Poisson Integrals for Homogeneous, Rank 1 Koszul Manifolds

Let X be a complex manifold which is homogeneous under the real analytic Lie group G where G is solvable with codimension one nilradical. Suppose that X has a G-invariant measure m = K dx where dx is Lebesgue measure. The Koszul form H is del-delbar of K. (If X is bounded, then H becomes the Bergmann metric.) We say that X is a Koszul domain if H is non-degenerate. This makes X into a pseudo-Riemannian manifold. In this work we study the eigenvalue problem for the corresponding Laplace-Beltrami operator. Despite the highly non-elliptic nature of the problem, we succeed in constructing boundary values and Poisson transformations. With suitable restrictions, the boundary map is one to one. Our techniques involve a fascinating relationship between the Laplace-Beltrami operator of X and the Casimir operator for Sl(2, R). Our techniques are motivated by those of Kashiwara et al.