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?