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?



Borel--Bott--Weil for the Grassmannians
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.


