weyl group and not the root system (an important distinction in types B, C). In any case, the answer to your question is yes. It's useful to search MO for terms like "flag variety", "Springer correspondence", "Soergel bimodule" to get into the extensive related literature. – Jim Humphreys Dec 23 '13 at 0:48

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 $H^*(G/B)$.