The Penrose transform for Sp(n,R) and singular unitary representations

摘要

We give a general definition of the Radon-Penrose transform for a Zuckerman-Vogan derived functor module of a reductive Lie group G, which maps from the Dolbeault cohomology group over a pseudo-Kahler homogeneous manifold into the space of smooth sections of a vector bundle over a Riemannian symmetric space. Furthermore, we formulate a functorial property between two Penrose trans- forms in the context of the Kobayashi theory of discretely decomposable restrictions of unitary representations. Based on this general theory, we study the Penrose transform for a family of sin- Gular unitary representations of Sp(n,R) in details.