Nilpotent Orbits in Semisimple Lie Algebras Mathematics Series
Series Editors Raoul H. Bott, Harvard University David Eisenbud, Brandéis University Hugh L. Montgomery, University of Michigan Paul J. Sally, University of Chicago Barry Simon, California Institute of Technology Richard P. Stanley, Massachusetts Institute of Technology
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David H. Collingwood University of Washington William M. McGovern University of Washington
VAN NOSTRAND REINHOLD ______New York Copyright © 1993 by Van Nostrand Reinhold
Library of Congress Catalog Card Number 92-30461 ISBN 0-534-18834-6
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Library of Congress Cataloging-in-Publication Data Collingwood, David H. Nilpotent orbits in semisimple Lie algebras / David H. Collingwood, William M. McGovern, p. cm. Includes bibliographical references and index. ISBN 0-534-18834-6 1. Lie algebras. 2. Orbit method. I. McGovern, William M., [date- ]. II. Title. QA252.3.C65 1992 512’.55—dc20 92-30461 CIP For Phyllis and Kathy
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 (see [11],[9]). 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 $¡2 theory. After reviewing the basic prerequisites, we will 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 will 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„C acts on its Lie algebra gl^ of all complex n X 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 sur- viii Preface
vey 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 fom chapters of Humphreys’s classic text [37]; and the elemen tary Lie group theory contained in F. Warner’s book [89]. 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 [33, §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; for more detailed overviews see the chapter introductions. 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 semisim ple Lie algebra; this turns out to be much simpler than classifying the nilpo tent 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 om 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 sin 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 canon ical 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 para metrization (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. Preface ix
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. The material in Chapters 1-7, plus portions of Chapter 10, comprised a one-quarter course attended by graduate students with a background in basic structure theory. Chapters 8 and 9, together with a development of the classification of real forms (which is only recalled here) accounted for the majority of the second course. With careful planning, the entire book could reasonably be covered in a one-semester course. Alternatively, one could cover Chapters 1-5, 8, and 9 in a one-quarter course, giving the classification in the real and complex cases without touching upon the topological issues. Both authors are grateful to a number of people for their careful com ments, suggestions, corrections, and improvements to the (often fiawed) prelim inary manuscripts. We are especially pleased to acknowledge our thanks to the manuscript reviewers: Dan Barbasch, Cornell University; James Humphreys, University of Massachusetts; Paul Sally, University of Chicago; Nicolas Spal- tenstein. University of Oregon; and David A. Vogan, Jr., Massachusetts In stitute of Technology. In addition, our colleagues Ron Irving, Zong-Zhu Lin, Hisayosi Matumoto, Tim Morrison, Hiroyuki Ochiai, John Sullivan, and Jun Joseph Yang offered numerous helpful comments. The work was done while the authors received support from the National Security Agency and the National Science Foundation (grants MDA-904-90-H-4041 and DMS-9107890). Each au thor blames the faults that remain on the other.
Notation and Conventions
Z, N, N"*", Q, R, C, H denote the integers, nonnegative integers, strictly positive integers, rational numbers, real numbers, complex numbers, and quaternions, respectively. The notations [•,•], dim, |.T| denote the bracket operation in a Lie algebra, complex dimension (unless otherwise specified), and the cardinality of the set A', respectively. Lowercase Gothic letters (such as g) without subscripts denote finite-dimensional complex Lie algebras. Lowercase Gothic letters with subscript R denote finite-dimensional real Lie algebras. Except in Chapter 9, all Lie algebras are assumed to be complex (and finite-dimensional).
Contents
1 Preliminaries 1.1 Nilpotent and Semisimple Elements 1 1.2 Adjoint Orbits 5 1.3 Coadjoint Orbits 11 1.4 Orbits as Symplectic Manifolds 15
2 Semisimple Orbits 19 2.1 Some Structme Theory 19 2.2 Classification of Semisimple Orbits 24 2.3 Basic Topology of Semisimple Orbits 28
3 The Dynkin-Kostant Classification 29 3.1 Type A„ 30 3.2 Strategy 32 3.3 The Jacobson-Morozov Theorem 37 3.4 Theorems of Kostant and Mal’cev 40 3.5 Weighted Dynkin Diagrams 45 xii Contents
3.6 Weighted Dynkin Diagrams for Type An 47 3.7 Centralizer Structure 49 3.8 Parabolic Subalgebras 51
4 Principal, Subregular, and Minimal Nilpotent Orbits 55 4.1 The Principal Nilpotent Orbit 56 4.2 The Subregular Nilpotent Orbit 59 4.3 The Minimal Nilpotent Orbit 61 4.4 The Exponents of a Semisimple Group 64
5 Nilpotent Orbits in the Classical Algebras 69 5.1 Partition Type Classifications 69 5.2 Explicit Standard Triples 74 5.3 Weighted Dynkin Diagrams 80 5.4 Partitions Corresponding to Opr%n, Ogubreg, and Omin 85
6 Topology of Nilpotent Orbits 87 6.1 The Fundamental Group and A {0) 87 6.2 The Closure Ordering 93 6.3 Special Nilpotent Orbits in the Classical Algebras 97
7 Induced Nilpotent Orbits 105 7.1 Basic Results 105
7.2 Induced Nilpotent Orbits in Type A 110 Contents xiii
7.3 Induced and Rigid Orbits in the Classical Algebras 113
8 The Exceptional Cases and Bala-Carter Theory 119 8.1 Levi Subalgebras Containing Nilpotent Elements 119 8.2 Distinguished Nilpotent Elements and Parabolic Subalgebras 121 8.3 Connections with Induction 126 8.4 Tables 127
9 Real Nilpotent Orbits 135 9.1 Survey of Real Simple Algebras 135 9.2 The Jacobson-Morozov Theorem Revisited 137 9.3 Nilpotent Orbits in Classical Algebras 139 9.4 Cayley and Normal Triples; Basic Conjugacy Results 145 9.5 Sekiguchi’s Bijection and Weighted Dynkin Diagrams 147 9.6 Tables of the Real Exceptional Orbits 150
10 Advanced Topics 165 10.1 The Springer Correspondence 165 10.2 Associated Varieties of Primitive Ideals 170 10.3 Classification of Primitive Ideals 172 10.4 Primitive Ideals and the Geometry of Orbits 175
Bibliography 179 Index 184
1 Preliminaries
We begin by reviewing certain fundamental concepts from linear algebra. As innocent as this may seem, all that follows can be traced back to the elementary linear algebraic ideas surrounding the study of similarity between matrices. This review material is followed by a primer on symplectic geometry and its relevance to the study of adjoint orbits. The material in this chapter should be viewed as a quick review (with ref erences) of important elementary concepts from linear algebra, Lie theory, and diflFerential geometry. We need to know that every element of a semisimple Lie algebra can be expressed as the sum of a nilpotent element and a semisimple element. In tm*n, this leads to the notions of nilpotent and semisimple orbits in a Lie algebra (or its dual) and allows us to formulate the classification problems central to this book. In §1.4, we include a brief tour of the symplectic geometry relevant to our needs. We ultimately want to know that the previously men tioned orbits are symplectic manifolds and to understand the basic implications of having such a geometric structure.
