The [Borel--Bott--Weil Theorem](http://en.wikipedia.org/wiki/Borel%E2%80%93Weil%E2%80%93Bott_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?



Analogue of Borel--Bott--Weil for General Equivariant Vector Bundles
The [article](http://www.math.tamu.edu/~jml/kostant61.pdf) 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.


