One studies the eigenspaces \({\mathcal E}_{\lambda}\) of the Laplace- Beltrami operator \(\Delta\) of a hyperbolic space \(X=G/H\), \[ {\mathcal E}_{\lambda}=\{f\in C^{\infty}(X)| \quad \Delta f=(\lambda^ 2- \rho^ 2)f\}\quad, \] where \(G=U(p,q; {\mathbb{F}})\), \(H=U(1, {\mathbb{F}})\times U(p-1,q; {\mathbb{F}})\), \({\mathbb{F}}\) is one of the classical fields \({\mathbb{R}}\), \({\mathbb{C}}\) or \({\mathbb{H}}\), \(\rho =(dp+dq-2)\) \((d=\dim_{{\mathbb{R}}} {\mathbb{F}})\), (and also for the exceptional space \(F^ 1_ 4/Spin(1,8)\) for which one may say that \(p=2\), \(q=1\), and \({\mathbb{F}}={\mathbb{O}}\), the algebra of the Cayley numbers).
One describes explicitly the closed C-invariant subspaces of \({\mathcal E}_{\lambda}\) in terms of the K-types occuring in such a subspace, where \(K=U(p; {\mathbb{F}})\times U(q; {\mathbb{F}})\), a maximal subgroup of G. In particular one determines for which \(\lambda\) the space \({\mathcal E}_{\lambda}\) is irreducible, and also the K-types occuring in the discrete series.
The range of the Poisson transform \({\mathcal P}_{\lambda}\) is a subspace of \({\mathcal E}_{\lambda}\). One gives the values of \(\lambda\) for which this range is a proper subspace of \({\mathcal E}_{\lambda}\).