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

Abstract

The basic setup consists of a complex flag manifold $Z=G/Q$ where $G$ is a complex semisimple Lie group and $Q$ is a parabolic subgroup, an open orbit $D=G_0(z)\subset Z$ where $G_0$ is a real form of $G$, and a $G_0$--homogeneous holomorphic vector bundle $\mathbb{E}\to D$. The topic here is the double fibration transform $\mathcal{P}:H^q(D;\mathcal{O}(\mathbb{E}))\to H^0({\mathcal{M}}_D;\mathcal{O}({\mathbb{E}}^{\prime }))$ where $q$ is given by the geometry of $D$, ${\mathcal{M}}_D$ is the cycle space of $D$, and ${\mathbb{E}}^{\prime }\to {\mathcal{M}}_D$ 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.