Progress in Mathematics Volume 123

Series Editors J. Oesterle A. Weinstein Lie Theory and Geometry In Honor of Bertram Kostant

Jean-Luc Brylinski Ranee Brylinski Victor Guillemin Victor Kac Editors

Springer Science+Business Media, LLC Jean-Luc Brylinski Ranee Brylinski Department of Mathematics Department of Mathematics Penn State University Penn State University University Park, PA 16802 University Park, PA 16802

Victor Guillemin Victor Kac Department of Mathematics Department of Mathematics MIT MIT Cambridge, MA 02139 Cambridge, MA 02139

Library of Congress Cataloging In-Publication Data Lie theory and geometry : in honor of Bertram Kostant / Jean-Luc Brylinski... [et al.], editors. p. cm. - (Progress in mathematics ; v. 123) Invited papers, some originated at a symposium held at MIT in May 1993. Includes bibliographical references. ISBN 978-1-4612-6685-3 ISBN 978-1-4612-0261-5 (eBook) DOI 10.1007/978-1-4612-0261-5 1. Lie groups. 2. Geometry. I. Kostant, Bertram. II. Brylinski, J.-L. (Jean-Luc) III. Series: Progress in mathematics : vol. 123. QA387.L54 1994 94-32297 512'55~dc20 CIP

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ISBN 978-1-4612-6685-3

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987654321 Bertram Kostant Table of Contents

Preface . . . . . ix

Normality of Some Nilpotent Varieties and Cohomology of Line Bundles on the Cotangent Bundle of the Flag Variety Bram Broer 1

Holomorphic Quantization and Unitary Representations of the Teichmiiller Group Jean-Luc Brylinski and Dennis McLaughlin ...... 21

Differential Operators on Conical Lagrangian Manifolds Ranee Brylinski and Bertram Kostant ...... 65

Groups and the Buckyball Fan R. K. Chung, Bertram Kostant, and Shlomo Sternberg . .... 97

Spinor and Oscillator Representations of Quantum Groups Jintai Ding and Igor B. Frenkel...... 127

Familles coherentes sur les groupes de Lie semi-simples et restriction aux sous-groupes compacts maximaux Michel Duflo and Michele Vergne ...... 167

The Differential Geometry of Fedosov's Quantization Claudio Emmrich and Alan Weinstein . ... 217

Closedness of Star Products and Cohomologies Moshe Flato and Daniel Sternheimer .... 241

The Algebra of Chern-Simons Classes, the Poisson Bracket on it and the Action of the Gauge Group Israel M. Gelfand and Mikhail M. Smirnov ...... 261

A Distinguished Family of Unitary Representations for the Exceptional Groups of Real Rank = 4 Benedict H. Gross and Nolan R. Wallach .... 289 Reduced Phase Spaces and Riemann-Roch Victor Guillemin ...... 305 viii Table of Contents

The Invariants of Degree up to 6 of all n-ary m-ics Roger Howe ...... 335

Tensor Products of Modules for a Vertex Operator Algebra and Vertex Tensor Categories Yi-Zhi Huang and James Lepowsky ..... 349

Enveloping Algebras: Problems Old and New Anthony Joseph ...... 385

Integrable Highest Weight Modules Over Affine Superalgebras and Number Theory Victor G. Kac and Minoru Wakimoto ...... 415

Quasi-Equivariant V-Modules, Equivariant Derived Category, and Representations of Reductive Lie Groups Masaki Kashiwara and Wilfried Schmid ...... 457

Meromorphic Monoidal Structures David Kazhdan ...... 489

The Nil Hecke Ring and Singularity of Schubert Varieties Shrawan Kumar 497

Some Classical and Quantum Algebras Bong H. Lian and Gregg J. Zuckerman 509

Total Positivity in Reductive Groups George Lusztig ...... 531

The Spectrum of Certain Invariant Differential Operators Associated to a Hermitian Symmetric Space Siddhartha Sahi...... 569

Compact Subvarieties in Flag Domains Joseph A. Wolf ...... 577 The Work of Bertram Kostant

