For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than the flag manifolds: Take a homogeneous space $M := G/K$, with $G$ compact, such that $M$ has been given a complex structure. In this setting, does $G$ still have a representation on the holomorphic sections of the holomorphic vector bundles over $M$? It seems to me that this will only happen if the action of $G$ on $M$ is holomorphic. However, I can't see for sure that nothing goes wrong, and I don't see that this condition is obviously true in the special case of the flag manifolds. Can anyone help?