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?

`$G$`

to be a compact semisimple Lie group, though the Borel-Weil (and Bott) theorems also have complex group formulations, translated nicely by Demazure into algebraic language. In your situation, some kind of holomorphic assumption does seem necessary (and sufficient) to get a Lie group representation of G as in the flag manifold case where this structure arises for line bundles from characters of a maximal torus There are standard books indluding a graduate text by Joseph Taylor which should provide the needed foundations, at least in the complexified case. – Jim Humphreys Feb 5 '13 at 22:33