##
**Nilpotent orbits in semisimple Lie algebras.**
*(English)*
Zbl 0972.17008

New York, NY: Van Nostrand Reinhold Company. xiii, 186 p. (1993).

We cite from the authors’ preface: “Over the last decade, a circle of ideas has emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra. Any attempt to understand or exploit these connections ultimately presupposes a good understanding of each of these objects. Unfortunately, many of the fundamental results are scattered throughout the literature or shrouded in folklore, severely limiting accessibility by the nonexpert. The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from the common ground of basic structure theory. The techniques we use are elementary and in the toolkit of any graduate student hoping to enter the field of representation theory. In fact, one could argue that much of what follows is an elegant illustration of the power of \(\mathfrak{sl}_2\) theory.

After reviewing the basic prerequisites, we develop the Dynkin-Kostant and Bala-Carter classifications of complex nilpotent orbits, derive the Lusztig-Spaltenstein theory of induction of nilpotent orbits, discuss some basic topological questions, and classify real nilpotent orbits. We emphasize the special case of classical Lie algebras throughout; here the theory can be simplified by using the combinatorics of partitions and tableaux. We conclude with a survey of advanced topics related to the above circle of ideas.

The general linear group \(GL_n\mathbb{C}\) acts on its Lie algebra \(\mathfrak{gl}_n\) of all complex \(n\times n\) matrices by conjugation; the orbits are of course similarity classes of matrices. The theory of the Jordan form gives a satisfactory parametrization of these classes and allows us to regard two kinds of classes as distinguished: those represented by diagonal matrices, and those represented by strictly upper triangular matrices. In particular, there are only finitely many similarity classes of nilpotent matrices; more precisely, the set of such classes is parametrized by partitions of \(n\). Also, there is a very similar parametrization of nilpotent orbits in any classical semisimple Lie algebra.

The book is divided into three parts. In the first two, we give a more or less self-contained exposition with complete proofs; in the third, we give a survey of various results without proof. In the first part, consisting of Chapters 1-8, we work with complex semisimple Lie algebras. There are two prerequisites for understanding this material: the basic structure theory of such algebras, as given in the first four chapters of Humphreys’s classic text [J. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Math. 9, Springer-Verlag, New York (1972; Zbl 0254.17004)] and the elementary Lie group theory contained in F. Warner’s book [F. Warner, Foundations of Differentiable Manifolds and Lie Groups., 2d ed., Springer-Verlag, New York (1983; Zbl 0516.58001)]. In the second part (Chapter 9); we work with real semisimple Lie algebras. We assume familiarity with their classification and basic structure theory, as given in Helgason’s book [S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978; Zbl 0451.53038), reprint AMS (2001), §VI,X]. Finally, in the last part (Chapter 10), we discuss various advanced topics. It is here that we describe the connections between nilpotent orbits and representation theory.

Here is a brief outline of the content of each chapter.

In Chapter 1, we give the basic definitions and review the basic facts that we need. In Chapter 2, we classify the semisimple orbits in any complex semisimple Lie algebra; this turns out to be much simpler than classifying the nilpotent orbits. We also mention a few of the elementary topological properties of semisimple orbits and give references for their proofs.

Chapter 3 is the heart of the book; in it we derive the classical Dynkin-Kostant classification of complex nilpotent orbits. While this classification does not actually yield a parametrization (a parametrization is obtained in Chapter 8), it does prove one of our fundamental results: there are only finitely many nilpotent orbits. It also gives a uniform method for labeling nilpotent orbits.

In Chapter 4 we construct three canonical nilpotent orbits in any (complex) simple algebra. They may be specified by their positions in the Hasse diagram of nilpotent orbits relative to a certain partial order which we define in the chapter introduction.

All of the remaining theory in the book turns out to be much more concrete and explicit for the classical algebras than in general (and even simpler for \({\mathfrak sl}_n\) than for the other classical algebras). In Chapter 5 we refine the Dynkin-Kostant classification for classical algebras to an explicit parametrization in terms of partitions. We also give a formula for the labels attached in Chapter 3 to the orbit corresponding to a given partition.

Chapter 6 serves as an introduction to the topology of nilpotent orbits. We compute the fundamental groups of classical nilpotent orbits and give an explicit formula for the partial order relation of Chapter 4 in the classical cases. We also define an order-reversing involution on the set of so-called special orbits.

Chapter 7 continues the theme of Chapter 4 by constructing certain canonical nilpotent orbits. Instead of starting from nothing, however, we start from nilpotent orbits in smaller semisimple algebras. Once again we are able to give explicit formulas for the behavior of our construction in the classical cases.

As mentioned above, we refine the Dynkin-Kostant classification to a parametrization (this time for general semisimple algebras) in Chapter 8, following the approach of Bala and Carter. We give tables of the exceptional nilpotent orbits there.

Finally, in the last two chapters, we redo the theory of the preceding eight chapters (mostly the material in Chapters 3, 4, and 5) for real nilpotent orbits and then give a survey of some related representation theory. Topics in the last chapter include the Springer correspondence, associated varieties, and the classification of primitive ideals in enveloping algebras.

