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Full Description
This book presents a treatment of the theory of $L$-functions developed by means of the theory of Eisenstein series and their Fourier coefficients, a theory which is usually referred to as the Langlands-Shahidi method. The information gathered from this method, when combined with the converse theorems of Cogdell and Piatetski-Shapiro, has been quite sufficient in establishing a number of new cases of Langlands functoriality conjecture; at present, some of these cases cannot be obtained by any other method. These results have led to far-reaching new estimates for Hecke eigenvalues of Maass forms, as well as definitive solutions to certain problems in analytic and algebraic number theory. This book gives a detailed treatment of important parts of this theory, including a rather complete proof of Casselman-Shalika's formula for unramified Whittaker functions as well as a general treatment of the theory of intertwining operators. It also covers in some detail the global aspects of the method as well as some of its applications to group representations and harmonic analysis. This book is addressed to graduate students and researchers who are interested in the Langlands program in automorphic forms and its connections with number theory.
Contents
Chapters
Introduction
Chapter 1. Reductive groups
Chapter 2. Satake isomorphisms
Chapter 3. Generic representations
Chapter 4. Intertwining operators
Chapter 5. Local coefficients
Chapter 6. Eisenstein series
Chapter 7. Fourier coefficients of Eisenstein series
Chapter 8. Functional equations
Chapter 9. Further properties of $L$-functions
Chapter 10. Applications to functoriality
Appendices: Tables of Dynkin diagrams
