Geometric analysis on symmetric spaces

Among Riemannian manifolds, symmetric spaces (in the sense of Cartan) provide an abundant supply of elegant examples, the structures of which are enhanced by the rich theory of semisimple Lie groups. On these spaces, global analysis, particularly integration theory and partial differential operators, arises in a natural way by the requirement of geometric invariance. In Euclidean space these two subjects are related by the Fourier transform. The Peter-Weyl theory for compact groups, and Cartan's refinement of it, provides a way to develop harmonic analysis on compact symmetric spaces. The noncompact symmetric spaces, however, present a multitude of new and natural problems. This book is devoted to geometric analysis on noncompact Riemannian spaces. The exposition in this book is accessible to readers with modest background in semisimple Lie group theory. In particular, familiarity with representation theory is not needed.