# Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)

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?

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P.S. Maybe add a tag lie-groups? – Jim Humphreys Feb 5 '13 at 20:04
Keywords "cohomological induction" may be useful: the Borel-Wallach book of that title, and also the Knapp-Vogan book, for example. – paul garrett Feb 5 '13 at 21:37
I guess you mean $G$ 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
As soon as the Dolbeault operator is $G$-invariant (i.e. it commutes with the $G$-action on spaces of sections of a given vector bundle), you will have a representation of $G$ on its kernel, i.e. the space of holomorphic sections. Now Dolbeault is invariant as soon as you have a $G$-invariant complex structure on $G/K$, so the question is really about the existence of such a structure. – Alain Valette Feb 5 '13 at 23:02
That seems to answer my question. Please enter it as an answer and so I can accept it. One more thing thing though: Is this an iff situation, ie if the complex structure is not $G$-invariant, then will it happen that the holomorphic sections no longer carry a representation of $G$? – Jean Delinez Feb 6 '13 at 15:44