© 1989

Nilpotent Orbits, Primitive Ideals, and Characteristic Classes

A Geometric Perspective in Ring Theory


Part of the Progress in Mathematics book series (PM, volume 78)

Table of contents

  1. Front Matter
    Pages i-vii
  2. W. Borho, J-L. Brylinski, R. MacPherson
    Pages 1-9
  3. W. Borho, J-L. Brylinski, R. MacPherson
    Pages 31-44
  4. W. Borho, J-L. Brylinski, R. MacPherson
    Pages 45-67
  5. W. Borho, J-L. Brylinski, R. MacPherson
    Pages 68-94
  6. W. Borho, J-L. Brylinski, R. MacPherson
    Pages 95-123
  7. Back Matter
    Pages 124-134

About this book


1. The Subject Matter. Consider a complex semisimple Lie group G with Lie algebra g and Weyl group W. In this book, we present a geometric perspective on the following circle of ideas: polynomials The "vertices" of this graph are some of the most important objects in representation theory. Each has a theory in its own right, and each has had its own independent historical development. - A nilpotent orbit is an orbit of the adjoint action of G on g which contains the zero element of g in its closure. (For the special linear group 2 G = SL(n,C), whose Lie algebra 9 is all n x n matrices with trace zero, an adjoint orbit consists of all matrices with a given Jordan canonical form; such an orbit is nilpotent if the Jordan form has only zeros on the diagonal. In this case, the nilpotent orbits are classified by partitions of n, given by the sizes of the Jordan blocks.) The closures of the nilpotent orbits are singular in general, and understanding their singularities is an important problem. - The classification of irreducible Weyl group representations is quite old.


Algebra Cohomology Group representation Irreducibility cls homomorphism ring theory

Authors and affiliations

  1. 1.BUGH - FB7Wuppertal 1Federal Republic of Germany
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  3. 3.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Bibliographic information


"…Most of the results in this book are not new. Instead the aim has been to use geometric (in place of the more traditional algebraic) methods in the constructions and proofs. This sheds new lights on the close connection between the three topics. The book is not self-contained. It relies heavily on previous work by the authors as well as on many basic facts both from algebraic groups, topology and representation theory. However, the authors have taken great care to make the book readable to people without complete background in these theories..."

--Zentralblatt Math