Semisimple Lie groups, and their algebraic analogues over fields other than the reals, are of fundamental importance in geometry, analysis, and mathematical physics. Three independent, self-contained volumes, under the general title Lie Theory, feature survey work and original results by well-established researchers in key areas of semisimple Lie theory.
Unitary Representations and Compactifications of Symmetric Spaces, a self-contained work by A. Borel, L. Ji, and T. Kobayashi, focuses on two fundamental questions in the theory of semisimple Lie groups: the geometry of Riemannian symmetric spaces and their compactifications; and branching laws for unitary representations, i.e., restricting unitary representations to (typically, but not exclusively, symmetric) subgroups and decomposing the ensuing representations into irreducibles.
Ji's introductory chapter motivates the subject of symmetric spaces and their compactifications with carefully selected examples. A discussion of Satake and Furstenberg boundaries and a survey of the geometry of Riemannian symmetric spaces in general provide a good background for the second chapter, namely, the Borel-Ji authoritative treatment of various types of compactifications useful for studying symmetric and locally symmetric spaces. Borel-Ji further examine constructions of Oshima, De Concini, Procesi, and Melrose, which demonstrate the wide applicability of compactification techniques.
Kobayashi examines the important subject of branching laws. Important concepts from modern representation theory, such as Harish-Chandra modules, associated varieties, microlocal analysis, derived functor modules, and geometric quantization are introduced. Concrete examples and relevant exercises engage the reader.
Knowledge of basic representation theory of Lie groups as well as familiarity with semisimple Lie groups and symmetric spaces is required of the reader.
"The present volume consists of three chapters, and the corresponding material is based on lectures delivered by the authors to various European Schools in Group Theory. The first chapter...includes a very nice discussion of some of the basic ideas in the theory of symmetric spaces and their compactifications, starting from the fundamental examples of the Poincare disc and the bidisc. The third chapter...is a very good and most welcome introduction to a circle of ideas in representation theory centered on branching laws. The exposition includes many concrete examples...The above presentation of the contents is certainly too short to do justice to all beautiful ideas containe din its three chapters. This book should appeal to whoever has a taste for the beauty of the idea of symmetry in mathematics. If there is anyone asking only for the specific usefulness of the techniques developed here, thn we shall answer that these techniques are extremely useful to the graduate students, as well as to other people working in differential geometry, Lie theory, representation theory, or analysis on homogeneous spaces." ---Revue Roumaine de Mathematiques Pures et Appliquees