  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.
