The G-Fredholm Property of the partial derivative-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 -> 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 Laplacian square = partial derivative*partial derivative + partial derivative partial derivative* on M has the following properties: The kernel of square restricted to the forms Lambda(p,) (q) with q > 0 is a closed, G-invariant subspace in L(2)(M, Lambda(p,) (q)) of finite G-dimension. Secondly, we show that if q > 0, then the image of square contains a closed, G-invariant subspace of finite G-codimension in L(2)(M, Lambda(p,) (q)). These two properties taken together amount to saying that square is a G-Fredholm operator. It is a corollary of the first property mentioned that the reduced L(2)-Dolbeault cohomology spaces L(2)(H) over bar (p, q)(M) of M are finite G-dimensional for q > 0. The boundary Laplacian square(b) has similar properties.