Eisenstein Series and Automorphic Representations

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.

• 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.

'This book provides a bridge between two very active and important parts of mathematics and physics, namely the theory of automorphic forms on reductive groups and string theory. The authors have masterfully presented both aspects and their connections, and have provided examples and details at all levels to make the book available to a large readership, including non-experts in both fields. This is a valuable contribution and a welcoming text for graduate students as well.' Freydoon Shahidi, Purdue University, Indiana