Automorphic forms on SL₂(R)

✍ Scribed by Armand Borel

• Publisher: Cambridge University Press
• Year: 1997
• Tongue: English
• Leaves: 204
• Series: Cambridge tracts in mathematics, 130
• Edition: First Edition
• Category: Library

✦ Synopsis

This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup ^DG of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; this is introduced in the last sections, making this connection explicit. The topics treated include the construction of fundamental domains, the notion of automorphic form on ^DG\G and its relationship with the classical automorphic forms on X, Poincaré series, constant terms, cusp forms, finite dimensionality of the space of automorphic forms of a given type, compactness of certain convolution operators, Eisenstein series, unitary representations of G, and the spectral decomposition of L2(^D*G/G). The main prerequisites are some results in functional analysis (reviewed, with references) and some familiarity with the elementary theory of Lie groups and Lie algebras

✦ Table of Contents

Content: Part I. Basic Material On SL2(R), Discrete Subgroups and the Upper-Half Plane: --
1. Prerequisites and notation --
2. Review of SL2(R), differential operators, convolution --
3. Action of G on X, discrete subgroups of G, fundamental domains --
4. The unit disc model --
Part II. Automorphic Forms and Cusp Forms: --
5. Growth conditions, automorphic forms --
6. Poincare series --
7. Constant term:the fundamental estimate --
8. Finite dimensionality of the space of automorphic forms of a given type --
9. Convolution operators on cuspidal functions --
Part III. Eisenstein Series: --
10. Definition and convergence of Eisenstein series --
11. Analytic continuation of the Eisenstein series --
12. Eisenstein series and automorphic forms orthogonal to cusp forms --
Part IV. Spectral Decomposition and Representations: --
13.Spectral decomposition of L2(G\G)m with respect to C --
14. Generalities on representations of G --
15. Representations of SL2(R) --
16. Spectral decomposition of L2(G\SL2(R)): the discrete spectrum --
17. Spectral decomposition of L2(G\SL2(R)): the continuous spectrum --
18. Concluding remarks.