Notes on homogeneous vector bundles over complex flag manifolds

Let P be a parabolic subgroup of a semisimple complex Lie group G defined by a subset ∑ ⊂ ∏ of simple roots of G, and let E ϕ be a homogeneous vector bundle over the flag manifold M = G/P corresponding to a linear representationϕ of P. Using Bott’s theorem, we obtain sufficient conditions on ϕ in terms of the combinatorial structure of ∑ ⊂ ∏ for cohomology groups H q(M,ε ϕ ) to be zero, where ε ϕ is the sheaf of holomorphic sections of E ϕ . In particular, we define two numbers d(P), ℓ(P) ∈ ℕ such that for any ϕ obtained by natural operations from a representation φ~ of dimension less than d(P) one has H q(M,ε ϕ ) = 0 for 0 < q < ℓ(P). Applying this result to H 1(M,εϕϕ), we see that the vector bundle E ϕ, is rigid.