Over a period of more than four decades Kostant has produced some extremely creative, important and influential mathematics. It continues to be very striking to all of us that, among his many papers, virtually all are pioneering with deep consequences, many an article spawning a whole field of activity. While there have been many papers that either further develop or generalize papers by him, few have improved on his treatment of any topic. Most of Kostant's work is related to Lie theory and its many facets, including algebraic groups and invariant theory, the geome• try of homogeneous spaces, representation theory, geometric quantization and symplectic geometry, Lie algebra cohomology, Hamiltonian mechanics, modular forms, etc... In fact, one striking consequence of Kostant's body of work is that Lie theory (and symmetry in general) now occupies a larger role in mathematics than it did before him. Kostant's depth of range ex• tends to major papers in algebra and mathematical physics, and includes a number of unpublished papers which have also been very influential. Much of Kostant's mathematics demonstrates the value and the im• portance of achieving the necessary level of generality at the same time as new ground is broken. By the same token, however, his work illustrates his own belief that there are certain objects and structures in mathematics which are so beautiful and so fundamental that they justify many years of effort to unearth their hidden treasures. The present volume is a tribute to Bert Kostant. The 22 high level articles assembled here reflect the diversity of his influence on many fields. While the excellence of the articles will speak for themselves, we will at• tempt to give an overview of Kostant's work to date (early July, 1994). We have chosen to divide the exposition according to the decades of his scien• tific activity, as this seems to lead to a more lively picture of the ensemble of his work. We have attempted to at least illustrate the variety of the papers, their origin, their influence on mathematics (and also on physics); we have not tried to discuss the works of other mathematicians except to the extent that they are directly related to or have influenced Kostant's work and his thinking. No doubt much has been omitted and many con• nections have escaped our notice. We hope that this overview can serve as a guide, as experience shows again and again that so much is gained from reading the original source of a discovery, and Kostant's papers are very well-written. There was a Symposium in honor of Kostant at MIT in May, 1993. Some (but hardly all) of the papers in this volume grew out of talks x Preface

given there. The Symposium, entitled "Lie Theory, Algebra and Geomet• ric Quantization", seems to be a fitting descriptive synthesis of Kostant's works. This summary was prepared by the Editors. We thank David Vogan for giving us his account of Kostant's work in representation theory up to the mid 1970s. Any inaccuracies are of course the responsibility of the Editors.

