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The Borel--Bott--Weil Theorem is usually stated for the complete flag manifold of $SU(N)$. Does an analogue hold for the other flags, for example the Grassmannians?

More precisely, suppose $G(\mathbf C)$ is a complex reductive group, and $P(\mathbf C)$ is a parabolic subgroup. Characters $\lambda$ of $P(\mathbf C)$ give rise to line bundles $\mathcal{L}(\lambda)$ on $G(\mathbf C)/P(\mathbf C)$. When is $H^i(G(\mathbf C)/P(\mathbf C),\mathcal{L}(\lambda))$ nonzero, and, in terms of the parabolic $P(\mathbf C)$, which irreducible representations of $G(\mathbf C)$ arise from this construction?

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    $\begingroup$ mathoverflow.net/questions/178783/… $\endgroup$
    – Carlo Beenakker
    Commented Mar 16, 2016 at 20:04
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    $\begingroup$ Yes. The theorem is exactly the same. The highest weights which appear are the ones that extend to characters of $P$. The proof is just pushing forward under the obvious map, and noting that those line bundles are trivial on the fibers. $\endgroup$
    – Ben Webster ♦
    Commented Mar 16, 2016 at 20:55
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    $\begingroup$ Also, there is a useful extension for vector bundles corresponding to the irreducible representations of the parabolic subgroup. $\endgroup$
    – Sasha
    Commented Mar 17, 2016 at 13:55

2 Answers 2

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Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. Math. 74 (1961), 329-387.

W. Schmid, Homogeneous complex manifolds and representations of semisimple Lie groups, Proceedings of the International Congress of Mtahematicians: Helsinki 1978 (ed. O. Lehto) 195-208.

You could also look at R. Baston, M. Eastwood, The Penrose Transform: Its Interaction with Representation Theory, Oxford, 1989.

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Lars, search for a paper of Kostant ora paper of Griffiths-Schmid, you will find a complete answer to your question, even when \lambda is just an irreducible representation. best regards

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