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

Let \(X=Sp(1,n)/Sp(n)\) $X=Sp(1,n)/Sp(n)$ 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 \(\Sigma \) $\mathrm{\Sigma }$ . In this paper, we focus on the Plancherel formula on X associated with the Poisson transform of vector-valued \(L^2\) $L^2$ -space on \(\Sigma \) $\mathrm{\Sigma }$ . Through the Fourier-Jacobi transform and the Fourier-Poisson transform, we derive the Plancherel decomposition of the unitary representation of Sp(1, n) on \(L^2(X)\) $L^2(X)$ .