Penrose transformation on flag domains

Building on our recent work, we construct the Penrose transformations of the cohomology groups of homogeneous line bundles on flag domains $D = G_\R / T$, where $G_\R$ is of Hermitian type. We provide sufficient conditions for the injectivity of the Penrose transformation and identify conditions under which the Penrose transformation of the automorphic cohomology groups on compact quotients of flag domains is an isomorphism. Finally, we prove that the higher automorphic cohomology groups of certain homogeneous line bundles are isomorphic to the groups of automorphic forms on the Hermitian symmetric domain, and we apply this result to the cup products of the automorphic cohomology groups.