  Let G --> G' be an embedding of semisimple complex Lie groups, let B and B'
be a pair of nested Borel subgroups, and let f:G/B --> G'/B' be the associated
equivariant embedding of flag manifolds. We study the pullbacks of cohomologies
of invertible sheaves on G'/B' along the embedding f. Let O' be a
G'-equivariant invertible sheaf on G'/B', and let O be its restriction to G/B.
Consider the G-equivariant pullback on cohomology p : H(G'/B',O') --> H(G/B,O).
The Borel-Weil-Bott theorem implies that the two cohomology spaces above are
irreducible modules of G' and G respectively. By Schur's lemma, p is either
surjective or zero. In this paper we establish a necessary and sufficient
condition for nonvanishing of p, and apply it to the study of regular and
diagonal embeddings. We also prove a structure theorem about the set of
cohomological pairs of highest weights. We also study in detail two cases of
embeddings which are neither regular nor diagonal. The first is the case of
homogeneous rational curves in complete flag manifolds, and the second is the
embedding of the complete flag manifold of G into the complete flag manifold of
G'=SL(Lie(G)), via the adjoint representation of G. We show that the generators
of the algebra of invariants in the polynomial algebra on Lie(G) can be
realized as cohomological components. Our methods rely on Kostant's theory of
Lie algebra cohomology.
