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?
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asked Oct 13, 2019 at 14:18
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2$\begingroup$ (I'm not an expert on representation theory, so you can take this with a grain of salt.) My understanding is that line bundles correspond to vectors in the weight lattice, and the dominant weights correspond to the ample ones. I suppose that you meant for your $G$ to be semisimple. $\endgroup$– Donu ArapuraCommented Oct 13, 2019 at 21:03
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$\begingroup$ Yes, the assumption is that $G$ is semisimple. $\endgroup$– Fofi KonstantopoulouCommented Oct 13, 2019 at 21:52
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1$\begingroup$ @Donu: Of course, it is not just dominant weights, it has been edited accordingly. $\endgroup$– Fofi KonstantopoulouCommented Oct 13, 2019 at 21:56
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1$\begingroup$ Two comments: 1) A search of this site for something like "ample line bundle" reveals many overlapping questions here, such as 99361. 2) It's probably better to use Chevalley's classification to pass to the algebraic setting over any algebraically closed field, since the answer to the question has little to do with the complex setting (or with combinatorics). $\endgroup$– Jim HumphreysCommented Oct 16, 2019 at 16:16
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1$\begingroup$ Also, no need for "semisimple" here, since the radical is contained in every Borel subgroup. $\endgroup$– Jim HumphreysCommented Oct 25, 2019 at 15:36
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1 Answer
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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 .
answered Oct 14, 2019 at 13:09
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$\begingroup$ As you say, "This result is pretty standard and can be found in many places." No need for Google here, since this site already has answers to the .question. $\endgroup$ Commented Oct 25, 2019 at 15:31