Representation theory and complex geometry. (English) Zbl 0879.22001

Chriss, Neil; Ginzburg, Victor

Boston, MA: Birkhäuser. x, 495 p. (1997).

This book treats the study of the geometry associated with a complex semisimple Lie group, such as the geometry of flag varieties, nilpotent conjugacy classes, Springer resolutions etc. For this study, the main tools used are symplectic geometry, equivariant algebraic \(K\)-theory and convolution operation in homology. The book also presents a uniform geometric approach to the classification of finite-dimensional irreducible representations of Weyl groups, the Lie algebra \({\mathfrak sl} (\mathbb{C})\) and affine Hecke algebras. This book is a nice blend of Algebraic Geometry, Symplectic Geometry and Representation Theory.

Reviewer: V.Lakshmibai (Boston)

MSC:

22-02	Research exposition (monographs, survey articles) pertaining to topological groups
17-02	Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry

Keywords:

algebraic \(K\)-theory; algebraic geometry; semisimple Lie group; flag varieties; Springer resolutions; symplectic geometry; irreducible representations; Weyl groups; Lie algebra; Hecke algebras