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

verfasst von: Roelof W. Bruggeman, Roberto J. Miatello

Verlag: Springer Nature Switzerland

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In the Introduction we sketch the context of automorphic forms on Lie groups and symmetric spaces, and discuss the choice of the Lie group \({\mathrm {SU}}(2,1)\) for this book. We summarize the main results on Fourier term modules in four theorems. We give an overview of applications to automorphic forms, considering also automorphic forms with moderate exponential growth.

Roelof W. Bruggeman, Roberto J. Miatello

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

Abstract

The aim of this work is to understand the modules involved in the Fourier expansions of functions on \(\Gamma \backslash {\mathrm {SU}}(2,1)\) for cofinite discrete subgroups \(\Gamma \).

In this preparatory chapter, we fix a standard realization \(G\subset {\mathrm {SL}}_3(\mathbb {C})\) of \({\mathrm {SU}}(2,1)\), and consider the representation theory of the maximal unipotent subgroup N and of the maximal compact subgroup K in an Iwasawa decomposition \(G=NAK\). We need to understand the realizations of irreducible representations of N and of K in spaces of functions on N and K, respectively.

Roelof W. Bruggeman, Roberto J. Miatello

Chapter 3. Fourier Term Modules

Abstract

We start the study of Fourier term modules, containing the Fourier terms of automorphic forms. For general values of the spectral parameters all Fourier term modules are a direct sum of submodules isomorphic to principal series representations.

Our approach is based on the description of functions in Fourier term modules as a tensor product of functions on N, A and K in an Iwasawa decomposition \(G=NAK\). This description enables us to use computer calculations, which are essential in some of the proofs in this work.

Roelof W. Bruggeman, Roberto J. Miatello

Chapter 4. Submodule Structure

Abstract

We turn to Fourier term modules for those spectral pairs for which the principal series modules are reducible. Then also the submodule structure of the Fourier term modules becomes much more complicated. Moreover, the lattice of submodules is not determined only by the spectral parameters. Specially the non-abelian cases show a large variation in the structure of the lattice of submodules.

In the final section of this chapter we list the irreducible modules that allow a unitary structure.

Roelof W. Bruggeman, Roberto J. Miatello

Chapter 5. Application to Automorphic Forms

Abstract

Finally, we use the knowledge concerning Fourier term modules to understand better the Fourier expansion of automorphic forms.

Usually, automorphic forms are required to have at most polynomial growth at the cusps. Here we also define automorphic forms with moderate exponential growth. A growth condition on the modular form implies properties of the Fourier expansion. For \({\mathrm {SL}}_2(\mathbb {R})\), an automorphic form with Fourier terms that have polynomial growth has polynomial growth itself. For \({\mathrm {SU}}(2,1)\) this does not necessarily hold.

We consider also the Fourier expansion of families of automorphic forms, and of generating vectors of irreducible automorphic modules.

Roelof W. Bruggeman, Roberto J. Miatello

Backmatter

Titel: Representations of SU(2,1) in Fourier Term Modules
verfasst von: Roelof W. Bruggeman
Roberto J. Miatello
Copyright-Jahr: 2023
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-031-43192-0
Print ISBN: 978-3-031-43191-3
DOI: https://doi.org/10.1007/978-3-031-43192-0

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