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 $t \rightarrow \infty$?



Closure relations between Bruhat cells on the flag variety
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 $v\_{s\_{\alpha}w\lambda}$. 


