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?
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.
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