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I've recently been reading about flag varieties and their cohomology. I'm mainly interested in representation theory, and I've heard that flag varieties are important objects, especially in Lie representation theory, but the connection between the two is still somewhat esoteric to me.

Can anyone briefly explain what the connections are? References to books/papers would be welcome.

I'd also like to know whether cohomology rings of flag varieties carry any representation theoretic information.

Thanks for help!

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  • $\begingroup$ How much geometry do you know? Do you know Chern classes of vector bundles? Other cohomology theories besides singular (Borel-Moore, Chow)? $\endgroup$ Commented Dec 2, 2013 at 17:02
  • $\begingroup$ @MichaelJoyce I'm in the process of learning basics of Chern classes and Borel-Moore homology. $\endgroup$ Commented Dec 2, 2013 at 17:48
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    $\begingroup$ Also, a minor point of decorum -- it's generally frowned upon to post the same question to mathoverflow and mathstackexchange. I'd have answered your question below differently if I had seen your mathoverflow post. As it is, I suspect the answer below is mostly telling you things you already know, but I'll leave it up for any other people who are interested in your question but have less background. $\endgroup$ Commented Dec 2, 2013 at 20:30

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Briefly, the answer is yes, a lot of representation-theoretic information is available from the geometry of flag varieties. A (complete) flag variety is a variety of the form $G/B$ where $G$ is a (complex, say) reductive algebraic group and $B$ is a Borel subgroup of $G$. The classical flag variety corresponds to $G = GL_n$ and $B$ the subgroup of upper triangular matrices in $GL_n$. Fix a maximal torus $T$ of $G$ contained in $B$. In the case of the classical flag variety, you can take $T$ to be the subgroup of diagonal matrices.

Let $\mathcal{X}(T) = \text{Hom}_{\text{alg grps}}(T, \mathbb{C}^*)$ and $\mathcal{X}(B) = \text{Hom}_{\text{alg grps}}(B, \mathbb{C}^*)$ be the character groups of $T$ and $B$ respectively. It is not hard to show that the restriction map $\mathcal{X}(B) \rightarrow \mathcal{X}(T)$ is an isomorphism, so people often abuse notation and identify $\mathcal{X}(T) = \mathcal{X}(B)$. Associated to any $\lambda \in \mathcal{X}(T)$, one constructs a line bundle $L_{\lambda}$ on the flag variety $G/B$ that is $G$-equivariant. It follows that the space of global sections of $L_{\lambda}$, $\Gamma(G/B, L_{\lambda})$, becomes a $G$-representation.

The Borel-Weil-Bott Theorem says that every irreducible, finite-dimensional representation of $G$ can be obtained by this construction. More precisely,

  • $\Gamma(G/B, L_{\lambda}) \neq \{ 0 \}$ if and only if $\lambda$ is a dominant weight.
  • If $\lambda$ is dominant, then as a $G$-representation, $\Gamma(G/B, L_{\lambda})$ is the irreducible representation of $G$ with highest weight $\lambda$.

This is a first illustration of how the flag variety encodes representation-theoretic information. There is also additional, very deep, representation theoretic information that is partly understood. For example, Schubert calculus and Kazhdan-Lusztig theory both obtain information about the representation theory of Hecke algebras and their specializations by studying the geometry of the flag variety. Basically, Schubert calculus is the study of the ordinary cohomology of the Schubert varieties on a flag variety, while Kazhdan-Lusztig theory is the study of the intersection cohomology of Schubert varieties on a flag variety. However, to a large extent, the combinatorics and representation theory is better understood than the geometry, so many of the open problems are to find geometric explanations / interpretations of the representation theoretic and combinatorial theorems in the area.

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