We give a general method for constructing Poincaré series on higher rank groups satisfying automorphic differential equations, by winding up solutions to differential equations of the form (Δ - λ)^ν u = θ on the underlying Riemannian symmetric space G/K, where Δ is the Laplacian, λ is a complex parameter, ν is an integral power, and θ a compactly supported distribution. To obtain formulas that are as explicit as possible we restrict ourselves to the case in which G is a complex semi-simple Lie group, and we consider two simple choices for θ, namely θ = δ, the Dirac delta distribution at the basepoint, and θ = S_b, the distribution that integrates along a shell of radius b around the basepoint. We develop a global zonal spherical Sobolev theory, which enables us to use the harmonic analysis of spherical functions to obtain integral representations for the solutions. In the case θ = δ, we obtain an explicit expression for the solution, allowing relatively easy estimation of its behavior in the eigenvalue parameter λ, necessary for applications involving the associated Poincaré series. The behavior of the solution corresponding to θ = S_b is considerably subtler, even in the simplest possible higher rank cases; nevertheless, global automorphic Sobolev theory ensures the existence and uniqueness of an automorphic spectral expansion for the associated Poincaré series in a global automorphic Sobolev space, which is sufficient for the applications we have in mind.

The author was partially supported by the Doctoral Dissertation Fellowship from the Graduate School of the University of Minnesota, by NSF grant DMS-0652488, and by a research grant from the University of St. Thomas.