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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Kenneth D. Johnson
Journal: Trans. Amer. Math. Soc. 215 (1976), 269-283
MSC: Primary 22E45; Secondary 43A80
MathSciNet review: 0385012
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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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Keywords: $ F_4^1$, exceptional simple Jordan algebra, spherical harmonics, Poisson kernel, transvection operator, composition series, intertwining operators
Article copyright: © Copyright 1976 American Mathematical Society