Composition series and intertwining operators for the spherical principal series. II

Johnson, Kenneth D.

doi:10.1090/S0002-9947-1976-0385012-X

Composition series and intertwining operators for the spherical principal series. II
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Trans. Amer. Math. Soc. 215 (1976), 269-283 Request permission

Abstract:

In this paper, we consider the connected split rank one Lie group of real type ${F_4}$ which we denote by $F_4^1$. We first exhibit $F_4^1$ 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 $F_4^1/{\text {Spin}}(9)$ as the unit ball in ${{\mathbf {R}}^{16}}$ with boundary ${S^{15}}$. After decomposing the space of spherical harmonics under the action of ${\text {Spin}}(9)$, we obtain the matrix of a transvection operator of $F_4^1{\text {/Spin}}(9)$ 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 $F_4^1$.

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