# Invariant differential operators and eigenspace representations on an affine symmetric space

### Abstract

Let $G/H$ be an affine symmetric space of split rank $r$. Let $\mathbf{D}$ 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 $\mathbf{D}$ on $G/H$ form an admissible $(\mathfrak{g},K)$-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 $\tau$-spherical eigenfunctions on $G/H$ by Eisenstein integrals.

Editor’s Note July 2017: DOI https://doi.org/10.4007/annals.2017.186.1.9

Jing-Song Huang