Eisenstein Series and Automorphic Representations: With Applications in String Theory

✍ Scribed by Philipp Fleig, Henrik P. A. Gustafsson, Axel Kleinschmidt, Daniel Persson

• Publisher: Cambridge University Press
• Year: 2018
• Tongue: English
• Leaves: 588
• Series: Cambridge Studies in Advanced Mathematics 176
• Category: Library

✦ Synopsis

This introduction to automorphic forms on adelic groups G(A) emphasises the role of representation theory. The exposition is driven by examples, and collects and extends many results scattered throughout the literature, in particular the Langlands constant term formula for Eisenstein series on G(A) as well as the Casselman–Shalika formula for the p-adic spherical Whittaker function. This book also covers more advanced topics such as spherical Hecke algebras and automorphic L-functions. Many of these mathematical results have natural interpretations in string theory, and so some basic concepts of string theory are introduced with an emphasis on connections with automorphic forms. Throughout the book special attention is paid to small automorphic representations, which are of particular importance in string theory but are also of independent mathematical interest. Numerous open questions and conjectures, partially motivated by physics, are included to prompt the reader's own research.

✦ Table of Contents

Cover
Series Page
Eisenstein Series and
Automorphic Representations
with Applications in String Theory
Copyright
Dedications
Contents
List of Definitions and Theorems
List of Examples
Preface
1 Motivation and Background
PART ONE: AUTOMORPHIC REPRESENTATIONS
2 Preliminaries on p-adic and Adelic Technology
3 Basic Notions from Lie Algebras and Lie Groups
4 Automorphic Forms
5 Automorphic Representations and Eisenstein Series
6 Whittaker Functions and Fourier Coefficients
7 Fourier Coefficients of Eisenstein Series on SL(2,𝔸)
8 Langlands Constant Term Formula
9 Whittaker Coefficients of Eisenstein Series
10 Analysing Eisenstein Series and Small Representations
11 Hecke Theory and Automorphic L-functions
12 Theta Correspondences
PART TWO: APPLICATIONS IN STRING THEORY
13 Elements of String Theory
14 Automorphic Scattering Amplitudes
15 Further Occurrences ofAutomorphic Forms in String Theory
PART THREE: ADVANCED TOPICS
16 Connections to the Langlands Program
17 Whittaker Functions, Crystals and Multiple Dirichlet Series
18 Automorphic Forms on Non-split Real Forms
19 Extension to Kac–Moody Groups
APPENDICES
Appendix A: SL(2, ℝ) Eisenstein Series and PoissonResummation
Appendix B: Laplace Operators on G/K and Automorphic Forms
Appendix C: Structure Theory of 𝔰𝔲(2, 1)
Appendix D: Poincaré Series and Kloosterman Sums
References
Index