Van den Ban–Schlichtkrull–Wallach asymptotic expansions on nonsymmetric domains

Abstract

if it is annihilated by every element of DG(X), the algebra of all G-invariant differential operators without constant term. One of the most beautiful results in the harmonic analysis of symmetric spaces is the Helgason conjecture, which states that on a Riemannian symmetric space of noncompact type, a function is harmonic if and only if it is the Poisson integral of a hyperfunction over the Furstenberg boundary G/Po where Po is a minimal parabolic subgroup. (See [14], [17].) One of the more remarkable aspects of this theorem is its generality; one obtains a complete description of all solutions to the system of invariant differential operators on X without imposing any boundary or growth conditions. If X is a Hermitian symmetric space, then one is typically interested in complex function theory, in which case one is interested in functions whose boundary values are supported on the Shilov boundary rather than the Furstenberg boundary. (The Shilov boundary is G/P where P is a certain maximal parabolic containing Po.) In this case, it turns out that the algebra of G invariant differential operators is not necessarily the most appropriate one for