Cusp forms for locally symmetric spaces of infinite volume

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 $\mathcal{C}(\Gamma \backslash G)$ on $\Gamma \backslash G$ for torsion-free geometrically finite subgroups $\Gamma$ of $G$. We show that it has a Fréchet 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 $\mathcal{C}(\Gamma \backslash G)$. Moreover, we define the space of cusp forms $°\mathcal{C}(\Gamma \backslash G)$ on $\Gamma \backslash G$, which is a geometrically defined subspace of $\mathcal{C}(\Gamma \backslash G)$. 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.