50s

Some of Kostant's first papers ([1] [4] [5]) are devoted to a study of the holonomy groups of homogeneous spaces. Kostant described the Lie al• gebra of the holonomy group in terms of certain operators Ax associated with Killing fields X. In [4] and [5] the holonomy group is described for a large class of homogeneous spaces. In [10] Kostant gives a characteri• zation of those affine connections which admit locally a transitive group of connection preserving transformations. These results are reproduced in the treatise of Kobayashi and Nomizu. In [3] it is proved that for any Rie• mannian metric on the sphere sn, the holonomy group is always the full rotation group. This basic result is derived from a representation-theoretic fact proved in [10], namely, that if a subgroup G of SO(n) (for n ~ 5) has no invariants in I\i jRn for all 0 < i < n, then G = SO(n). Kostant gave in [2] a complete classification of real Cartan subalgebras of a real simple Lie algebra. The classification is done in terms of suitable families of orthogonal roots. In the second part of [2] there are complete tables for all the Cartan subalgebras. This second part was widely circu• lated, but remained unpublished as the PNAS complained that the tables were too complicated! The first (and independent) publication of the lists was in a paper by Sugiura (the lists of Kostant and Sugiura coincide). In [8] Kostant gave his famous formula for the multiplicity of a weight J.L in a finite dimensional representation V,x of a simple Lie algebra g. The mul• tiplicity is the alternating sum over w in the Weyl group of the expression S;P(w· (,X + p) - (J.L + p», where s;P denotes the Kostant partition function, which counts the number of expressions of a weight as a sum of posi• tive roots. The Kostant multiplicity formula is a precursor to the theory of Verma modules and to the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional representation. It served as a model for many subse• quent character formulas (for instance, Blattner's formula). The paper [8] also contains some other curious identities, which deserve to be studied (see the review of [8] by Freudenthal). There is much work of Kostant on generalizations of [8] which has remained unpublished - notably he has a formula for the restriction of a representation to an arbitrary reductive Preface xi subgroup, and another result deals with a subalgebra which is the fixed point algebra of some involution. In [7] Kostant gave a beautiful infinitesimal characterization of the standard representations of the classical groups; they are characterized by the fact that the Lie algebra representation contains operators of rank 1 or 2. Combined with his unpublished results mentioned above, this led him to prove that the compact pairs (SU(n), SU(n -1)), and (SO(n), SO(n -1)) are the only ones such that any irreducible representation of the big algebra restricts to a multiplicity free representation of the small algebra. In the 50s Kostant wrote the first [9] of a series of fundamental papers on Lie theory which have changed the subject. In [9] Kostant studied the principal three-dimensional subalgebra (TDS) .6 (which was introduced by Dynkin and by Freudenthal) of a complex simple Lie algebra g, and described the structure of g as a representation of .6 under the adjoint action. This .6-module structure is related to the Betti numbers of the corresponding Lie group G and to the heights of the roots of g. Kostant was thus able to give a theoretical proof of an empirical observation of Shapiro and Steinberg, relating these two sets of quantities. More precisely, g decomposes under .6 as the direct sum of representations of dimension 2mi + 1, where the mi's are the exponents of g. Here the exponents are defined in such a way that ili (1 + t 2mi +l) is the Poincare polynomial of G. Equivalently, if (e, h, J) is the standard basis of the Lie algebra .6, the mi are the eigenvalues of h on ge. The principal TDS is nowadays a standard tool in Lie theory. Another important new object introduced in [9] is a distinguished con• jugacy class A of regular elements of order equal to the Coxeter number h; the elements of this class restrict to the Coxeter element, which is the product of the simple reflections in the Weyl group, on a suitable Cartan subalgebra. The exponents of G are described from the adjoint action of the Coxeter element on g. Kostant gives a uniform theoretical proof of the equality hl = 2r where h is the Coxeter number, I is the rank, and r is the number of positive roots; this equality was observed empirically by Cole• man. The conjugacy class A turns out to be of fundamental importance in a variety of domains, as evidenced by the work of Kostant in the 70s. It is proved in [9] that any regular element of G has order at least h, and any element of G of order exactly h belongs to the Coxeter class. The paper [9] was presented by Koszul in Bourbaki Seminar no. 191. In the 50s Kostant was also actively working on Lie algebra cohomol• ogy theory. While studying the G-invariants in f\P g* which correspond to primitive cohomology classes in H 2P(BG), Kostant found a strange identity which he related to old results of Frobenius on the representation theory xii Preface

of the alternating group [6]. This identity turned out to be the Amitsur• Levitski theorem, which says that the the skew-symmetrized 2n-fold prod• uct of 2n matrices of size (n, n) is O. Kostant also gave in [6] a new identity for skew-symmetric matrices. The paper [6] was the subject of Bourbaki Seminar no. 243 by Dieudonne. It is probably in the late 50s that Kostant proved the important theo• rem that any orbit of a unipotent algebraic group acting on an affine alge• braic variety is closed. The proof can be found in M. Rosenlicht "On quo• tient varieties and the affine embed dings of certain homogeneous spaces", Trans AMS 101 (1961), 211-223; cf. also D. Birkes, Ann. of Math. 93 (1971), 459-475, and Rosenlicht's book Algebraic Groups.