1.1 Nilpotent and Semisimple Elements Consider a linear transformation X of a finite-dimensional complex vector space V to itself; i.e., an element of End(F). Recall that, by definition, we say that X is a semisimple operator if every X-invariant subspace has an X-invariant complement. We say that X is a nilpotent operator if X ’’ = X o • • • o X = 0 for some r > 0. Since the base field is C, it follows that X is semisimple if and only if X is diagonalizable (i.e. V admits a basis of eigenvectors for X) [36]. We emphasize that when working over R (or any non-algebraically-closed field), the notions of semisimplicity and diagonalizability differ. Our first order of business is the definition of nilpotent and semisimple elements in a complex semisimple Lie algebra 9. Such elements are easily defined, if we first recall the adjoint representation 2 CHAPTER 1 Preliminaries
ad : 0 End(0); a d x ^ = [X, Y].
We say that an element X £ gis nilpotent if adx is a nilpotent endomorphism of the vector space 9. In a similar spirit, X G 9 is said to be semisimple if adx is a semisimple operator on 9. Our definitions of nilpotent and semisimple elements in a Lie algebra give initial preference to the adjoint representation. You might change these defini tions by picking a favorite finite dimensional representation
p : 9 End(y)
and defining nilpotence or semisimplicity of X € 9 in terms of the nilpotence or semisimplicity of the operator p(X) on V. This approach would pose an obvious problem if X E ker(p). (Recall that the kernel of the adjoint representation on any Lie algebra is just the algebra’s center, which must be trivial in the semi simple case; consequently, the adjoint representation is faithful.) But, assuming p is faithful, it is perhaps hard to argue at the outset why our “adjoint definition” is to be preferred to the “p definition”. In fact, since the classical Lie algebras and sOn are defined directly as subalgebras of some End(F), that V, rather than 9, might seem the most natural one to use; see [37, §1.2]. As you would expect, our original definition and these proposed variants will ultimately be equivalent. To prove this, begin with an elementary result from linear algebra [37, §4.2], T heorem 1.1.1 (Jordan Decomposition). Let X he an endomorphism of a fínite dimensional complex vector space V. (i) There exist unique operators Xg^Xn G End(F) satisifying the conditions: X = Xg Xn^Xg is semisimple, Xn is nilpotent, Xg and Xn commute. (ii) Each of the operators Xg and Xn are polynomials in X with zero constant term. In particular, Xg and Xn commute with any operator commuting with X and stabilize any subspace of V that X stabilizes. Recall that End(F) is a Lie algebra under the usual commutator bracket operation; we will denote this Lie algebra by 9l(V). Of course, the careful reader will notice that gl{V) is not semisimple! Viewing X £ 9l(V) as an element of End(V), we can directly apply the Jordan decomposition theorem to find X = Xg Xn where Xg is semisimple, Xn is nilpotent and the two operators commute. Alternatively, we can apply the Jordan decomposition theorem to the operator adx viewed as an element in End(9l(V)) and obtain
adx = (adx)s + (adx)n,
where (adx)« is semisimple, (adx)n is nilpotent, and the two operators commute. The connection between these two observations is straightforward [37, §4.2]. 1.1 Nilpotent and Semisimple Elements 3
Lem m a 1.1.2. Let X G ^ -f Xn is the Jordan decomposition of X in End(y), then adx =adx^H-adxn is the Jordan decomposition of adx in End(0i(y)). The previous lemma is used to prove our proposed alternate definitions of semisimplicity and nilpotence coincide, but there is one subtle point needing attention. Since a semisimple Lie algebra g can be realized as a subalgebra of End(g) (via the adjoint representation), we can use (1.1.2) to decompose each XEgasX = Xad- X„, where [Xg^Xn] = 0; we provisionally refer to this as the abstract Jordan decomposition of X and define Xg (resp. Xn) to be the semisimple (resp. nilpotent) part of X. However, a priori^ we do not know that the elements Xg and Xn lie in the subalgebra g! The fact that this is true justifies our terminology and insures that the abstract and usual Jordan decompositions of X coincide [37, §6.4]. With all these ingredients in hand, we are led to the following result [14,1.6.3], which ties together the various approaches to defining nilpotent and semisimple elements. Proposition 1.1.3. Let g be a complex semisimple Lie algebra. If X G 0, then the following are equivalent: (i) X is a semisimple (resp. nilpotent) element of g. (ii) For every Gnite dimensional representation (p, V) of g, p{X) is a semisimple (resp. nilpotent) element of End{V). (iii) If g is a Lie subalgebra of End(F), then X is semisimple (resp. nilpotent) as an element of End{V). In the sequel, we will frequently make use of the spectral decomposition (or generalized eigenspace decomposition) of an operator X G End(F). Con sequently, we review the relevant linear algebra; details can be found in [36]. Recall that the annihilator of X in the polynomial ring C[t] is defined to be
Annc[t]X = {feC[t]\f{X) = 0}.
