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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.

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  • $\begingroup$ You should follow up the tag 'flag-verieties' here, for instance the related earlier question: mathoverflow.net/questions/120984 $\endgroup$ – Jim Humphreys Dec 2 '13 at 19:52
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    $\begingroup$ 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. $\endgroup$ – Jim Humphreys Dec 23 '13 at 0:48
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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.

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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).

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