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 our setting that the line bundle parameter is outside the good range, we use some techniques from algebraic representation theory.