Since the annihilator of X is a nonzero principal ideal, it is generated by a monic polynomial /x , referred to as the minimal polynomial of X. By the Fundamental Theorem of Algebra, we can factor
/x (i) = (t-Ai)^^...(t-Afe)^*,
where Ai,..., A^ G C, ri,..., G N"*". In this context, it is useful to recall that an endomorphism X is semisimple if and only if the minimal polynomial fx has distinct roots. Let
Wi = {veV\(x-Xi-i)^^v = 0},
which is just the kernel of the operator (X — A^ • 7)^*. 4 CHAPTER 1 Preliminaries
Lemma 1.1.4 (Spectral Decomposition). We have V = W'l 0 ... 0 Wk- For each 1 < i < k, Wi is invariant under X and the minimal polynomial of X\wi is
We refer to the decomposition in (1.1.4) as the spectral or generalized eigenspace decomposition of V determined by X. Example 1.1.5. Recall that a key example of the above ideas is the weight space decomposition of a finite dimensional representation. Namely, let (p, V) be a finite dimensional representation of a semisimple Lie algebra g and f) an abelian subalgebra consisting of semisimple elements. For each X G If), by (1.1.3)(ii), p{X) is a semisimple operator on F, leading to a spectral decomposition:
{Ai(X),...,At(X)}GC
where V\.(x) = {v e V\p{X)v = A¿(X)t;} is the A¿(X)-eigenspace of p{X). As we vary X over the abelian algebra Í), we can simultaneously diagonalize the commuting family {p{X)\X € Í)}. In so doing, the eigenvalues become eigenfunctionals on f) and we have
V = E i'A.. {Ai,...,At}Gi)*
where V\^ = {v G V\p{X)v = Xi{X)v^ X G f)}. This is the famiUar f)-weight space decomposition of V. In the sequel, it will be important to work in a slightly more general setting than that of a semisimple Lie algebra. Roughly speaking, this generality will be forced upon us so that certain inductive constructions will be functorial. A complex finite-dimensional Lie algebra g is said to be reductive if the adjoint representation ad: g »-► End(fl) is completely reducible. An alternate characterization can be given once we recall some terminology. Denote by c the center of g; i.e.,
c = { X € 0 |[X ,0]=O}.
Obviously c is an ideal of g, as is the derived algebra [g,g]. A proof of the following lemma can be found in [14,1.6.4]. Lemma 1.1.6. The following conditions on a complex Lie algebra g are equivalent: (i) g is reductive. (ii) g = c 0 [g, g] with [g, g] a semisimple ideal of g. (iii) g has a finite-dimensional representation whose associated trace form (see [14]) is nondegenerate. 1.2 Adjoint Orbits 5
The most basic example of a reductive Lie algebra is the set of n x n matrices over C, denoted 9!^. Every semisimple Lie algebra is trivially reductive. The upper triangular matrices in 9!^ offer an example of a Lie algebra which is not reductive. If 9 is reductive, we define X G 9 to be semisimple if adx is a semisimple endomorphism of 9. Since the kernel of ad is c, this definition requires elements of c to be semisimple. Although adx is (trivially) a nilpotent endomorphism for X G c, such an X will not be considered nilpotent unless it is zero. Rather, we will say X G 9 is nilpotent if a,dx is a nilpotent endomorphism of 9 and X G [9,9]. Of course, when 9 is semisimple, our definitions collapse to the original ones. Example 1.1.7. Suppose 9 = SI2 = { X G 9I2 1 trace(X) = 0 } and X G 5(2* Then the characteristic polynomial for
X
is (t — Ai)(t — A2), where Ai and A2 are complex numbers. Case 1: Suppose Ai ^ A2. Then the minimal polynomial of X is fx{t) = (^ — Ai)(i — A2). Since the roots of the minimal polynomial are distinct, X is diagonalizable with eigenvalues Ai and A2. Since elements of ${2 have trace zero, we see that Ai = — A2. Case 2: Suppose Ai = A2 = A. Since X has trace zero, we get A = 0. Then the minimal polynomial for X is either fx{t) = t ot fx{i) If t is the minimal polynomial, X is the zero matrix. If is the minimal polynomial, then X is similar to its Jordan form In summary, we have verified the following lemma.
Lemma 1.1.8 . Any X G SI2 is similar to either for some (0 -A > A G C. In the former case, X is nilpotent; in the latter, it is semisimple, In addition, X is both semisimple and nilpotent if and only if it is zero.
1.2 Adjoint Orbits One of our ultimate goals is to devise some sort of classification scheme for the nilpotent and semisimple elements in a complex semisimple Lie algebra 9. With this in mind, let
S = { semisimple elements of9 }, jV = { nilpotent elements of 9 }.
In view of the previous section, it is clear that 6 CHAPTER 1 Preliminaries
= {0}.
We are now led to a fundamental problem in the structure theory of reductive Lie algebras, which will be refined momentarily: P rob lem . Classify the elements in S and M. Rather than focusing oiu* attention on the individual elements in S and we will take a cue from (1.1.8) and study conjugacy classes of elements using the natural action of a Lie group attached to our Lie algebra. The power of this viewpoint will become clearer in the sequel, once the correct group is isolated. Roughly speaking, the geometry of a conjugacy class is much richer than the geometry of a single element. In fact, we will later see that these conjugacy classes carry the structure of symplectic manifolds. Given these remarks and accepting the existence of a natural Lie group action on the sets of semisimple and nilpotent elements (spelled out carefully below), we refine the above problem: Refined Problem. Classify the conjugacy classes in S and M.
Example 1.2.1. Before proceeding in the general case, let’s consider the case 0 = s[2. We introduce the parameter sets:
A. = {C/{A--A}}, An = {0,1}. (1.2.2)
Denote by GL2 the set of invertible complex 2 x 2-matrices. For each X E si2, form the conjugacy class
Ox = GL2-X = {A X A-^lAeGL2}.
Note that since trace(AB)=trace(BA), the GL2-conjugate of an element in SI2 is again in $l2. If A € A« (resp. A G A„), we put
^ (^ ^ = (0 -A )(resp.F(A)=(j ^ )).
Then, by (1.1.8), we have a partition of SI2 into GLa-conjugacy classes
6I2 = IJ ^x(x) I U (Oy(o) U ^y(i)) (1.2.3) k A € A .\{0} '
with fhe properties:
(i) A conjugacy class Ox is semisimple if and only if Ox = Ox{\), some A € A,. 1.2 Adjoint Orbits 7
(ii) A conjugacy class Ox is nilpotent if and only if Ox = Oy {\)^ some A G (iii) A conjugacy class is simultaneously nilpotent and semisimple if and only
itO^=Ox(o) = 0^(«) = O o = ( j “).
