Cusp forms for locally symmetric spaces of infinite volume

TL;DR

This paper develops a framework for cusp forms on infinite volume locally symmetric spaces, defining Schwartz spaces and analyzing their structure, including a decomposition related to the Plancherel theorem for certain groups.

Contribution

It introduces a new Schwartz space and cusp form space for geometrically finite groups acting on rank-one symmetric spaces, with a decomposition compatible with harmonic analysis.

Findings

Defined the Schwartz space on $\Gamma ackslash G$ with a Fréchet structure

Established the density of smooth compactly supported functions in the Schwartz space

Proved a direct sum decomposition of cusp forms respecting the Plancherel decomposition

Abstract

Let $ G $ be a real simple linear connected Lie group of real rank one. Then, $ X := G/K $ is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, $X$ is a real/complex/quaternionic hyperbolic space or the Cayley hyperbolic plane. We define the Schwartz space $ \mathscr{C}(\Gamma \backslash G) $ on $ \Gamma \backslash G $ for torsion-free geometrically finite subgroups $ \Gamma $ of $G$. We show that it has a Fr\'echet space structure, that the space of compactly supported smooth functions is dense in this space, that it is contained in $ L^2(\Gamma \backslash G) $ and that the right translation by elements of $G$ defines a representation on $ \mathscr{C}(\Gamma \backslash G) $. Moreover, we define the space of cusp forms $ \deg\mathscr{C}(\Gamma \backslash G) $ on $ \Gamma \backslash G $, which is a geometrically defined subspace of $ \mathscr{C}(\Gamma \backslash G) $. It consists of the Schwartz functions which have vanishing "constant term" along the ordinary set $\Omega_\Gamma \subset \partial X$ and along every cusp. We show that these two constant terms are in fact related by a limit formula if the cusp is of smaller rank (not of full rank). The main result of this thesis consists in proving a direct sum decomposition of the closure of the space of cusp forms in $ L^2(\Gamma \backslash G) $ which respects the Plancherel decomposition in the case where $ \Gamma $ is convex-cococompact and noncocompact. For technical reasons, we exclude here that $X$ is the Cayley hyperbolic plane.