Notes on homogeneous vector bundles over complex flag manifolds

Abstract

Let P be a parabolic subgroup of a semisimple complex Lie group G defined by a subset Σof simple roots of G, and let E_ϕbe a homogeneous vector bundle over the flag manifold G/P corresponding to a linear representation ϕof P. Using Bott's theorem, we obtain sufficient conditions on ϕin terms of the combinatorial structure of Σfor some cohomology groups of the sheaf of holomorphic sections of E_ϕto be zero. In particular, we define two numbers d(P), l(P) such that for any ϕobtained by natural operations from a representation of dimension less than d(P) the q-th cohomology group of E_ϕis zero for 0<q<l(P). We prove also that in this case the vector bundle E_ϕis rigid.