Holomorphic Laplacian on the Lie ball and the Penrose transform

Abstract

We prove that any holomorphic function $f$ on the Lie ball of even dimension satisfying $\Delta f=0$ is obtained uniquely by the higher-dimensional Penrose transform of a Dolbeault cohomology for a twisted line bundle of a certain domain of the Grassmannian of isotropic subspaces. To overcome the difficulties arising from that the line bundle parameter is outside the {\it{good range}}, we use some techniques from algebraic representation theory.