Cusp forms and parabolic cohomology classes for symmetric spaces of rank one

Abstract

For any rank-one Riemannian symmetric space S of non-compact type and any discrete, cofinite, non-cocompact, torsion-free group $Γ$ of orientation-preserving Riemannian isometries on S, we develop a cohomological interpretation for the cusp forms of $Γ$. To that end, we identify certain $Γ$-submodules of smooth semi-analytic vectors in the spherical principal series representation with spectral parameter $ν$ as well as certain subspaces of parabolic cohomology spaces of $Γ$ of degree dim S-1 with these $Γ$-submodules. We provide explicit isomorphisms between the spaces of cusp forms of spectral parameter $ν$ 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.