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Birkhäuser

Lie Theory

Unitary Representations and Compactifications of Symmetric Spaces

  • Book
  • © 2005

Overview

Editors:
  1. Jean-Philippe Anker

    1. Laboratoire de Mathématiques (MAPMO), Université d’Orléans, Orléans Cedex 2, France

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  2. Bent Orsted

    1. Department of Mathematics and Computer Science, University of Southern Denmark, Odense M, Denmark

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  • 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

Part of the book series: Progress in Mathematics (PM, volume 229)

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Table of contents (3 chapters)

  1. Front Matter

    Pages i-x
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  2. Introduction to Symmetric Spaces and Their Compactifications

    • Lizhen Ji
    Pages 1-67

  3. Compactifications of Symmetric and Locally Symmetric Spaces

    • Armand Borel, Lizhen Ji
    Pages 69-137

  4. Restrictions of Unitary Representations of Real Reductive Groups

    • Toshiyuki Kobayashi
    Pages 139-207
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Keywords

About this book

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 and provides 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. 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.

Reviews

"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 Poincaré 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 Mathématiques Pures et Appliquées

Editors and Affiliations

  • Laboratoire de Mathématiques (MAPMO), Université d’Orléans, Orléans Cedex 2, France

    Jean-Philippe Anker

  • Department of Mathematics and Computer Science, University of Southern Denmark, Odense M, Denmark

    Bent Orsted

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