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Automorphic Representations and L-Functions for the General Linear Group

Book Description

This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. Definitions are kept to a minimum and repeated when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. The book includes concrete examples of global and local representations of GL(n), and presents their associated L-functions. In Volume 1, the theory is developed from first principles for GL(1), then carefully extended to GL(2) with complete detailed proofs of key theorems. Several proofs are presented for the first time, including Jacquet's simple and elegant proof of the tensor product theorem. In Volume 2, the higher rank situation of GL(n) is given a detailed treatment. Containing numerous exercises by Xander Faber, this book will motivate students and researchers to begin working in this fertile field of research.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Introduction
  6. Preface
  7. 12 The classical theory of automorphic forms for GL(n, R)
    1. 12.1 Iwasawa decomposition for GL(n, R)
    2. 12.2 Congruence subgroups of SL(n, Z)
    3. 12.3 Automorphic functions of arbitrary weight, level, and character
    4. Exercises for Chapter 12
  8. 13 Automorphic forms and representations for GL(n, AQ)
    1. 13.1 Cartan, Bruhat decompositions for GL(n, R)
    2. 13.2 Iwasawa, Cartan, Bruhat decompositions for GL(n, QP)
    3. 13.3 Strong approximation for GL(n)
    4. 13.4 Adelic lifts and automorphic forms for GL(n, AQ)
    5. 13.5 The Fourier expansion of adelic automorphic forms
    6. 13.6 Adelic automorphic representations for GL(n, AQ)
    7. 13.7 Tensor product theorem for GL(n)
    8. 13.8 Newforms for GL(n)
    9. Exercises for Chapter 13
  9. 14 Theory of local representations for GL(n)
    1. 14.1 Generalities on representations of GL(n, QP)
    2. 14.2 Generic representations of GL(n, QP)
    3. 14.3 Parabolic induction for GL(n, QP)
    4. 14.4 Supercuspidal representations of GL(n, QP)
    5. 14.5 The Bernstein-Zelevinsky classification for GL(n, QP)
    6. 14.6 Classification of smooth irreducible representations of GL(n, QP) via the growth of matrix coefficients
    7. 14.7 Unitary representations of GL(n, QP)
    8. 14.8 Generalities on (g, K∞)-modules of GL(n, R)
    9. 14.9 Generic representations of GL(n, R)
    10. 14.10 Parabolic induction for GL(n, R)
    11. 14.11 Classification of the unitary and the generic unitary representations of GL(n, QP)
    12. 14.12 Unramified representations of GL(n, QP) and GL(n, R)
    13. 14.13 Unitary duals and other duals
    14. 14.14 The Ramanujan conjecture for GL(n, AQ)
    15. Exercises for Chapter 14
  10. 15 The Godement-Jacquet L-function for GL(n, AQ)
    1. 15.1 The Poisson summation formula for GL(n, AQ)
    2. 15.2 The global zeta integral for GL(n, AQ)
    3. 15.3 Factorization of the global zeta integral for GL(n, AQ)
    4. 15.4 The local functional equation for GL(n, QP)
    5. 15.5 The L-function and local functional equation for the supercuspidal representations of GL(n, QP)
    6. 15.6 The local functional equation for tensor products
    7. 15.7 The local zeta integral for a parabolically induced representation of GL(n, QP)
    8. 15.8 The local zeta integral for discrete series (square integrable) representations of GL(n, QP)
    9. 15.9 The local zeta integral for irreducible unitary generic representations of GL(n, R)
    10. Exercises for Chapter 15
  11. Solutions to Selected Exercises
  12. Reference
  13. Symbols Index
  14. Index