Composition Series and Intertwining Operators for the Spherical Principal Series. II

In this paper, we consider the connected split rank one Lie group of real type which we denote by . We first exhibit as a group of operators on the complexification of A. A. Albert's exceptional simple Jordan algebra. This enables us to explicitly realize the symmetric space <!-- MATH $F_4^1/{\text{Spin}}(9)$ --> as the unit ball in <!-- MATH ${{\mathbf{R}}^{16}}$ --> with boundary . After decomposing the space of spherical harmonics under the action of <!-- MATH ${\text{Spin}}(9)$ --> , we obtain the matrix of a transvection operator of <!-- MATH $F_4^1{\text{/Spin}}(9)$ --> acting on a spherical principal series representation. We are then able to completely determine the Jordan Holder series of any spherical principal series representation of .