# Analogue of Borel--Bott--Weil for General Equivariant Vector Bundles

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?

• 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. – Sasha Nov 10 '14 at 18:38
• Could you give a reference for this? – Tomasz Köner Nov 10 '14 at 18:40
• It follows easily from the equivalence of the category of equivariant bundles and the category of representations of Borel subgroup, since Borel is solvable. – Sasha Nov 10 '14 at 19:32
• Bott's 1957 Annals article, well worth reading, analyzes the general $G/P$ situation, too. – Dave Anderson Nov 11 '14 at 3:23
• @user36087: Since a Borel subgroup is solvable, its module category is far from being semi-simple. – Jim Humphreys Nov 11 '14 at 14:13

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 $G/P$.