A partial generalization of the Helgason Conjecture to general bounded homogeneous domains

Abstract

Let X = G/K be a Riemannian symmetric space and ⊂ DG(X) a co-finite ideal. A function F on X is -harmonic if F = 0. A result of Oshima and Sekiguchi [Advanced Studies in Pure Math. 4: 391–432, 1984] says such a function is the Poisson integral of a distribution over the Furstenberg boundary G/P if and only if it has moderate growth. We prove a partial generalization of this result to general, non-symmetric, bounded homogeneousdomains in ℂn. Instead of DG(X), we use an algebra of geometrically defined differential operators, Dgeo(X) which, in the symmetric case, is a subalgebra of DG(X). We prove the existence of an asymptotic expansion for -harmonic functions that reduces in the symmetric space case to the expansions due to Wallach [Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals, Academic Press, 1988] and van den Ban and Schlichtkrull [J. Reine Angew. Math. 380: 108–165, 1987]. We prove a convergence theorem for these expansions that seems to be new even in the symmetric space case. These expansions are used to define boundary values for F which uniquely determine F. An algorithm constructing F from its boundary values is given.