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

TL;DR

This paper establishes a cohomological framework for understanding cusp forms on rank-one symmetric spaces, providing explicit isomorphisms via integral transforms that are uniform across these spaces.

Contribution

It introduces a novel cohomological interpretation of cusp forms, linking them to specific subspaces of parabolic cohomology through explicit integral transforms.

Findings

Explicit isomorphisms between cusp forms and cohomology spaces are constructed.

The integral transform used for the isomorphism has a reproducing property.

Results are uniform across all rank-one symmetric spaces, independent of classification.

Abstract

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.