Prom these remarks, we see that there are an infinite number of distinct semi simple GL2-conjugacy classes, but just two nilpotent GL2“Conjugacy classes. In this example, S \ {0} (resp. M) coincides with the first (resp. second) term in (1.2.3). Next, let SL2 be the subgroup of GL2 consisting of determinant one matrices and PSL2 = 5L2/Z , where Z = { /, —I } is the center of 5L2. Observe that GL2-, 5L2, and P5L2-conjugacy classes coincide, since
-1 AXA~^ = •X* (1.2.4) \/det(A)^ V^det(A)^
for all X G sl2?^ ^ GL2- Thus we have described the semisimple and nilpotent elements of SI2 up to GL2, SL2^ and P 5'L2-conjugacy. These arguments can be generalized to show the following: Let X G sin? where
si„ = { y e 0l„ I trace(y) = 0 },
then the GL^^SLu and PS'Ln-conjugacy classes of X all coincide; here, GLn denotes the n x n invertible complex matrices, SLn the subgroup of elements having determinant one and PSLn the group SLn modulo its center. The key point is to replace the term y^det(A) by the root of det(A) in (1.2.4). The careful reader will recognize that (1.2.1) is a special case of a more general theme developed in elementary linear algebra: the GLn-conjugacy class of a matrix X is just its similarity class. Thus, the “refined classification prob lem” posed above is a natural generalization to the setting of a semisimple Lie algebra of the similarity problem in linear algebra. In om* classification of nilpotent and semisimple elements, we want to form conjugacy classes using some natural Lie group. Recall that to each complex Lie algebra g we can canonically associate its adjoint group Gach which is a connected complex Lie group. It is defined to be the connected subgroup of GL(g) with Lie algebra adg. If 0 is semisimple, we can realize this group more explicitly. We begin with the automorphism group
Aut(fl) = {4>e gl(b) I ^(y)] = 4>[x,y], for aii A ,y e 0}. (1.2.5) of 0. Notice that for (f)-adx- For this reason, the semisimplicity (or nilpotence) of an element X is equivalent to the semisimplicity (or nilpotence) of all the elements (X),0 £ Aut(g). Con sequently, when classifying the semisimple or nilpotent elements in a Lie algebra, we could approach the problem by classifying such elements “up to the action of Aut(g)”; i.e., by classifying the sets Aut(0) - X = {0(A)1(^G A ut(g)}. This is one of our ultimate goals, at least when X is semisimple or nilpotent, but that is getting ahead of the game, for Aut(9) is not in general connected. More precisely, we have [39, IX,exercises] Aut(0) = Aut(0)'' X T{g) where A ut{gy is the identity component of Aut(9) and T{g) is a finite group (the graph automorphism group); whenever g is simple, this finite group is either trivial, Z/2Z, or 53. We can then also define the adjoint group Gad to be Gad = Aut(0)''. (1.2.7) An alternate approach to the adjoint group is as follows. Let G be any connected semisimple complex Lie group with Lie algebra g; then we can use the differentials of the automorphisms of conjugation to define the adjoint representation of G [89] Ad : G I—► Aut(0). (1.2.8) One can now define AdG=Image(Ad) to be the adjoint group of g. This is consistent, in view of the following fundamental result [33]. Lemma 1.2.9. Let G be a connected semisimple complex Lie group with Lie algebra g. Then Ado =Aut(g)® = Gad- The center Z of G is a Unite subgroup and Z=ker(Ad). If G is reductive, then its derived group [G, G] is semisimple and we have Gad — [G^G]ad- In particular, notice that (1.2.7) may fail in the reductive case (e.g., if g is abelian, then Gad = {1} and Aut^(g) = GL{g)). A final fact about the adjoint group will prove important in Chapter 6. Ul timately, we need to understand the collection of connected semisimple complex Lie groups having the same Lie algebra g. The theory of covering groups tells us that there exists a simply connected complex Lie group Gsc with Lie algebra g, 1.2 Adjoint Orbits 9 and every other connected complex Lie group G with Lie algebra g is a quotient of Gsc by a (finite) central subgroup. In particular, there are only finitely many such G; the adjoint group is the smallest group with Lie algebra g, while Gsc is the largest. Example 1.2.10. Consider the case of g = 5I2. Then the group SL2 is diffeomor- phic to KAN^ where K = SU {2) = { x e S L 2\x ~^ = X* }; . = { ( $ I ) | . > 0 }; All three of the groups K , A, N are simply connected, so SL2 is simply connected. In other words, Gsc = SL2 in this example. Moreover, one can check that { ± /} is the center of 5L2, so that Gad = P SL2. Iffl is a semisimple Lie algebra and X G g, we consider the action of the adjoint group on the vector space g and define the orbit through X by Ox := Gad-X = {(X)l For emphasis in the sequel, we may sometimes refer to these as adjoint orbits or conjugacy classes in g. An adjoint orbit Ox is called semisimple (resp. nilpotent) if X is semisimple (resp. nilpotent) in g. It follows from (1.2.6) that an orbit is semisimple (resp. nilpotent) if and only if every element in the orbit is semisimple (resp. nilpotent). Caution 1.2.12. The Gad-orbits and the Aut(g)-orbits need not coincide, in general. Of course, the problem is that two different Gad-conjugacy classes could coalesce under the action of the full automorphism group, leading to a smaller number of orbits. In the case of SL2, P SL 2 = Aut(sl2)- However, consider the case of SI3. The map (p defined by (f{A) = — where (*)^ represents the transpose operation, is the composition of two anti-automorphisms of 5I3, hence Now, 0 \ /< - 2 0 0 0 = 0 - 3 0 - 5 / '^ 0 0 5 but these matrices have different eigenvalues. Consequently, the matrices 0 0 ' 0 0 \ / 3 0 1 an d 0 - 3 0 0 - 5 ,/ V 0 0 5 10 CHAPTER 1 Preliminaries can’t be conjugate under Gad, since such conjugation preserves eigenvalues. In fact, in this example, one can show that Aut(5l3) = Aut(5l3)'^ X It is important to notice that an adjoint orbit Ox is a homogeneous complex manifold of the form Ox = GadlG^i, where G^^ = {xeGad\x^X = X] (1.2.13) is the centralizer of X in Gad', here (and elsewhere in the sequel) we use the notation x • X to denote Ad^^A. For a quick review of the relevant theory