The Spectrum of Kleinian Manifolds

✍ Scribed by Ulrich Bunke; Martin Olbrich

• Book ID: 102972607
• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 530 KB
• Volume: 172
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3494

✦ Synopsis

We obtain the Plancherel theorem for L 2 (1"G), where G is a classical simple Lie group of real rank one and 1/G is convex cocompact discrete subgroup, and deduce its consequences for the spectrum of locally invariant differential operators on bundles over Kleinian manifolds. As the main tool, we develop a geometric version of scattering theory which, in particular, contains the meromorphic continuation of the Eisenstein series for this situation. The central role played by invariant distribution sections supported on the limit set is emphasized.

2000 Academic Press

Contents 1. Introduction. 2. Geometric preparations. 3. Analytic preparations. 4. Push-down and extension. 5. Meromorphic continuation of ext. 6. Invariant distributions on the limit set. 7. Consequences of unitarity. 8. Abstract harmonic analysis on 1 "G. 9. Tempered invariant distribution vectors. 10. Eisenstein series, wave packets, and scalar products. 11. The Plancherel theorem and spectral decompositions.

L 2 (1 "G, .