Abstract
Let G/H be an affine symmetric space of split rank r. Let D be a preferred polynomial algebra of G-invariant differential operators on G/H generated by r elements. We show that the space of K-finite joint eigenfunctions of D on G/H form an admissible (g, K)-module which is called an eigenspace representation. The main content of this paper is description of the algebras of invariant differential operators and determination of the eigenspace representations on G/H. We also obtain a Poisson transform for T-spherical eigenfunctions on G/H by Eisenstein integrals.
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