Eisenstein series and automorphic representations

Abstract

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of the Langlands constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker function associated to unramified automorphic representations of G(Q_p). In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore also introduce some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In particular, we provide a detailed treatment of supersymmetry constraints on string amplitudes which enforce differential equations of the same type that are satisfied by automorphic forms. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics which go beyond the scope of this book.

326 pages. Detailed and example-driven exposition of the subject with highlighted applications to string theory. v2: 375 pages. Substantially extended and small corrections

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Cited by in corpus (29)

• Automorphic Forms and Fermion Masses
• Tensor network and ($p$-adic) AdS/CFT
• All-order differential equations for one-loop closed-string integrals and modular graph forms
• On supersymmetric E11 exceptional field theory
• Higher Derivative Supergravities in Diverse Dimensions
• Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms
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• Massive deformations of Maass forms and Jacobi forms
• To the cusp and back: Resurgent analysis for modular graph functions
• Fourier Coefficients for Degenerate Eisenstein Series and the Descending Decomposition
• No-boundary solutions are robust to quantum gravity corrections
• $D^6{R}^4$ curvature corrections, modular graph functions and Poincaré series
• 1/8-BPS Couplings and Exceptional Automorphic Functions
• Zero mode of the Fourier series of some modular graphs from Poincare series
• Notes on massless scalar field partition functions, modular invariance and Eisenstein series
• Calculus of conformal fields on a compact Riemann surface
• The E10 Wheeler-DeWitt operator at low levels
• A template method for Fourier coefficients of Langlands Eisenstein series
• 3D $\mathcal{N}=1$ supergravity from Virasoro TQFT: Gravitational partition function and Out-of-time-order correlator
• Counting Discrete, Level-$1$, Quaternionic Automorphic Representations on $G_2$
• Modular invariance in finite temperature Casimir effect
• Sketch of a Program for Universal Automorphic Functions to Capture Monstrous Moonshine
• Global Iwasawa-decomposition of SL($n$, ${\mathbb{A}}_{\mathbb{Q}}$)
• Fourier coefficients attached to small automorphic representations of ${\mathrm{SL}}_n(\mathbb{A})$
• Modular properties of massive scalar partition functions
• Massive Theta Lifts
• On the structure of Kac-Moody algebras
• Kac--Moody groups and automorphic forms in low dimensional supergravity theories
• Manifest Modular Invariance in the Near-Critical Ising Model