Injectivity of the Double Fibration Transform for Cycle Spaces of Flag Domains

The basic setup consists of a complex flag manifold $Z=G\mathrm{/}Q$ where $G$ is a complex semisimple Lie group and $Q$ is a parabolic subgroup, an open orbit $D = G_0(z) \subset Z$where$G_0$isarealformof$G$,anda$G_0$–homogeneous holomorphic vector bundle $\mathbb{E}\to D$. The topic here is the double fibration transform ${\cal P}: H^q(D;{\cal O}(\mathbb E)) \to H^0({\cal M}_D;{\cal O}(\mathbb E’))$where$q$isgivenbythegeometryof$D$,${\cal M}_D$isthecyclespaceof$D$,and$\mathbb E’ \to {\cal M}_D$ is a certain naturally derived holomorphic vector bundle. Schubert intersection theory is used to show that $\mathcal{P}$ is injective whenever $\mathbb{E}$ is sufficiently negative.