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Representations of SU(2,1) in Fourier Term Modules

  • Book
  • © 2023

Overview

Authors:
  1. Roelof W. Bruggeman

    1. Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherlands

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  2. Roberto J. Miatello

    1. FaMAF-CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina

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    You can also search for this author in PubMed   Google Scholar

  • Describes the structure of Fourier term modules
  • Gives complete Fourier-Jacobi expansions for SU(2,1)
  • Provides computations in the Mathematica notebook

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2340)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xi
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  2. Introduction

    • Roelof W. Bruggeman, Roberto J. Miatello
    Pages 1-9

  3. The Lie Group SU(2,1) and Subgroups

    • Roelof W. Bruggeman, Roberto J. Miatello
    Pages 11-32

  4. Fourier Term Modules

    • Roelof W. Bruggeman, Roberto J. Miatello
    Pages 33-84

  5. Submodule Structure

    • Roelof W. Bruggeman, Roberto J. Miatello
    Pages 85-166

  6. Application to Automorphic Forms

    • Roelof W. Bruggeman, Roberto J. Miatello
    Pages 167-193

  7. Back Matter

    Pages 195-210
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Keywords

About this book

This book studies the modules arising in Fourier expansions of automorphic forms, namely Fourier term modules on SU(2,1), the smallest rank one Lie group with a non-abelian unipotent subgroup. It considers the “abelian” Fourier term modules connected to characters of the maximal unipotent subgroups of SU(2,1), and also the “non-abelian” modules, described via theta functions. A complete description of the submodule structure of all Fourier term modules is given, with a discussion of the consequences for Fourier expansions of automorphic forms, automorphic forms with exponential growth included.

These results can be  applied to prove a completeness result for Poincaré series in spaces of square integrable automorphic forms.


Aimed at researchers and graduate students interested in automorphic forms, harmonic analysis on Lie groups, and number-theoretic topics related to Poincaré series, the book will also serve as a basic reference on spectral expansion with Fourier-Jacobi coefficients. Only a background in Lie groups and their representations is assumed.

Authors and Affiliations

  • Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherlands

    Roelof W. Bruggeman

  • FaMAF-CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina

    Roberto J. Miatello

About the authors

Roelof W.  Bruggeman was born in Zwolle, the Netherlands. He obtained his PhD at Utrecht University in 1972, and was a postdoctoral fellow at Yale University (1972–73). He has worked at Utrecht University since 1980, now as a guest after his retirement in 2005. In 2022 he became a corresponding member of the Academia Nacional de Ciencias in Córdoba, Argentina. The main research themes in his work are the spectral theory of Maass forms, the study of families of automorphic forms as a function of complex parameters, and the relation between automorphic forms and cohomology.

Roberto J. Miatello was born in Buenos Aires, Argentina. After studying at FaMAF, UNCordoba, Argentina (1965–1970) he obtained his PhD from Rutgers University in 1976. He was a professor at UFPe, Brazil (1977-1980), a member of the IAS, Princeton, USA (1980-81), and has been a professor at FaMAF (UNC) since 1982, becoming Professor Emeritus in 2016. He is a member of Conicet and (since 1996) the National Academy of Sciences of Argentina. His research focuses on geometry, the spectral theory of locally symmetric varieties, and automorphic forms.

Bibliographic Information

  • Book Title: Representations of SU(2,1) in Fourier Term Modules

  • Authors: Roelof W. Bruggeman, Roberto J. Miatello

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-031-43192-0

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Softcover ISBN: 978-3-031-43191-3Published: 07 November 2023

  • eBook ISBN: 978-3-031-43192-0Published: 06 November 2023

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XI, 210

  • Number of Illustrations: 15 b/w illustrations, 51 illustrations in colour

  • Topics: Number Theory, Fourier Analysis, Topological Groups, Lie Groups

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