of homogeneous spaces, see [89, §3], [83, §2]. For now, we emphasize that the group need not be connected; understanding the consequences of disconnectedness will be important in Chapter 6. The Lie algebra of is given by = {Yeg\[X,Y] = 0} (1.2.14) and is referred to as the centralizer of X in g. We have the following elementary lemma, which is the first of many results illustrating the important role played by centralizers. Lemma 1.2.15. Let Ox be the adjoint orbit containing the element X. Then is a subalgebra of g and dim{Ox) = dim(g) — dim(g^). We close this section with a review of the classical groups. The most fundamental example of a complex reductive Lie group is the complex general linear group GLn = { invertible n x n matrices A = {uij), aij eC}. The Lie algebra of GLn is just the set of n x n complex matrices; i.e., gl(n). Recall the operations (-)^? (*) of transpose and conjugate of a complex matrix. Define Ip to be the p x p identity matrix and put J _ i 0 /n \ [-In 0 )■ We now recall the definitions of the classical complex groups: 1.3 Coadjoint Orbits 11 SLn = {A e GLn\det{A) = 1}; S02n-\-i = {A £ SL2n-\-i \ A^ • A = / 2T1+1 }; (1.2.16) ^P2n = { -A G GI/2n \A • Jji • A = S02n = {AeSL2n\A^-A = l2n}. These are all examples of connected complex semisimple Lie groups; their corre sponding Lie algebras are denoted 502n+ii ^p2n^ [37, §1.2]. 1.3 Coadjoint Orbits We need to extend the notions of semisimplicity and nilpotence to orbits and elements of g*, the algebraic dual of a reductive Lie algebra g. Unfortunately, there is a sticky point at the outset, since g* has no Lie algebra structure. However, just as the action of the adjoint group leads to an orbit partitioning of g, the coadjoint representation leads to an analogous partitioning of g*. In detail, let G be a connected complex reductive Lie group with Lie algebra g and recall the coadjoint representation of G on g*. Ad* ; G ^ Aut(0*); Ad:(/)(F) = f{Ad,-^{Y)). (1.3.1) The differential of Ad* yields the coadjoint representation of g on g* ad* : g End(g*); ad^^/(y) = -f(adxY). (1.3.2) The coadjoint orbit of / G g*, denoted O f, is precisely Of = { A d :(/) \xgG} = { Adlif) \xeGad }. (1.3.3) The vector space g* now breaks up into a disjoint union of coadjoint orbits and we can pose the problem of classifying such orbits. However, addressing the issue of defining nilpotent or semisimple elements (or coadjoint orbits) does not fit into the context of §1.1; it makes no sense to realize g* as a Lie subalgebra of some End(U)! There are two ways to deal with this dilemma, each of which deserves discussion. Lazy approach. If 0 is semisimple, the Killing form is a nondegener ate invariant symmetric bilinear form; recall that a bilinear form ^ is invariant if 'ip{AdxY,Z) = -0 ( y , Adaj-iZ) for all x G Gad^Y.Z G g, or equivalently if V^([X, F],Z) = -0(X, [y,Z]) for all X,Y^Z G g. In the reductive case, we have the following lemma [14, 1.6.4]. Lemma 1.3.4. If g is reductive, there exists an invariant nondegenerate symmetric bilinear form on g. Moreover, if xp is such a form and $ is a simple summand of [g, g], then xp{-, •)|fi is a nonzero constant multiple of the Killing form on s. 12 CHAPTER 1 Preliminaries Beginning with a reductive Lie algebra 0, we can identify g* with g via an invariant nondegenerate symmetric bilinear form V^(*, •). Given such a form t/; : 0 X 0 C, we induce a vector space map = lp{X, ■). (1.3.5) The nondegeneracy of ip forces to be a bijection (since 0 is finite-dimensional). Furthermore, by the invariance of the form, i^{Ad,X) = V>(Ad,X, •) = A d ,-i(-)) = Ad:(i^(x)). This implies that H{Ox ) = (1.3.6) whence we get a one-to-one correspondence between adjoint orbits in 0 and coadjoint orbits in 0*. The correspondence in (1.3.6) can now be used to transfer the notion of semisimple or nilpotent adjoint orbits over to coadjoint orbits. In turn, we say an element / G 0* is semisimple (resp. nilpotent) if and only if / lies in with X a semisimple (resp. nilpotent) element of 0. We will provisionally refer to these definitions as the lazy deSnitions of semisimple and nilpotent elements (or orbits) in 0*. S oph isticated approach. An alternative more natural approach to working with coadjoint orbits is as follows. Given / G 0*, define 0/ = { y G 0|ad^/ = O}, (1.3.7) called the stabilizer of / in 0. Since / maps 0 into C, it makes sense to restrict / to 0*^. We say / is nilpotent if = 0. We say / is semisimple if 0-^ is reductive in 0 (i.e., the adjoint representation of 0 restricted to the subalgebra 0*^ is completely reducible). We refer to these as the sophisticated definitions of nilpotent and semisimple elements in g*. We then define a coadjoint orbit Of to he semisimple (resp. nilpotent) if / is semisimple (resp. nilpotent) in the sophisticated sense. One should note that a coadjoint orbit is semisimple (resp. nilpotent) in the sophisticated sense if and only if every element in the orbit is semisimple (resp. nilpotent) in the sophisticated sense. We would like to claim that there is no difference between the lazy and the sophisticated definitions described above. We need an elementary lemma. Lemma 1.3.8. Let g be reductive^ xp a nondegenerate symmetric invariant bilinear form on g and X e g. Then (i) = 0^. 