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Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.

How do we prove the closure relations between the cells, which are indexed by the Weyl group elements?

I was thinking about constructing a curve that connects two $T-$fixed points, using the exponential map. Let $e_{\alpha}$ be a root in the Lie algebra. How do we think about the action of $\exp(t \cdot e_{\alpha})$ on $G/B$, as $t \rightarrow \infty$?

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The place I've read about this is in the paper "Schubert cells, and the cohomology of the spaces $G/P$" by Berstein, Gelfand, and Gelfand. The flag variety can be identified with the projectivization of a representation of $G$, $V_{\lambda}$. Then you compute the limit of the 1-psg $exp(t\cdot e_{\alpha})$ as it acts on weight vector $v_{w\lambda}$. As $t\to \infty$, you get $v_{s_{\alpha}w\lambda}$.

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  • $\begingroup$ The three authors (then in Moscow) also known as BGG were Joseph Bernstein together with his teacher Israel Gelfand and Gelfand's son Sergei. The paper is available online in Russian: mathnet.ru/php/… $\endgroup$ Dec 17, 2014 at 17:33
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For a first introduction you can read Michel Brion's "http://arxiv.org/pdf/math/0410240v1.pdf". He gives a nice introduction (for G=GL(n)) in Section 1.

I'm not sure whether your curve method works but your strategy is similar to that in this question: Is there a Morse theory proof of the Bruhat decomposition?.

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