60s

In 1961 Kostant published the second in his series of major papers on Lie theory [11]. The paper [11] establishes a firm link between representation theory and Lie algebra cohomology. For any parabolic subalgebra p of g, with nilpotent radical m and Levi subalgebra gl, Kostant determines the action of gl on the Lie algebra cohomology HP(m, VA»' The representations of gl occur with multiplicity one, and the representations 1Tu which do occur are parametrized by a E WI, where WI c W is a subset of the Weyl group. This implies both a generalization of the Weyl character formula and its interpretation in terms of Lie algebra cohomology. Lie algebra cohomology was to play an ever increasing role in representation theory. Lie algebra cohomology is fundamental in the algebraic representation theory of semisimple Lie algebras, starting with Vogan's algebraic classification theory for Harish-Chandra modules. It is central as well in the localization theory of U(g)-modules (where the fibers of V-modules are Lie algebra cohomology groups). The Hecht-Schmid character formula is expressed in terms of Lie algebra cohomology, very much in the spirit of [11]. The formula of Kostant for Hi(m,VA ) made its appearance in the theory of automorphic forms and of locally symmetric spaces in Zucker's paper on L2-cohomology and intersection cohomology; it is now an important tool in this field. The method of proof of Kostant's formula is very interesting, as it uses methods similar to Hodge's in Lie algebra cohomology. Finally, [11] has been generalized by Garland and Lepowsky to Kac-Moody Lie algebras; see their article in Invent. Math. 34 (1976) for this generalization and its applications to the Macdonald-Kac formula. A version of Lie algebra cohomology in the affine Lie algebra case (mostly in degree 0) is used in constructing the conformal blocks in the WZW (Wess-Zumino-Witten) conformal field theory, and in the construction of tensor categories due to Preface xiii

Kazhdan and Lusztigj see the article by Kazhdan in this volume for more recent developments. It is also a crucial standard tool in gauge theories and in string theory, namely BRS theory, on which Kostant and Sternberg worked in the 80s. The paper [16], which is a continuation of [11], solves the mystery dimHi(n) = dimH2i(GIB), where n is the nilpotent radical of b. This equality had been proved by Bott using topological methods and the Riemann-Roch theorem, and the question arose to find a direct proof. Kostant introduced a new kind of laplacian with the help of which he set up an isomorphism between [Hi(m)®Hi(m)*)]91 and H2i(GIP). Both are parametrized by WI j moreover, Kostant proved the remarkable fact that the class in H2i (G I P) of the Schubert cycle Zn associated to any a E WI corresponds under this isomorphism to the canonical (identity) tensor in 7rn ® 7r; C Hi(m) ® Hi(m)*. This generalizes a classical formula relating the Schubert calculus with multiplication of Schur functions. In particular, Kostant was able to determine which harmonic polynomials on the Cartan subalgebra ~ correspond to Schubert cycles in G I B, under the isomorphism between harmonic polynomials and H* (G I B). The paper [15] (announced in [14]) is another milestone in Lie theory and geometry, and its many fundamental results are of constant use. One main construction is that of the graded space H of harmonics inside the symmetric algebra of a semisimple Lie algebra g, as the space annihilated by all positive degree invariant constant coefficient differential operators. One of the main results in [15] is the tensor splitting 8(g) = I ® H, where I is the algebra of invariants (which is a polynomial algebra by Cheval• ley's theorem). [15] also gives the analogous splitting for U(g). Kostant proved in [15] that H is the space spanned by all powers xk of nilpotent elements of g. Under the action of g, H decomposes with finite multipli• cities, and each irreducible finite-dimensional representation V occurs with multiplicity equal to the dimension of its zero-weight space. Kostant used sophisticated commutative algebra to prove that the nilcone 91 in 9 (the cone of nilpotent elements) is a normal variety. An important point here is the use of the cross-section e + gf for the set of regular (Le., maximal dimensional) adjoint orbits, where (e, h, J) is the basis of a principal TDS. The normality is an extremely important result. It implies that all regular functions on the unique orbit Oe of regular nilpotents extend to regular functions on 91 so that R(Oe) = R(91). Moreover it implies that the sym• metric algebra 8(g) maps onto the algebra R(O) of regular functions for each regular orbit 0, and that R(O) identifies with H as a representation of g. The surjectivity implies easily that the global sections of the sheaf V G / B of differential operators over GIB is a quotient of U(g), a fact first xiv Preface

