In this article the images of the Poisson transform on the degenerate series representations attached to the Shilov boundary of a bounded symmetric domain of tube type are considered. We characterize the images by means of second-order differential equations.
1996 Academic Press, Inc.
1. Introduction
Let X=GรK be a Riemannian symmetric space of the noncompact type. Kashiwara et al. [9] proved the so-called Helgason's conjecture; the Poisson transform maps the space B(GรP, L * ) of hyperfunction-valued sections of a spherical principal series representation, which attaches to the Furstenberg boundary GรP of X, bijectively onto an eigenspace A(GรK, M * ) of invariant differential operators on X under certain condition on the parameter * # a c *. Here P is a minimal parabolic subgroup of G.
Hereafter we assume that X is an irreducible bounded symmetric domain of tube type. We consider the problem of characterizing the image of the Poisson transform on the space B(GรP 5 ; s) (s # C) of hyperfunction-valued sections of a degenerate principal series representation attached to the Shilov boundary GรP 5 of X. Here P 5 is a certain maximal parabolic subgroup of G. The degenerate series representation B(GรP 5 ; s) is a G-submodule of a spherical principal series representation B(GรP, L * s ) for some * s # a c * and its image under the Poisson transform is a G-submodule of an eigenspace A(GรK, M * s ) of invariant differential operators.
Let r denote the rank of X and assume that r>1. Let m$ denote the multiplicity of the short restricted roots. Let g and k be the Lie algebras of G and K respectively and g=k+p be a Cartan decomposition. Let z be the center of k. We choose an element Z # -&1 z such that (ad Z) 2 is 1 on p c . article no.