2017/11/30 by Gilles Becker, Becker, Gilles
Mathematics · #22E40 #22E46 #58C40 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1711.11272
openalex publication_date 2017/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathscrC(Γ\backslash G) on Γ\backslash G for torsion-free geometrically finite subgroups Γ 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 L2(Γ\backslash G) and that the right translation by elements of G defines a representation on \mathscrC(Γ\backslash G) . Moreover, we define the space of cusp forms °\mathscrC(Γ\backslash G) on Γ\backslash G , which is a geometrically defined subspace of \mathscrC(Γ\backslash G) . It consists of the Schwartz functions which have vanishing "constant term" along the ordinary set ΩΓ⊂ ∂ 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 L2(Γ\backslash G) which respects the Plancherel decomposition in the case where Γ is convex-cococompact and noncocompact. For technical reasons, we exclude here that X is the Cayley hyperbolic plane.