Cusp forms and parabolic cohomology classes for symmetric spaces of rank one
For any rank-one Riemannian symmetric space S of non-compact type and any discrete, cofinite, non-cocompact, torsion-free group $\Gamma$ of orientation-preserving Riemannian isometries on S, we develop a cohomological interpretation for the cusp forms of $\Gamma$. To that end, we identify certain $\Gamma$-submodules of smooth semi-analytic vectors in the spherical principal series representation with spectral parameter $\nu$ as well as certain subspaces of parabolic cohomology spaces of $\Gamma$ of degree dim S-1 with these $\Gamma$-submodules. We provide explicit isomorphisms between the spaces of cusp forms of spectral parameter $\nu$ and these specific cohomology subspaces. The isomorphisms from cusp forms to cohomology are given by an integral transform, and the explicit form of the inverse isomorphism takes advantage of a certain reproducing property of the integral transform. The result is uniform for all these symmetric spaces and does not rely on their classification.