Harmonic analysis and special functions on symmetric spaces

the long version

This sharply focused book treats the large class of hypergeometric functions used in the solution of differential equations and in physics. Although known from the time of Euler, these functions can now be understood for the first time in the context of harmonic analysis and symmetric spaces. Divided into two parts, the material in Harmonic Analysis and Special Functions on Symmetric Spaces is based on lectures given for the "European School of Group Theory," an advanced course on current developments in group theory. The authors provide students and researchers with a thorough and thoughtful overview, elaborating on the topic with clear statements of definitions and theorems and augmenting these with time-saving examples. An extensive set of notes supplements the text. The book leads readers from the fundamentals of semisimple symmetric spaces to the Reimannian case. The 19th century work of Euler, Gauss, Kummer, Riemann, and Klein on hypergeometric functions is linked to root systems and symmetric spaces. Algebraic and analytic methods are used, with many connections made in the geometric context of symmetric spaces. This volume will interest harmonic analysts, those working on or applying the theory of symmetric spaces, and those with an interest in special functions.