noted by J.-L. Brylinski and Kashiwara, which is important to their so• lution of the Kazhdan-Lusztig conjecture and of course also to the more general Beilinson-Bernstein localization theory. Kostant also established that the G-orbits in !Jt (the so-called "nilpo• tent orbits") are finite in number. Kostant's results inspired much further work on the geometry of nilpotent orbits, due to Barbasch, Borho, Broer, J.-L. Brylinski, R. Brylinski, DeConcini, Ginsburg, Graham, Guillemin, Hesselink, Hinich, Joseph, Kraft, Levasseur, Lusztig, MacPherson, McGov• ern, D. Peterson, Procesi, Richardson, Slodowy, Smith, Stafford, Sternberg, Vogan, Wolf and many others. The normality of nilpotent orbits was proved for sl(n) by Kraft and Procesi, Closures of conjugacy classes are normal, Invent. Math. 53 (1979), 227-247, but for other types there are non-normal nilpotent orbits. The well-known Springer correspondence between nilpotent orbits and representations of the Weyl group is strongly connected to Kostant's work in [151. Furthermore the moment map T*G / B - 9 defines Springer's resolution of singularities T*G / B - !Jt. It follows using again the normality of!Jt that R(!Jt) = R(T*G/ B). Building on [91, Kostant introduced in [15] the notion of generalized ex• ponents for a finite-dimensional representation V as the eigenvalues Ai (V) e of h on the space vg • By [9], for the adjoint representation one recovers the usual exponents of g. The generalized exponents of V are equal (with the correct multiplicities) to the degrees in which V occurs in H. Hesselink and D. Peterson (independently) proved that the generating function of the generalized exponents is, up to a simple shift, equal to the q-analog of O-weight multiplicity, where the latter is defined by substituting a natural q-analog of Kostant's partition function into Kostant's weight multiplicity formula. There is a wealth of beautiful results on generalized exponents and on the q-analogs of weight multiplicities, due to Hesselink, D. Peter• son, Lusztig, Kato, R. Brylinski (some of them under her former name R. Gupta) and others. Kostant's work on functions on regular semisimple and nilpotent or• bits and generalized exponents was extended to spaces of "twisted" func• tions (sections of homogeneous line bundles) by R. Brylinski in two articles (in JAMS 2 (1989) and in the Dixmier Festschrift, Birkhiiuser PM 92). There the orbits are considered as varieties fibered over G / B. The reg• ular semisimple orbits are realized as, in Kostant's terminology, shifted cotangent bundles, while of course 0 e C T* G / B is open. Corresponding to each irreducible finite-dimensional g-representation VI' with non-zero zero-weight space Brylinski constructs a natural ideal II' of R(!Jt) with the property that the multiplicity of any VA in II' is equal to the dimension Preface xv of the weight space V,:; there is a more general construction for general V,. using the simply-connected cover of De. Broer subsequently studied in detail these ideals, proved Brylinski's conjecture on the generation of the ideals, and obtained strong results on the vanishing of cohomology groups of line bundles on T*G / B- see Invent. Math. 113 (1993) and his paper in this volume. Furthermore Broer obtained striking results on the subregular orbit very much in the spirit of [15). The paper [15) was presented by Godement in Bourbaki Seminar no. 260. The splitting theorem of Kostant was extended to quantized envelop• ing algebras by A. Joseph and G. Letzter. Reference to this work and a discussion of various problems related to Kostant's work can be found in Joseph's paper in this volume. The paper [29) with S. Rallis (announced in [23) [24)) generalizes many of the results of [15) to symmetric pairs (g, t). Let G be the complex adjoint group, and let Ko C G be the subgroup fixed by the Cartan involution B. They study the action of the complex group Ko on the complexified vector space Pc, and show that the algebra of Ko-invariant polynomial functions on Pc is a polynomial algebra in I homogeneous generators Pi>"" Pl' For each A E C l , we have a closed algebraic subvariety Pc,'\ defined by the ideal generated by the Pi - Ai' The algebraic group Ko acts algebraically on each PC,'\. For general A, this is a smooth subvariety of PC,,\, which is a single Ko-orbit. For arbitrary A, Kostant and Rallis study the structure of each Pc,'\, and show that it decomposes into finitely many Ko-orbits; there is exactly one closed Ko-orbit, consisting of semisimple elements, and one open orbit (which is not always connected). The variety Pc,o is the nilcone, and consists of the nilpotent elements in Pc. Kostant and Rallis prove an analog of the Jacobson-Morozov theorem, which says that any nilpotent e E Pc is contained in a simple three-dimensional subalgebra which is B• stable. Unlike the situation in [15), here the nilcone may have several irreducible components and these components may be non-normal. The paper [29) also gives the generalization to symmetric pairs of the tensor splitting theorem of [15). The papers [15) and [29) have greatly influenced the development of "effective" invariant theory due to Vinberg, Popov, Kac and others. It turns out that if one replaces the involution by an arbitrary automorphism of finite order a of a complex simple Lie algebra gc, the representation of the fixed Lie algebra gc on each eigenspace of a has remarkable properties similar to those established for involutions in [29). Moreover these examples almost exhaust all finite-dimensional representations V of simple Lie groups with nice invariant-theoretic properties (such as freeness of the algebra the xvi Preface

