The \(G\)-Fredholm property of the \(\bar{\partial}\)-Neumann problem (Q1032605)

Let \(M\) be a complex manifold with boundary which is strongly pseudoconvex. Let \(G\) be a unimodular Lie group acting freely by holomorphic transformations on \(M\) so that \(M/G\) is compact. It is shown that, for \(q>0,\) the Kohn Laplacian \(\square = \overline\partial^* \, \overline \partial + \overline \partial \, \overline \partial^*\) in \(L^2(M, \Lambda^{p,q})\) is \(G\)- Fredholm, which means that \({\text{dim}}_M {\text{ker}}(\square ) < \infty.\) As a corollary one obtains that the reduced \(L^2\)- Dolbeault cohomology spaces \(L^2 \tilde H^{p,q} (M)\) of \(M\) are finite \(G\)- dimensional for \(q>0.\) The boundary Laplacian \(\square_b\) has similar properties.