Eisenstein Series and Automorphic Representations

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.