algebra of polynomial invariants, finiteness of the number of orbits in each subvariety VA of V). Furthermore [15] and [29] gave a strong impetus for further work in invariant theory of reductive group actions in the sense of Hermann Weyl. In addition to the mathematicians mentioned above particularly relevant here is the work of Brion, Carrell, Dixmier, Howe, Knop, Littleman, Luna, Vust and many others. Early in the 60s, Kostant started to develop a major theory called geometric quantization, for the purpose of geometrically constructing uni• tary representations of Lie groups. Kostant was influenced by a number of geometric constructions of representations done in the 50s and the early 60s, to wit: - the Borel-Weil-Bott theorem, which in particular constructs finite• dimensional representations of a complex reductive group as the space of holomorphic sections of a line bundle over the flag manifold G / B; - the Bargmann-Segal construction of the Stone-von Neumann repre• sentation of the Heisenberg group; - the constructions of the principal series representations by 1. Gelfand, M. Graev and S. Gindikin, and the construction of the discrete series by Harish-Chandra; - Chevalley's construction of the Spin representation; - Kirillov's complete description of the unitary dual of any unipotent group, in terms of the coadjoint orbits in his paper "Unitary representations of nilpotent groups", Uspekhi Math. Nauk 17, 1962. Soon after Kirillov's paper, Kostant saw that all these constructions could be fitted into a unique theory, and he conceived of his geometric quan• tization program which is summarized in the note [19]. An aspect of the theory which is crucial for applications to representation theory is the obser• vation that coadjoint orbits carry a canonical symplectic form. The skew• symmetric pairing on each tangent space to the orbit was written down in Kirillov's 1962 paper; this is an algebraic construction. The geometric fact is that the resulting 2-form on the coadjoint orbit is a symplectic form, i.e., a closed non-degenerate 2-form. This geometric construction was also part of the program developed independently by Souriau. The idea that for "integral orbits" the symplectic form is the curvature of a line bundle was conceived by Kostant near that time, as were the ideas of prequantization, the topological notion of integrality and its relation with induced represen• tations. Kostant lectured on all this already at MIT in 1964-65. Iwahori's own notes from these lectures were circulated widely both in the US and abroad and were referenced in the book Representations des Groupes de Lie Resolubles by Bernat, Conze, Duflo, Levy-Nahas, Rais, Renouard and Preface xvii