1.3 Coadjoint Orbits 13 (ii) If Z G g^, then adxadz= ad^adx- (iii) Set [fl, X] = {[W^, X] I € 0}. Then dim(0^ ) + dim([0, X]) = dim(0). Proof. Let be the vector space isomorphism in (1.3.5). Now Y £ g ^ ^ [X,F] = 0; ■<=>■ y], W) = 0, for all ty 6 0 (nondegeneracy); V’(-X’, [Y, W]) = 0, for all fy 6 0 (invariance); V(A-)([y,ty]) = 0, foralliy 6 0; -¿^(x)([y,iy]) = (ad^i^(j>i:)(iy) = o, for aiify 6 0; ^ Y e 0*'^w . This proves (i), while (ii) follows from the Jacobi identity. For part (iii), consider the linear map adx ^ 0 •-► 0; observe that is the nullspace, and [g, X] is the range. □ Remark. It is worth noting that (iii) does not hold on the level of a vector space decomposition; see the proof of the Jacobson-Morozov Theorem in §3.3. We can now establish the equivalence of our two approaches to defining nilpotent elements in g*. We point out that part of the argument given will be used in the proof of the Jacobson-Morozov Theorem, which is a central result in all that follows. Lemma 1.3.9. Let g be a reductive Lie algebra. The lazy and the sophisticated definitions of nilpotent elements in g* coincide. Proof. Construct a vector space isomorphism z\/, as in (1.3.5), with respect to some fixed nondegenerate invariant symmetric bilinear form on g; such exists by (1.3.4). Let / G g*, then there exists a unique X e g such that / = i^{X). Begin by assuming / = z^(X) is nilpotent using the lazy definition. This means that X is a nilpotent element in g; i.e., X € [g,g] and adx is a nilpotent operator. We know that il){XyY) = /(T ), for all F G g. After possibly passing to the components of X in the simple summands of [g, g], we can appeal to (1.3.4) and assume f{Z) = “0(X , Z) — Ac(X, Z) = trace(adxad^), Z G g^. (1.3.10) By (1.3.8), (adxadz)^ = ad ^ ad |, for all A: G N, Z G g^. Since ad^ is zero for large fc, it follows that ad^ad^ is nilpotent; so its trace is zero. This implies (1.3.10) is zero so that f\gf = 0. This shows / is nilpotent in the sophisticated sense. Conversely, suppose / = z^(X) is nilpotent in the sophisticated sense, so that f\,f = 0. (1.3.11) 14 CHAPTER 1 Preliminaries By invariance, X], W) = “0(9, [X, VF]) = 0 (9 ,a d x i^ ) = 0, for all W € g^. This shows that [9,^ ] is a subspace of 9 which is 0-orthogonal to 9"^. By the nondegeneracy of the form and (1.3.8)(iii), dim(9^)-^ = dim(9) — dim(9^) = dim([9,X]). This shows that [g,X] = (9^)"^. By (1.3.11), X G (9^)"^, so X G [9, X] C [9, 9]. It remains to show that the operator adjv is nilpotent. Since X G [9, X], we know there exists i f G 9, so that X = [H^X]. Consider the operator ad/f : 0 9 and decompose 9 into generalized adif-eigenspaces. In other words, for each a G C, define 0a = { ^ 0 1 (adff - a • lyW = 0, some s = s« G N }, then 0 = 0 0 a- (1.3.12) aec Let W E 0a‘, then by definition (adif - a • ly w = 0, for somes G N+. (1.3.13) Using the Jacobi identity, we observe that for all Z G 0 adn[X, Z] = [F, [X, Z]] = -[X , [Z, H]] - [Z, [ii, X]] = [X, ad^Z] + [X, Z]. Using the binomial theorem and repeated applications of this identity we find (adH - (a + 1) • iy[X, W] = j 2 ( l ) (-!)'’■''(« + k=0 ^ ^ = )ad*HW^](-l)*-''(a + 1)*-" fc=0 ^ ^ i=0 ^ ' = [X,{adH-a-iyW]. By (1.3.13), this last term is zero, showing [X, W] G 0a+i; hence adx0a C 0a+i- Since the sum in (1.3.12) only involves a finite number of terms, we see that (adx)^ = 0, for some t; i.e., adx is a nilpotent operator. This proves that / is nilpotent in the lazy sense. □ For emphasis, we state (without proof) the equivalence of our two defini tions of semisimplicity in 9*. This result will not be needed in the sequel, but you can provide a proof after reviewing the structure theory discussed in Chapter 2. 1.4 Orbits as Symplectic Manifolds 15 Lemma 1.3.14. Let g be a reductive Lie algebra. The lazy and the sophisticated deSnitions of semisimple elements in g* coincide. 1.4 Orbits as Symplectic Manifolds Certain arguments can be simplified if we exploit the differential geometry of a coadjoint orbit. In this section, we review the relevant concepts, ultimately describing a complex symplectic structure on any coadjoint orbit. Among other useful properties, the adjoint and coadjoint orbits in the sequel always have even (complex) dimension. All that follows can be developed over the base field R, but we will stick to the complex setting. To begin with, a symplectic vector space is a pair (V, u) consisting of a com plex (finite-dimensional) vector space V and a nondegenerate skew-symmetric bilinear form u on V. One typically refers to a; as a symplectic structure on V. Example 1.4.1. Let V = and view vectors in V as column matrices. Define the 2 x 2 matrix and construct the 2n x 2n matrix ( s 0 0 • •• 0 0 5 0 • • 0 B^ = 0 0 s •••• 0 u 0 0 • • • s which is the block diagonal sum of n copies of S. It is easily checked that Lu{v, w) = V* • Buj • w defines a symplectic structure on V. Symplectic vector spaces are set apart from standard Euclidean space due to the presence of self-orthogonal subspaces. If VF is a subspace of a symplectic vector space (V,u;), we write W'^ = {i; G V\uj{v^w) = 0, for allw e W }; this is the orthogonal subspace of W, relative to uj. We say that W is isotropic if W C W-^. For example, the skew-symmetry of the form implies that the 16 CHAPTER 1 Preliminaries one-dimensional subspace generated by a nonzero vector is isotropic. For more details on the basic properties of symplectic vector spaces, including the next lemma, we refer the reader to [38]. Lemma 1.4.2. Let (V,uj) be a symplectic vector space. Then dim(F) = 2n for some n G N. IfW is an isotropic subspace of V, then dim(W') < n. We now shift to the setting of a complex manifold M. We will denote by T*{M) (resp. T*{M)) the complex cotangent space at x (resp. complex cotan gent bundle over M). In order to make sense of the following definition, we remind you that a smooth complex 2-form a; on M is a smooth section of the exterior bundle /\^{T*{M)). For each x € M, we have a canonical vector space isomorphism between and the vector space of skew-symmetric bi linear forms on Tx{M). Thus, given a section uj of /\^(T*(M )), we obtain a family {uJx}xeM of skew-symmetric bilinear forms. A symplectic manifold is a pair (M,a;), consisting of a complex manifold M and a smooth closed complex valued 2-form u on M, such that for all x G M, cux is nondegenerate. We re fer to a; as a symplectic structure on M. Otherwise put, for each x G M, we are provided with a nondegenerate skew-symmetric bilinear form Ux on Tx{M) and these fit together to form a smooth global complex valued 2-form on M. A submanifold AT of M is called an isotropic submanifold if Tx(N) C Tx(N)-^, for every x G iV; here, _L is relative to lJx- The following result is immediate from (1.4.2). Lemma 1.4.3. If {M,uj) is a symplectic manifold, then dim(M) is even. If N is an isotropic submanifold of M, then dim(AT) < I dim(M). Homogeneous complex manifolds M = G /H play a fundamental role in modern Lie theory. Quite often, these manifolds additionally carry a symplec tic