Abstract
In this paper, we consider the connected split rank one Lie group of real type F 4 {F_4} which we denote by F 4 1 F_4^1 . We first exhibit F 4 1 F_4^1 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 F 4 1 / Spin ( 9 ) F_4^1/{\text {Spin}}(9) as the unit ball in R 16 {{\mathbf {R}}^{16}} with boundary S 15 {S^{15}} . After decomposing the space of spherical harmonics under the action of Spin ( 9 ) {\text {Spin}}(9) , we obtain the matrix of a transvection operator of F 4 1 /Spin ( 9 ) 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 F 4 1 F_4^1 .
Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
