# Borel--Bott--Weil for the Grassmannians

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?

• mathoverflow.net/questions/178783/… – Carlo Beenakker Mar 16 '16 at 20:04
• 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. – Ben Webster Mar 16 '16 at 20:55
• Also, there is a useful extension for vector bundles corresponding to the irreducible representations of the parabolic subgroup. – Sasha Mar 17 '16 at 13:55