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The Orbit Method in Geometry and Physics

In Honor of A.A. Kirillov

  • Book
  • © 2003

Overview

Editors:
  1. Christian Duval

    1. Centre de Physique Théorique CNRS, Campus de Luminy, Marseille Cedex 9, France

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  2. Valentin Ovsienko

    1. Institut Girard Desargues, Université Claude Bernard Lyon 1, Villeurbanne Cedex, France

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  3. Laurent Guieu

    1. Département des Sciences Mathématiques, Université Montpellier II, Montpelier Cedex 5, France

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  • A wide collection of recent research articles describing the most recent developments in representation theory and related topics
  • Researchers will find in this volume a representative and "state-of-the-art" literature and references
  • These articles are written by the most distinguished and active mathematicians in the subject

Part of the book series: Progress in Mathematics (PM, volume 213)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xiii
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  2. A Principle of Variations in Representation Theory

    • A. Alekseev, F. Petrov
    Pages 1-7

  3. Finite Group Actions on Poisson Algebras

    • Jacques Alev, Daniel R. Farkas
    Pages 9-28

  4. Representations of Quantum Tori and G-bundles on Elliptic Curves

    • Vladimir Baranovsky, Sam Evens, Victor Ginzburg
    Pages 29-48

  5. Dixmier Algebras for Classical Complex Nilpotent Orbits via Kraft-Procesi Models I

    • Ranee Brylinski
    Pages 49-67

  6. Brèves remarques sur l’œuvre de A. A. Kirillov

    • Jacques Dixmier
    Pages 69-71

  7. Gerbes of Chiral Differential Operators. III

    • Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman
    Pages 73-100

  8. Defining Relations for the Exceptional Lie Superalgebras of Vector Fields

    • Pavel Grozman, Dimitry Leites, Irina Shchepochkina
    Pages 101-146

  9. Schur-Weyl Duality and Representations of Permutation Groups

    • Eugene Gutkin
    Pages 147-164

  10. Quantization of Hypersurface Orbital Varieties in sl n

    • Anthony Joseph, Anna Melnikov
    Pages 165-196

  11. Generalization of a Theorem of Waldspurger to Nice Representations

    • D. Kazhdan, A. Polishchuk
    Pages 197-242

  12. Two More Variations on the Triangular Theme

    • A. A. Kirillov
    Pages 243-258

  13. The Generalized Cayley Map from an Algebraic Group to its Lie Algebra

    • Bertram Kostant, Peter W. Michor
    Pages 259-296

  14. Geometry of GL n (ℂ) at Infinity: Hinges, Complete Collineations,Projective Compactifications, and Universal Boundary

    • Yurii A. Neretin
    Pages 297-327

  15. Why Would Multiplicities be Log-Concave?

    • Andrei Okounkov
    Pages 329-347

  16. Point Processes Related to the Infinite Symmetric Group

    • Grigori Olshanski
    Pages 349-393

  17. Some Toric Manifolds and a Path Integral

    • W. Rossmann
    Pages 395-419

  18. Projective Schur Functions as Bispherical Functions on Certain Homogeneous Superspaces

    • Alexander Sergeev
    Pages 421-443

  19. Maximal Subalgebras of the Classical Linear Lie Superalgebras

    • Irina Shchepochkina
    Pages 445-472

  20. Back Matter

    Pages 473-474
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Keywords

About this book

The volume is dedicated to AA. Kirillov and emerged from an international con­ ference which was held in Luminy, Marseille, in December 2000, on the occasion 6 of Alexandre Alexandrovitch's 2 th birthday. The conference was devoted to the orbit method in representation theory, an important subject that influenced the de­ velopment of mathematics in the second half of the XXth century. Among the famous names related to this branch of mathematics, the name of AA Kirillov certainly holds a distinguished place, as the inventor and founder of the orbit method. The research articles in this volume are an outgrowth of the Kirillov Fest and they illustrate the most recent achievements in the orbit method and other areas closely related to the scientific interests of AA Kirillov. The orbit method has come to mean a method for obtaining the representations of Lie groups. It was successfully applied by Kirillov to obtain the unitary rep­ resentation theory of nilpotent Lie groups, and at the end of this famous 1962 paper, it was suggested that the method may be applicable to other Lie groups as well. Over the years, the orbit method has helped to link harmonic analysis (the theory of unitary representations of Lie groups) with differential geometry (the symplectic geometry of homogeneous spaces). This theory reinvigorated many classical domains of mathematics, such as representation theory, integrable sys­ tems, complex algebraic geometry. It is now a useful and powerful tool in all of these areas.

Reviews

"…the volume might be useful to a large number of potential readers interested in various fields, like representation theory of Lie groups, symplectic geometry, differential equations, combinatorics, etc. It is noteworthy that the history of mathematics can also be added to this list of topics, due to the nice article authored by J. Dixmier."

—Romanian Journal of Pure and Appl. Math.

Editors and Affiliations

  • Centre de Physique Théorique CNRS, Campus de Luminy, Marseille Cedex 9, France

    Christian Duval

  • Institut Girard Desargues, Université Claude Bernard Lyon 1, Villeurbanne Cedex, France

    Valentin Ovsienko

  • Département des Sciences Mathématiques, Université Montpellier II, Montpelier Cedex 5, France

    Laurent Guieu

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