Eisenstein Series and Automorphic Representations With Applications in String Theory Cambridge Studies in Advanced Mathematics Series

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.

1. Motivation and background; Part I. 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 II. Applications in String Theory: 13. Elements of string theory; 14. Automorphic scattering amplitudes; 15. Further occurrences of automorphic forms in string theory; Part III. 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; Appendix A. SL(2, R) Eisenstein series and Poisson resummation; 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.

Philipp Fleig is a Postdoctoral Researcher at the Max-Planck-Institut für Dynamik und Selbstorganisation, Germany.
Henrik P. A. Gustafsson is a Postdoctoral Researcher in the Department of Mathematics at Stanford University, California.
Axel Kleinschmidt is a Senior Scientist at the Max-Planck-Institut für Gravitationsphysik, Germany (Albert Einstein Institute) and at the International Solvay Institutes, Brussels.
Daniel Persson is an Associate Professor in the Department of Mathematical Sciences at Chalmers University of Technology, Gothenburg.