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Clifford algebras. --- Eisenstein series.

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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.

Eisenstein series. --- Automorphic functions. --- String models.

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Operator theory --- Eisenstein series. --- Spectral theory (Mathematics) --- Decomposition (Mathematics) --- Eisenstein, Séries d' --- Théorie spectrale (mathématiques) --- Décomposition (mathématiques) --- Eisenstein series --- Series, Eisenstein --- Automorphic functions --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Mathematics --- Probabilities --- Eisenstein, Séries d'.

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The decomposition of the space L2(G(Q)G(A)), where G is a reductive group defined over Q and A is the ring of adeles of Q, is a deep problem at the intersection of number and group theory. Langlands reduced this decomposition to that of the (smaller) spaces of cuspidal automorphic forms for certain subgroups of G. This book describes this proof in detail. The starting point is the theory of automorphic forms, which can also serve as a first step towards understanding the Arthur-Selberg trace formula. To make the book reasonably self-contained, the authors also provide essential background in subjects such as: automorphic forms; Eisenstein series; Eisenstein pseudo-series, and their properties. It is thus also an introduction, suitable for graduate students, to the theory of automorphic forms, the first written using contemporary terminology. It will be welcomed by number theorists, representation theorists and all whose work involves the Langlands program.

Eisenstein series. --- Automorphic forms. --- Spectral theory (Mathematics) --- Decomposition (Mathematics)

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Chevalley, Groupes de --- Chevalley groups --- Eisenstein, Séries d'. --- Eisenstein series --- Groupes finis --- Représentations de groupes --- Geometrie algebrique --- Groupes algebriques --- Eisenstein, Séries d'.