structure. A fundamental result ensures that locally, all such homogeneous symplectic manifolds of the same finite dimension “look the same”. We will now isolate this local structure by exhibiting an important class of symplectic manifolds. We consider a connected complex Lie group G with Lie algebra g and consider the coadjoint orbits of G in 9*; although (1.3.3) was couched in the reductive setting, the notion of such orbits makes sense in general. Let O / be the coadjoint orbit in g* through /. The element / determines the subgroup Gf = {xeG\Adlf = f} (1.4.4) having Lie algebra (1.3.7). Define (1.4.5) Using the Jacobi identity, we see that cj/ is a skew-symmetric bilinear form on g. Observe that this form is degenerate and has radical Uf= g^. We conclude that 1.4 Orbits as Symplectic Manifolds 17 (jf defines a nondegenerate symplectic form on We make the coadjoint orbit into a complex homogeneous manifold using the structure it inherits under the identification Of ^ G/G^. (1.4.6) Consequently, we see that ujf provides a symplectic vector space structure on TfiOf) S Tx(G/G^) S Q/gf. The main result is Theorem 1.4.7. Let Of be a coadjoint orbit in q*. Then the symplectic bilinear form Uf on QIextends to a complex symplectic structure on Of, For a proof, we refer you to [88] or [32]. The following useful corollary is an immediate consequence. Corollary 1.4.8. The dimension of a coadjoint orbit is even. The same holds for adjoint orbits if 0 is reductive. As a final point of interest, we now refine the motivational remark preceed- ing our discussion of coadjoint orbits, highlighting their geometric importance. Since this result is not needed in the sequel, we refer you to [32, IV. 7]. Theorem 1.4.9. Each homogeneous complex symplectic manifold (relative to a semi simple Lie group) is a covering of some coadjoint orbit. References A. Alexeevsky . Component groups of centralizer for unipotent elements in semisimple algebraic groups [in Russian], Trudy Tbilissi. Mat. Inst. Razmazde Akad. Nauk. Gruzin SSR 62, (1979), 5-27. P. Bala and R. Carter . Classes of unipotent elements in simple algebraic groups I, Math. Proc. Camb. Phil. Soc. 79 (1976), 401â425. P. Bala and R. Carter . Classes of unipotent elements in simple algebraic groups II, Math. Proc. Camb. Phil. Soc. 80 (1976), 1â18. D. Barbasch and D. Vogan . Primitive ideals and orbital integrals in complex classical groups, Math. Ann. 259 (1982), 153â199. D. Barbasch and D. Vogan . Primitive ideals and orbital integrals in complex exceptional groups, J. Alg. 80 (1983), 350â382. D. Barbasch and D. Vogan . Unipotent representations of complex semisimple groups, Ann. of Math. 121 (1985), 41â110. A. Borel . Topics in the homology theory of fiber bundles. Lecture Notes in Math. 36, Springer-Verlag, New York, 1967. W. Borho . Ãber Schichten halbeinfacher Lie-Algebren, Inv. Math. 65 (1981), 283â317. W. Borho . A survey of enveloping algebras of semisimple Lie algebras, I, in Lie Algebras and Related Topics, Canad. Math. Soc. Conf. Proc. 5, Amer. Math. Soc. for CMS, Providence, 1986, 19â50. W. Borho and J. Brylinski . Differential operators on homogeneous spaces I: irreducibility of the associated variety, Inv. Math. 69 (1982), 437â476. W. Borho , J. Brylinski , and R. MacPherson . Nilpotent Orbits, Primitive Ideals, and Characteristic Classes: A Geometric Perspective in Ring Theory. Progress in Math. 78, Birkhäuser, Boston, 1989. W. Borho and H. Kraft . Ãber Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comm. Math. Helv. 54 (1979), 61â104. R. Bott . An application of Morse theory to the topology of Lie groups, Bull. Soc. Math. Fr. 84 (1956), 251â281. 180 N. Bourbaki . Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 1-3. Springer-Verlag, New York, 1989. N. Bourbaki . Ãléments de mathématique: Groupes et Algèbres de Lie; Chapitres 4-6, 7-8. Hermann, Paris, 1968. A. Broer . On the subregular nilpotent variety, preprint, Amsterdam, 1991. N. Burgoyne and R. Cushman . Conjugacy classes in the linear groups, J. Alg. 44 (1977), 339â362. R. Carter . Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. John Wiley & Sons, London, 1985. R. Carter . Simple Groups of Lie Type. Wiley Classics Library, John Wiley & Sons, London, 1989. A. Coleman . The Betti numbers of the simple Lie groups, Canad. J. Math. 10 (1958), 349â356. J. Dixmier . Enveloping Algebras. North-Holland, Amsterdam, 1977. D. DjokoviÄ . Closures of conjugacy classes in classical real linear Lie groups, in Algebra, Carbondale, 1980, Lecture Notes in Math. 848, Springer-Verlag, New York, 1980, 63â83. D. DjokoviÄ . Proof of a conjecture of Kostant, Trans. Amer. Math. Soc. 302 (1987), 577â585. D. DjokoviÄ . Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Alg. 112 (1988), 503â524. D. DjokoviÄ . Classification of nilpotent elements in the simple real Lie algebras E6(6) and E6(â26) and description of their centralizers, J. Alg. 116 (1988), 196â207. M. Duflo . Sur la classification des idéaux primitifs dans lâalgèbre enveloppante dâune algèbre de Lie semi-simple, Ann. of Math. 105 (1977), 107â120. E. Dynkin . Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. Ser. 2, 6 (1957), 111â245. D. Garfinkle . On the classification of primitive ideals for complex classical Lie algebras, I, Comp. Math. 75 (1990), 135â169. M. Gerstenhaber . On dominance and varieties of commuting matrices, Ann. of Math. 73 (1961), 324â348. M. Gerstenhaber . Dominance over the classical groups, Ann. of Math. 74 (1961), 532â569. W. Greub , S. Halperin , and R. Vanstone . Connections, Curvature and Cohomology II. Academic Press, New York, 1973. V. Guillemin and S. Sternberg . Geometric Asymptotics. Mathematical Surveys 14, American Mathematical Society, Providence, 1977. S. Helgason . Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, 1978. 