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 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 ) .