Let X be a projective real, complex, or quaternion hyperbolic space, realized as the pseudo-Riemannian symmetric space X ≅ G H with G = O(p, q), U(p, q), or Sp(p,q) (these are the classical isotropic symmetric spaces). Let Δ be the G-invariant Laplace-Beltrami operator on X. A complete description (by K-types), for each χ ∈ C, of all closed G-invariant subspaces of the eigenspace {f ∈ C^∞(X)|Δf = χf} is given. The eigenspace representations are compared with principal series representations, using "Poisson-transformations". Similar results are obtained also for the exceptional isotropic symmetric space. The Langlands parameters of the spherical discrete series representations are determined.