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 $G\to M\to X$ so that $M$ is also a strongly pseudoconvex complex manifold. In this work, we show that if $G$ acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on $M$ is infinite $G$-dimensional.