Representation Theory and Complex Geometry

This classic monograph provides an overview of modern advances in representation theory from a geometric standpoint. A geometrically-oriented treatment of the subject is very timely and has long been desired, especially since the discovery of D-modules in the early 1980s and the quiver approach to quantum groups in the early 1990s. The techniques developed are quite general and can be successfully applied to other areas such as quantum groups, affine Lie groups, and quantum field theory. The first half of the book fills the gap between the standard knowledge of a beginner in Lie theory and the much wider background needed by the working mathematician.

The book is largely self-contained. . . . There is a nice introduction to symplectic geometry and a charming exposition of equivariant K-theory. Both are enlivened by examples related to groups. . . . An attractive feature is the attempt to convey some informal 'wisdom' rather than only the precise definitions. As anumber of results is due to the authors, one finds some of the original excitement. This is the only available introduction to geometric representation theory. . . it has already proved successful in introducing a new generation to the subject.

--- Bulletin of the American Mathematical Society

The authors have tried to help readers by adopting an informal and easily accessible style. . . . The book will provide a guide to those who wish to penetrate into subject-matter which, so far, was only accessible in difficult papers. . . . The book is quite suitable as a basis for an advanced course or a seminar, devoted to the material of one of the chapters of the book.

--- Mededelingen van het Wiskundig Genootschap

Represents an important and very interesting addition to the literature.

--- Mathematical Reviews

• D-modules
• K-theory
• algebraic geometry
• complex geometry
• complex theory
• geometric representation theory
• mathematical physics
• quantum groups
• symplectic geometry

From the reviews:

"The authors have tried to help readers by adopting an informal and easily accessible style...to convey a sound intuitive grasp of the basic concepts and proofs... The book will provide a guide to those who wish to penetrate into subject matter which, so far, was only accessible in difficult papers... The book is quite suitable as a basis for an advanced course or a seminar." ---T.A. Springer (Mededelingen van het wiskundig genootschap)

"Represents an important and very interesting addition to the literature." --- Mathematical Reviews

"The book is largely self-contained.... There is a nice introduction to symplectic geometry and a charming exposition of equivariant K-theory. Both are enlivened by examples related to groups... An attractive feature is the attempt to convey some informal 'wisdom' rather than only the precise definitions. As a number of results is due to the authors, one finds some of the original excitement. This is the only available introduction to geometric representation theory...it has already proved successful in introducing a new generation to the subject." --- Bulletin of the AMS

“The material covered in this book is at the crossroads of algebraic geometry, symplectic geometry and ‘pure’ representation theory. … This volume provides a self-contained overview of some of the recent advances in representation theory from a geometric standpoint. … The techniques developed are quite general and can be successfully applied to other areas such as quantum groups, affine Lie groups, and quantum field theory.” (Vasily A. Chernecky, Zentralblatt MATH, Vol. 1185, 2010)

• Mathematical Finance Program, University of Chicago, Chicago, USA

  Neil Chriss

• Dept. Mathematics, University of Chicago, Chicago, USA

  Victor Ginzburg

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