For a complex Lie group $G$, with $B$ a choice of Borel subgroup. The line bundles over the flag manifold $G/B$ are indexed by elements of the weight lattice of $\frak{g}$. Which of these line bundles is ample?



Classifying ample line bundles over the flag manifold $G/B$
As Donu said, line bundles on $G/B$ correspond to elements of the weight lattice. The ample bundles correspond to the dominant regular weights. This means that the pairing of the weight $\lambda$ with any positive root is strictly positive (or strictly negative depending on your conventions for indexing the line bundles).


This result is pretty standard and can be found in many places... E.g. I found it here after a quick google search <https://pdfs.semanticscholar.org/af4c/14cebb25f9517eaa9a1349f00d760466fc90.pdf> , see Proposition 2.2.10 .