Vergne. Kostant's handwritten lecture notes from this course and from subsequent courses at MIT in the 1960's also circulated widely and had enormous impact on the development of the field from 1966 onward. In fact a copy still resides in the Mathematics Library of the University of Paris 7. Additional lectures by Kostant in this period were written up and cited often: Kostant's note [19) and his Phillips Lectures at Haverford in 1965 (the notes taken by Husemoller from these lectures will appear soon in book form). Kostant also introduced at that time the notion of polarization in its general form. This will be taken up later, because the important paper [28) appears in 1970, but we will illustrate here one significant way in which Kostant's approach differs from a more algebraic one, which was developed by Bernat, Conze, Dixmier, Dufio, Pukanszky, Rais, Vergne and others (cf. Dixmier's book Enveloping Algebras, the book Let;ons sur les Representations des Groupes by L. Pukanszky, and the aforementioned book by Bernat, et al.). For an algebraist, the main ingredient of the orbit method is that to each coadjoint orbit G . >. c g*, there should be attached a Lie subalgebra q containing gA, which is maximal isotropic with respect to the skew-form (this is the algebraic notion of polarization). Then >. gives a Lie algebra character of q, and if one can integrate it to a char• acter X of the corresponding Lie group Q, then the relevant representation of G is obtained by inducing X from Q to G. For Kostant, the notion of polarization means more generally an involutory lagrangian distribution of complex tangent spaces on the coadjoint orbit, and the vector space of the representation is obtained as the space of sections of the line bundle that are annihilated by the vector fields which belong to the distribution. A G-invariant lagrangian distribution of tangent spaces on G· >. ~ GIGA amounts to a GA-invariant subspace q of g containing gA, which is maxi• mally isotropic with respect to the symplectic form. The lagrangian dis• tribution is integrable if and only if q is a Lie subalgebra of g; thus one recovers the algebraic notion of polarization as a special case. In 1970 Kostant's seminal paper [28) appeared, to be followed by [30) [33) [37). The geometric quantization program of Souriau, which has many common points with that of Kostant, is discussed in his article in Comm. Math. Phys. 1 (1966), and his book Structure des Systemes Dynamiques, published in 1970. Souriau's book will soon be translated into English. We will have more to say later about Souriau's work as we present the paper [28). Kirillov developed his own program in the 60's: his paper in Punct. Anal. Appl. 1, no. 4 (1968) gives the geometric construction of the symplectic form on a coadjoint orbit; the article of Kirillov in Funct. Anal. Appl. 2 (1968) gives for nilpotent groups his famous formula expressing xviii Preface

the character of a unitary representation as the Fourier transform of the Liouville measure on a coadjoint orbit. Kirillov' s book Elements of the Theory of Representations was published in Russian by Nauka in 1972 and in English by Springer-Verlag in 1976. Kostant became very interested in Hopf algebras in the middle 60s, un• der the influence of the paper by Milnor and Moore on the classification of graded-commutative and cocommutative Hopf algebras over algebraically closed fields of characteristic zero. Kostant proved an extension of the Milnor-Moore theorem, stating that any cocommutative Hopf algebra over an algebraically closed field of characteristic zero is the smash-product of a group algebra and the enveloping algebra of a Lie algebra. This theorem and its proof were published by Moss Sweedler, who was Kostant's student, in his well-known book Hopf Algebras. This structure theorem was later to playa role in Kostant's definition of a super Lie group. The paper [20] introduces Kostant's Z-form Uz of the enveloping alge• bra of a semisimple Lie algebra. The construction uses the Chevalley basis, and in fact is in some sense dual to Chevalley's construction of algebraic groups over Z, nowadays called the Chevalleygroup schemes over Z. It involves divided powers, which previously were used mostly in algebraic topology, but which were to be a cornerstone of the theory of crystalline cohomology, developed by Grothendieck, Berthelot and Illusie in the late 60s. The theory of quantum groups over Z, which has recently been devel• oped by Lusztig, Kazhdan, S. Gelfand, DeConcini-Kac, Procesi, Andersen• Jantzen-Soergel, and others, can be viewed to some extent as a q-analog of the Kostant Z-form. The study of differential operators over flag vari• eties in positive characteristic, which is in its early stages, uses the Kostant Z-form as its starting point. The papers [12] and [13] give some major results on differential forms and cohomology. A theorem in [12] is that the Hochschild homology of a regular algebra A over a field of characteristic zero identifies with the regu• lar differential forms over A (in the sense of Grothendieck). This beautiful result left open the question of constructing the exterior differential on dif• ferential forms in the context of Hochschild homology. This problem was to be solved by A. Connes around 1983, with his introduction of the operator B and of cyclic cohomology, and dually by Tsygan and by Loday-Quillen, in the context of cyclic homology (which is in fact closer to the work [12]). In [13], Hochschild and Kostant prove that the ordinary cohomology of an affine algebraic group is always computed by the cohomology of the complex of regular differential forms. This was mentioned by Grothendieck as a motivation for his famous theorem that the cohomology of any smooth complex algebraic variety is the hypercohomology of the de Rham complex Preface xix of sheaves with respect to the Zariski topology.