181 W. Hesselink . Singularities in the nilpotent scheme of a classical group, Trans. Amer. Math. Soc. 222 (1979), 1â32. V. Hinich . On the singularities of nilpotent orbits, Israel J. Math. 73 (1991), 297â308. K. Hoffman and R. Kunze . Linear Algebra, 2d ed., Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1971. J. Humphreys . Introduction to Lie Algebras and Representation Theory, Graduate Texts in Math. 9, Springer- Verlag, New York, 1972. N. Jacobson . Basic Algebra I. W. H. Freeman and Company, New York, 1985. N. Jacobson . Lie Algebras. Dover, New York, 1962. G. D. James . Representation Theory of the Symmetric Groups, Lecture Notes in Math. 682, Springer-Verlag, New York, 1978. J. C. Jantzen . Einhüllende Algebren halbeinfacher Lie-Algebren. Ergebnisse der Mathematik 3, Springer-Verlag, New York, 1983. D. S. Johnston and R. W. Richardson . Conjugacy classes in parabolic subgroups of semisimple algebraic groups, II, Bull. Lond. Math. Soc. 9 (1977), 245â250. A. Joseph . On the annihilators of the simple subquotients of the principal series, Ann. Sci. Ec. Norm. Sup. (4) 10 (1977), 419â440. A. Joseph . Dixmierâs problem for Verma and principal series submodules, J. London Math. Soc. 20 (1979), 193â204. A. Joseph . Towards the Jantzen conjecture II, Comp. Math. 40 (1980), 69â78. A. Joseph . Goldie rank in the enveloping algebra of a semisimple Lie algebra, I,II, J. Alg. 65 (1980), 269â306. A. Joseph . On the associated variety of a primitive ideal, J. Alg. 93 (1985), 509â523. D. Kazhdan and G. Lusztig . Fixed point varieties on affine flag manifolds, Israel J. Math. 62 (1988), 129â168. G. Kempf . On the collapsing of homogeneous bundles, Inv. Math. 37 (1976), 229â239. G. Kempken . Induced conjugacy classes in classical Lie-algebras, Abh. Math. Sem. Univ. Hamb. 53 (1983), 53â83. A. W. Knapp . Lie groups, Lie algebras, and cohomology, Math. Lecture Notes 34, Princeton University Press, Princeton, 1988. B. Kostant . The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973â1032. B. Kostant . Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327â404. B. Kostant and S. Rallis . Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753â809. H. Kraft . Parametrisierung von Konjugationsklassen in s l n , Math. Ann. 234 (1978), 209â220. 182 H. Kraft and C. Procesi . On the geometry of conjugacy classes in classical groups, Comm. Math. Helv. 57 (1982), 539â602. G. Lusztig . A class of irreducible representations of a Weyl group, Indag. Math. 41 (1979), 323â335. G. Lusztig . A class of irreducible representations of a Weyl group II, Indag. Math. 44 (1982), 219â226. G. Lusztig . Characters of Reductive Groups over a Finite Field. Annals of Math. Studies 107, Princeton University Press, Princeton, 1984. G. Lusztig . Intersection cohomology complexes on a reductive group, Inv. Math. 75 (1984), 205â272. G. Lusztig . Sur les cellules gauches des groupes de Weyl, C. R. Acad. Sci. Paris (A) 302 (1986), 5â8. G. Lusztig and N. Spaltenstein . Induced unipotent classes, J. London Math. Soc. 19 (1979), 41â52. H. Matsumura . Commutative Ring Theory, tr. by M. Reid . Cambridge University Press, Cambridge, 1986. S. J. Mayer . On the characters of the Weyl group of type C, J. Alg. 33 (1975), 59â67. S. J. Mayer . On the characters of the Weyl group of type D , Math. Proc. Camb. Phil. Soc. 77 (1975), 259â264. W. McGovern . Rings of regular functions on nilpotent orbits and their covers, Inv. Math. 97 (1989), 209â217. W. McGovern . Unipotent representations and Dixmier algebras, Comp. Math. 69 (1989), 241â276. W. McGovern . Dixmier algebras and the orbit method, in Actes du colloque en lâhonneur de Jacques Dixmier. Progress in Math. 92, Birkhäuser, Boston, 1990, 397â416. G. D. Mostow . Some new decomposition theorems for semi-simple Lie groups, Mem. Amer. Math. Soc. 14 (1955), 31â54. T. Ohta . Closures of nilpotent orbits in classical symmetric pairs and their singularities, Tohoku Math. J. (2) 43 (1991), 161â213 H. Ozeki and M. Wakimoto . On polarizations of certain homogeneous spaces, Hiroshima Math. J. 2 (1972), 445â482. R. W. Richardson . Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London Math. Soc. 6 (1974), 21â24. R. W. Richardson . Derivatives of invariant polynomials on a semisimple Lie algebra, in Proceedings, Miniconference on Harmonic Analysis 15 (1987), Australian National University, Canberra, 1987, 228â242. J. Sekiguchi . Remarks on nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39 (1987), 127â138. J. P. Serre . Linear Representations of Finite Groups, tr. by Leonard Scott , Graduate Texts in Math. 42, Springer-Verlag, New York, 1977. 183 N. Spaltenstein . Classes Unipotentes et Sous-Groupes de Borel, Lecture Notes in Math. 946, Springer- Verlag, New York, 1982. N. Spaltenstein . Nilpotent orbits and conjugacy classes in Weyl groups, Astérisque 168 (1988), 191â217. T. A. Springer . A construction of representations of Weyl groups, Inv. Math. 44 (1978), 279â293. T. Springer . Linear Algebraic Groups. Progress in Mathematics 9, Birkhauser, Boston 1981. T. Springer and R. Steinberg . Conjugacy Classes, in Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math. 131, Springer-Verlag, New York, 1970, 167â266. R. Stanley . Enumerative Combinatorics. Advanced Mathematics Series, Wadsworth and Brooks/Cole, Pacific Grove, Calif. 1986. R. Steinberg . Conjugacy Classes in Algebraic Groups. Lecture Notes in Math. 366, Springer-Verlag, New York, 1974. V. Varadarajan . Lie Groups, Lie Algebras, and their Representations., 2d ed., Springer-Verlag, New York, 1984. D. Vogan . The orbit method and primitive ideals for semisimple Lie algebras, in Lie Algebras and Related Topics, Canad. Math. Soc. Conf. Proc. 5, Amer. Math. Soc. for CMS, Providence, 1986, 281â319. D. Vogan . Dixmier algebras, sheets, and representation theory, in Actes du colloque en lâhonneur de Jacques Dixmier. Progress in Math. 92, Birkhäuser, Boston, 1990, 333â397. D. Vogan . Associated varieties and unipotent representations, in Proceedings, Harmonic Analysis on Reductive Groups, Bowdoin College. Progress in Math. 101, Birkhäuser, Boston, 1991. G. Wall . On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austr. Math. Soc. 3 (1963), 1â62. N. Wallach . Symplectic Geometry and Fourier Analysis. Math Sci Press, Brookline, Massachusetts, 1977. F. Warner . Foundations of Differential Manifolds and Lie Groups., 2d ed., Springer-Verlag, New York, 1983. G. Warner . Harmonic Analysis on Semi-simple Lie groups I,II. Springer-Verlag, New York, 1972.