  We have in [1] proposed a definition of cusp forms on semisimple symmetric
spaces $G/H$, involving the notion of a Radon transform and a related Abel
transform. For the real non-Riemannian hyperbolic spaces, we showed that there
exists an infinite number of cuspidal discrete series, and at most finitely
many non-cuspidal discrete series, including in particular the spherical
discrete series. For the projective spaces, the spherical discrete series are
the only non-cuspidal discrete series. Below, we extend these results to the
other hyperbolic spaces, and we also study the question of when the Abel
transform of a Schwartz function is again a Schwartz function.

Representation Theory