This book is the by-product of a two-quarter graduate course taught by the authors at the University of Washington in 1991.”

An excellent book that is highly recommended for use in graduate courses.

After reviewing the basic prerequisites, we develop the Dynkin-Kostant and Bala-Carter classifications of complex nilpotent orbits, derive the Lusztig-Spaltenstein theory of induction of nilpotent orbits, discuss some basic topological questions, and classify real nilpotent orbits. We emphasize the special case of classical Lie algebras throughout; here the theory can be simplified by using the combinatorics of partitions and tableaux. We conclude with a survey of advanced topics related to the above circle of ideas.

The general linear group \(GL_n\mathbb{C}\) acts on its Lie algebra \(\mathfrak{gl}_n\) of all complex \(n\times n\) matrices by conjugation; the orbits are of course similarity classes of matrices. The theory of the Jordan form gives a satisfactory parametrization of these classes and allows us to regard two kinds of classes as distinguished: those represented by diagonal matrices, and those represented by strictly upper triangular matrices. In particular, there are only finitely many similarity classes of nilpotent matrices; more precisely, the set of such classes is parametrized by partitions of \(n\). Also, there is a very similar parametrization of nilpotent orbits in any classical semisimple Lie algebra.

The book is divided into three parts. In the first two, we give a more or less self-contained exposition with complete proofs; in the third, we give a survey of various results without proof. In the first part, consisting of Chapters 1-8, we work with complex semisimple Lie algebras. There are two prerequisites for understanding this material: the basic structure theory of such algebras, as given in the first four chapters of Humphreys’s classic text [J. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Math. 9, Springer-Verlag, New York (1972; Zbl 0254.17004)] and the elementary Lie group theory contained in F. Warner’s book [F. Warner, Foundations of Differentiable Manifolds and Lie Groups., 2d ed., Springer-Verlag, New York (1983; Zbl 0516.58001)]. In the second part (Chapter 9); we work with real semisimple Lie algebras. We assume familiarity with their classification and basic structure theory, as given in Helgason’s book [S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978; Zbl 0451.53038), reprint AMS (2001), §VI,X]. Finally, in the last part (Chapter 10), we discuss various advanced topics. It is here that we describe the connections between nilpotent orbits and representation theory.

Here is a brief outline of the content of each chapter.

In Chapter 1, we give the basic definitions and review the basic facts that we need. In Chapter 2, we classify the semisimple orbits in any complex semisimple Lie algebra; this turns out to be much simpler than classifying the nilpotent orbits. We also mention a few of the elementary topological properties of semisimple orbits and give references for their proofs.

Chapter 3 is the heart of the book; in it we derive the classical Dynkin-Kostant classification of complex nilpotent orbits. While this classification does not actually yield a parametrization (a parametrization is obtained in Chapter 8), it does prove one of our fundamental results: there are only finitely many nilpotent orbits. It also gives a uniform method for labeling nilpotent orbits.

In Chapter 4 we construct three canonical nilpotent orbits in any (complex) simple algebra. They may be specified by their positions in the Hasse diagram of nilpotent orbits relative to a certain partial order which we define in the chapter introduction.

All of the remaining theory in the book turns out to be much more concrete and explicit for the classical algebras than in general (and even simpler for \({\mathfrak sl}_n\) than for the other classical algebras). In Chapter 5 we refine the Dynkin-Kostant classification for classical algebras to an explicit parametrization in terms of partitions. We also give a formula for the labels attached in Chapter 3 to the orbit corresponding to a given partition.

Chapter 6 serves as an introduction to the topology of nilpotent orbits. We compute the fundamental groups of classical nilpotent orbits and give an explicit formula for the partial order relation of Chapter 4 in the classical cases. We also define an order-reversing involution on the set of so-called special orbits.

Chapter 7 continues the theme of Chapter 4 by constructing certain canonical nilpotent orbits. Instead of starting from nothing, however, we start from nilpotent orbits in smaller semisimple algebras. Once again we are able to give explicit formulas for the behavior of our construction in the classical cases.

As mentioned above, we refine the Dynkin-Kostant classification to a parametrization (this time for general semisimple algebras) in Chapter 8, following the approach of Bala and Carter. We give tables of the exceptional nilpotent orbits there.

Finally, in the last two chapters, we redo the theory of the preceding eight chapters (mostly the material in Chapters 3, 4, and 5) for real nilpotent orbits and then give a survey of some related representation theory. Topics in the last chapter include the Springer correspondence, associated varieties, and the classification of primitive ideals in enveloping algebras.

This book is the by-product of a two-quarter graduate course taught by the authors at the University of Washington in 1991.”

An excellent book that is highly recommended for use in graduate courses.

Reviewer: O.Ninnemann (Berlin)

### MSC:

17B08 | Coadjoint orbits; nilpotent varieties |

17B20 | Simple, semisimple, reductive (super)algebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

22E46 | Semisimple Lie groups and their representations |

22E60 | Lie algebras of Lie groups |

17B25 | Exceptional (super)algebras |