# Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups

I've recently read the following line in an interesting paper:

It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial functions on the Cartan sub algebra $\frak{h}$ by the ideal generated by the fundamental invariants $f_1 , . . . , f_r, r =$rank$({\frak h})$, of the Weyl group W, i.e. $$H^∗(G/B,{Q}) \simeq {\text Sym}_Q{\frak h}^∗/(f_1, . . . , f_r).$$

(1) Does this result extend to other fields, i.e. the complex and real case?

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There is a lot of related discussion along with references in a previous question 21651. In your question (2), the word "understandable" of course means different things to different people, so it's good to look at various treatments. – Jim Humphreys Nov 7 '12 at 14:46
P.S. Keep in mind that the concrete treatments by Fulton and Manivel focus on the most classical framework of Lie type $A$, where the Weyl group is a symmetric group, while Borel (and Hiller) give a more comprehensive treatment in the language of Lie groups or algebraic groups and Weyl groups. – Jim Humphreys Nov 7 '12 at 18:11
Jim's link: mathoverflow.net/questions/21651 – Allen Knutson Nov 8 '12 at 3:09

(1) Assuming you are referring to the coefficient field for your cohomology theory, then yes, the result immediately extends to $\mathbb{R}$ and $\mathbb{C}$ coefficients.