The BorelBottWeil 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?

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. $\endgroup$– SashaNov 10, 2014 at 18:38

$\begingroup$ Could you give a reference for this? $\endgroup$– Tomasz KönerNov 10, 2014 at 18:40

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. $\endgroup$– SashaNov 10, 2014 at 19:32

1$\begingroup$ Bott's 1957 Annals article, well worth reading, analyzes the general $G/P$ situation, too. $\endgroup$– Dave AndersonNov 11, 2014 at 3:23

1$\begingroup$ @user36087: Since a Borel subgroup is solvable, its module category is far from being semisimple. $\endgroup$– Jim HumphreysNov 11, 2014 at 14:13
2 Answers
The article Lie Algebra Cohomology and the Generalized BorelWeil 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.
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 AlFaisal

2$\begingroup$ 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. $\endgroup$ Nov 11, 2014 at 11:31