Restrictions of Quaternionic Representations

In a paper by Gross and Wallach [1996, J. Reine Angew. Math.481, 73–123] the K-types of the continuations of the quaternionic discrete series of a quaternionic Lie group G are associated with projective orbits O of certain subgroups in G(C). In this paper, we will show that the restrictions of the representations to quaternionic subgroups are closely related with the intersection of the Zariski closure of O with hyperplanes. We apply this to the minimal representations of the exceptional groups of real rank 4 and investigate the correspondences of certain compact dual pairs.