Fourier-Poisson Transforms Associated with the Principal Series Representations of Sp(1, n)

Let X = S p ( 1 , n ) / S p ( n ) be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where n is greater than or equal to 1. The Fürstenberg boundary of X is denoted as Σ . In this paper, we focus on the Plancherel formula on X associated with the Poisson tra...

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Bibliographic Details
Published in:Advances in applied Clifford algebras 2024-07, Vol.34 (3), Article 25
Main Authors: Fan, Xingya, He, Jianxun, Jia, Xiaoke
Format: Article
Language:English
Subjects:
Applications of Mathematics
Hyperbolic coordinates
Mathematical and Computational Physics
Mathematical Methods in Physics
Physics
Physics and Astronomy
Quaternions
Representations
Scalars
Theoretical
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Summary:Let X = S p ( 1 , n ) / S p ( n ) be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where n is greater than or equal to 1. The Fürstenberg boundary of X is denoted as Σ . In this paper, we focus on the Plancherel formula on X associated with the Poisson transform of vector-valued L 2 -space on Σ . Through the Fourier-Jacobi transform and the Fourier-Poisson transform, we derive the Plancherel decomposition of the unitary representation of Sp (1,  n ) on L 2 ( X ) .
ISSN:0188-7009
1661-4909
DOI:10.1007/s00006-024-01330-1