The Borel--Bott--Weil Theorem gives the dimensions of the cohomology groups of the equivariant line bundles over flag manifolds. Does there exist an analogous result for general equivariant vector bundles over the flag varieties, or even just for complex projective $n$-space?
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2$\begingroup$ On flag variety any equivariant vector bundle is an iterated extension of line bundles. So, its Euler characteristic can be computed by applying BBW to the factors and summing up. $\endgroup$– SashaCommented Nov 10, 2014 at 18:38
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$\begingroup$ Could you give a reference for this? $\endgroup$– Tomasz KönerCommented Nov 10, 2014 at 18:40
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5$\begingroup$ It follows easily from the equivalence of the category of equivariant bundles and the category of representations of Borel subgroup, since Borel is solvable. $\endgroup$– SashaCommented Nov 10, 2014 at 19:32
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1$\begingroup$ Bott's 1957 Annals article, well worth reading, analyzes the general $G/P$ situation, too. $\endgroup$– Dave AndersonCommented Nov 11, 2014 at 3:23
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1$\begingroup$ @user36087: Since a Borel subgroup is solvable, its module category is far from being semi-simple. $\endgroup$– Jim HumphreysCommented Nov 11, 2014 at 14:13
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2 Answers
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The article Lie Algebra Cohomology and the Generalized Borel-Weil theorem by Kostant contains generalization of the BBW theorem to equivariant vector bundles over $G/P$ associated to a $G$-representation, where $P$ is a parabolic subgroup. Actually, it contains a bit more, since Kostant goes a long way with just a "Lie summand" that reappeared in kind of a different generalization of BBW via the Kostant cubic Dirac operator.
answered Nov 10, 2014 at 18:47
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In this dissertation you can find the complete proof of generalized Borel Weil theorem over $G/P$.
On the Representation Theory of Semisimple Lie Groups by Faisal Al-Faisal
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3$\begingroup$ Thanks. One can find a complete proof in many places, even the original Kostant's article is quite complete and readable. But it is always nice to read a good distillation in a master thesis. $\endgroup$ Commented Nov 11, 2014 at 11:31