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Progress in Mathematics #229: Lie Theory: Unitary Representations, Number Theory, and Compactificationsby Jeanphilippe Anker
Synopses & ReviewsPublisher Comments: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, selfcontained volumes, under the general title Lie Theory, feature survey work and original results by wellestablished researchers in key areas of semisimple Lie theory. Unitary Representations and Compactifications of Symmetric Spaces, a selfcontained 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 BorelJi authoritative treatment of various types of compactifications useful for studying symmetric and locally symmetric spaces. BorelJi 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 HarishChandra 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.
Synopsis:* Focuses on two fundamental questions related to semisimple Lie groups: the geometry of Riemannian symmetric spaces and their compactifications, and branching laws for unitary representations * Wide applications of compactification techniques * Concrete examples and relevant exercises engage the reader * Knowledge of basic representation theory of Lie groups, semisimple Lie groups and symmetric spaces is required
Synopsis:Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces, a selfcontained 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 and provides a good background for the second chapter, namely, the BorelJi authoritative treatment of various types of compactifications useful for studying symmetric and locally symmetric spaces. Kobayashi examines the important subject of branching laws. Knowledge of basic representation theory of Lie groups and familiarity with semisimple Lie groups and symmetric spaces is required of the reader. Table of Contents* Preface * Ji: Introduction to Symmetric Spaces and Their Compactifications * Borel/Ji: Compactifications of Symmetric and Locally Symmetric Spaces * Kobayashi: Restrictions of Unitary Representations of Real Reductive Groups
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