NOTES ON HOMOGENEOUS VECTOR BUNDLES OVER COMPLEX FLAG MANIFOLDS

摘要

Let P be a parabolic subgroup of a semisimple complex Lie group G defined by a subset ∑ is contained in ∏ 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 representation Φ of P. Using Bott's theorem, we obtain sufficient conditions on Φ in terms of the combinatorial structure of ∑ is contained in ∏ 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,ε_(ΦΦ*)), we see that the vector bundle E_Φ is rigid.