The Levi Problem on Strongly Pseudoconvex G-bundles

Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle G → M → X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if G acts by holomorphic transformations in M , then the space of square-integrable holomorphic functions on M is infinite G-dimensional. We also establish the following: Let z be a point of the boundary M . Then there exists a holomorphic function with no smooth extension beyond z.