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 \Sigma of simple roots of G, and let E_\phi be a homogeneous vector bundle over the flag manifold G/P corresponding to a linear representation \phi of P. Using Bott's theorem, we obtain sufficient conditions on \phi in terms of the combinatorial structure of \Sigma for some cohomology groups of the sheaf of holomorphic sections of E_\phi to be zero. In particular, we define two numbers d(P), l(P) such that for any \phi obtained by natural operations from a representation of dimension less than d(P) the q-th cohomology group of E_\phi is zero for 0<q<l(P). We prove also that in this case the vector bundle E_\phi is rigid.