Spectral Decomposition and Eisenstein Series

The decomposition of the space L(superscript 2) (G(Q)G(A)), where G is a reductive group defined over Q and A is the ring of adeles of Q, is a deep problem at the intersection of number and group theory. Langlands reduced this decomposition to that of the (smaller) spaces of cuspidal automorphic forms for certain subgroups of G. This book describes this proof in detail. The starting point is the theory of automorphic forms, which can also serve as a first step towards understanding the Arthur-Selberg trace formula. To make the book reasonably self-contained, the authors also provide essential background in subjects such as: automorphic forms; Eisenstein series; Eisenstein pseudo-series, and their properties. It is thus also an introduction, suitable for graduate students, to the theory of automorphic forms, the first written using contemporary terminology. It will be welcomed by number theorists, representation theorists, and all whose work involves the Langlands' program.

A self-contained introduction to automorphic forms, and Eisenstein series and pseudo-series, this text proves some of Langlands' work at the intersection of number theory and group theory.

Preamble; Notation; 1. Hypotheses, automorphic forms, constant terms; 2. Decomposition according to cuspidal data; 3. Hilbertian operators and automorphic forms; 4. Continuation of Eisenstein series; 5. Construction of the discrete spectrum via residues; 6. Spectral decomposition via the discrete Levi spectrum; Appendix I. Lifting of unipotent subgroups; Appendix II. Automorphic forms and Eisenstein series on function fields; Appendix III. On the discrete spectrum of G2; Appendix IV. Non-connected groups; Bibliography; Index.