Subelliptic Boundary Value Problems and the G-fredholm Property Preliminary Version

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 complex manifold satisfying a local subelliptic estimate. In this work, we show that if G acts by holomorphic transformations in M , then the Laplacian = ∂̄∗∂̄ + ∂̄∂̄∗ on M has the following properties: The kernel of restricted to the forms Λ with q > 0 is a closed, G-invariant subspace in L(M,Λ) of finite G-dimension. Secondly, we show that if q > 0, then the image of contains a closed, G-invariant subspace of finite codimension in L(M,Λ). These two properties taken together amount to saying that is a G-Fredholm operator. In similar circumstances, the boundary Laplacian b has similar properties.