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Page 1 of 3

  • An orthogonality relation for $\mathrm {GL}(4, \mathbb R) $ (with an appendix by Bingrong Huang)

  • Part of
  • Dorian Goldfeld, Eric Stade, Michael Woodbury
  • Journal:
    Forum of Mathematics, Sigma / Volume 9 / 2021
    Published online by Cambridge University Press:
    07 June 2021, e47
    • Article
      • You have access
      • Open access
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  • Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.

Automorphic Representations and L-Functions for the General Linear Group
  • Volume 2
  • Dorian Goldfeld, Joseph Hundley
  • Published online:
    05 June 2012
    Print publication:
    21 April 2011
  • 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.

Automorphic Representations and L-Functions for the General Linear Group
  • Volume 1
  • Dorian Goldfeld, Joseph Hundley
  • Published online:
    16 May 2011
    Print publication:
    21 April 2011
  • 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.

  • Introduction

  • Dorian Goldfeld, Columbia University, New York, Joseph Hundley, Southern Illinois University, Carbondale
  • Book:
    Automorphic Representations and L-Functions for the General Linear Group
    Published online:
    16 May 2011
    Print publication:
    21 April 2011, pp xv-xviii

  • Summary

    The theory of L-functions is an old subject with a long history. In the 1940s Hecke and Maass rewrote the classical theory in the setting of automorphic forms, and it seemed as if the theory of L-functions had settled into a fairly final form. This view was effectively overturned with the publication of two major books: [Gelfand-Graev-Piatetski-Shapiro, 1969], [Jacquet-Langlands, 1970], where it was shown that the theory of L-functions could be recast in the language of infinite dimensional complex representations of reductive groups.

    Another milestone in the recent theory of L-functions was the book by Roger Godement and Hervé Jacquet, [Godement-Jacquet, 1972], which defined for the first time the standard L-functions attached to automorphic representations of the general linear group, and proved their key properties by generalizing the seminal ideas of [Tate, 1950], [Iwasawa, 1952, 1992]. The proofs in [Godement-Jacquet, 1972] made fundamental use of matrix coefficients associated to automorphic representations. The standard L-functions of the general linear group are often called Godement-Jacquet L-functions. Although several other techniques have since been discovered to obtain the main analytic properties of such L-functions, none is more beautiful and elegant than the method of matrix coefficients, originally devised by Godement and Jacquet, which is a major theme of this book.

    Modern research in the theory of automorphic representations and L-functions is largely focused in the direction of the Langlands program.

  • Introduction

  • Dorian Goldfeld, Columbia University, New York, Joseph Hundley, Southern Illinois University, Carbondale
  • Book:
    Automorphic Representations and L-Functions for the General Linear Group
    Published online:
    05 June 2012
    Print publication:
    21 April 2011, pp xv-xviii

  • Summary

    The theory of L-functions is an old subject with a long history. In the 1940s Hecke and Maass rewrote the classical theory in the setting of automorphic forms, and it seemed as if the theory of L-functions had settled into a fairly final form. This view was effectively overturned with the publication of two major books: [Gelfand-Graev-Piatetski-Shapiro, 1969], [Jacquet-Langlands, 1970], where it was shown that the theory of L-functions could be recast in the language of infinite dimensional complex representations of reductive groups.

    Another milestone in the recent theory of L-functions was the book by Roger Godement and Hervé Jacquet, [Godement-Jacquet, 1972], which defined for the first time the standard L-functions attached to automorphic representations of the general linear group, and proved their key properties by generalizing the seminal ideas of [Tate, 1950], [Iwasawa, 1952, 1992]. The proofs in [Godement-Jacquet, 1972] made fundamental use of matrix coefficients associated to automorphic representations. The standard L-functions of the general linear group are often called Godement-Jacquet L-functions. Although several other techniques have since been discovered to obtain the main analytic properties of such L-functions, none is more beautiful and elegant than the method of matrix coefficients, originally devised by Godement and Jacquet, which is a major theme of this book.

    Modern research in the theory of automorphic representations and L-functions is largely focused in the direction of the Langlands program.

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