The G-Fredholm Property of the barpartial-Neumann Problem

Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ be 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 in $M$, then the complex Laplacian $\square$ on $M$ has the following properties: The kernel of $\square$ restricted to the forms $\Lambda^{p,q}$ with $q$ positive is a closed, $G$-invariant subspace in $L^{2}(M,\Lambda^{p,q})$ of finite $G$-dimension. Secondly, we show that if $q$ is positive, then the image of $\square$ contains a closed, $G$-invariant subspace of finite codimension in $L^{2}(M,\Lambda^{p,q})$. These two properties taken together amount to saying that $\square$ is a $G$-Fredholm operator. The boundary Laplacian has similar properties.