Is there a generalization of Borel-Weil-Bott for partial flag varieties, i.e. homogeneous spaces of the form $G/P$ with $P$ parabolic and $G$ semisimple? If so, I would like a reference.



Borel–Weil–Bott for partial flag varieties
There are many places that give a complete answer to your question: One is a paper in the Annals of Math written by Kostant around the middle fifties, other more geometrical is due to Griffits-Schmid published in Acta Mathematica in the late sixties.
best regards