70s

In the 70s Kostant continued to make important strides in representation theory and to develop geometric quantization. Kostant's program on geo• metric quantization and the geometric construction of representations was developed in the address [27] to the IeM in Nice. The paper [28] (which is really a treatise on line bundles and prequantization) was published in 1970 but in fact represents the theory that Kostant had developed since the early 60s. Kostant developed systematically the theory of line bun• dles with connection (L, '\7) over a manifold, and emphasized that the pair (L, '\7) should be considered together. He presents a cohomological descrip• tion of the group of isomorphism classes of such pairs (L, '\7), using Cech cohomology. The sort of Cech cohomology that he considered corresponds to the complex of sheaves ~x ~A~, where ~x is the sheaf of smooth C* -valued functions on X, and A~ is the sheaf of smooth complex-valued I-forms. This complex of sheaves is the smooth analog of a complex in• troduced by Deligne a few years later to describe holomorphic line bundles with connection. Kostant then obtains two main general results on line bundles with connections: - a complex closed 2-form w appears as the curvature of some connec• tion on some line bundle if and only if it satisfies the quantization condition: the cohomology class of 2~i is integral; - if w satisfies the quantization condition, then the isomorphism elasses of pairs (L, '\7) with curvature w form a principal homogeneous space under the group Hl(X,C*). Such a pair (L, '\7) is called a quantum line bundle. Kostant then considers a symplectic manifold (X,w) and studies the problem of lifting a group G of symplectomorphisms to a group of diffeo• morphisms of the quantum line bundle L, which preserve the connection. This is the geometric stage of the prequantization method. Kostant first examines the infinitesimal problem, and finds (for X simply-connected) that there is a central extension of the Lie algebra H am(X) of Hamilto• nian vector fields, which lifts to a Lie algebra of vector fields on the line bundle. He then shows that the problem of lifting the action of g = Lie( G) ean be deeomposed in two parts: (1) first the action of g should be Hamiltonian; (2) then the central extension of II am(X), pulled-back to g, should split. Analyzing (1), Kostant is led to the concept of a moment map, and he xx Preface proved that the action is Hamiltonian if and only there exists a moment map for the central extension of g. He showed that for 9 finite-dimensional and semisimple, (1) and (2) automatically hold. These results are also found in the book Structure des Systemes Dyna• miques of J.-M. Souriau. However Kostant went further by showing that the obstruction to the problem of lifting the group G to a group of diffeo• morphisms of L is controlled by a central extension G of G. This beautiful result and the construction were found again in later years by many other authors, few of whom attribute it to [28]. The last step of the prequantization program consists in giving a canon• ical action of G on the space of sections of L. This representation will be unitary if the manifold is compact and the pair (L, V') has a hermitian struc• ture. There is very beautiful formula for the infinitesimal action, which is given in [28] and in the book of J.-M. Souriau; for the case of a cotangent bundle, this formula specializes to a formula of I. Segal. For the construction of representations of G in the spirit of the orbit method the relevant symplectic manifolds are the coadjoint orbits and their coverings. Kostant shows in [28] that these are exactly all the homogeneous symplectic manifolds for which the group action is Hamiltonian. Kostant then analyzes connections on a homogeneous line bundle over a coadjoint orbit G . A. Such a homogeneous line bundle corresponds to a character X of the stabilizer GA. The existence of such a group character implies an integrality condition on the corresponding Lie algebra character, which Kostant relates to his quantization condition on the curvature of a line bundle. To achieve true quantization, there remains the question of dividing by two by the number of variables on the orbit. For that purpose, one needs a polarization in the sense of Kostant, meaning a distribution of lagrangian subspaces on the symplectic manifold X. The quantization space should then, according to [26] [27] [28] [33], be the space of sections of L which are killed by any V' e, where the vector field e belongs to the distribution; this geometric idea achieves the halving of the variables. An example of a polarization is given by the tangent spaces to the fibers of a fibration X ---? M whose fibers are lagrangian. Such a lagrangian fibration is necessarily a shifted cotangent bundle in the sense of Kostant. In fact, Kostant more generally considers an involutory distribution V of complex lagrangian subspaces inside the complexified tangent bundle; it is necessary here to assume that V + V is also involutory. For instance, a Kaehler manifold admits such a complex polarization, and this already occurs in the Borel-Weil-Bott theorem, as well as non-positive complex polarizations. The advantage of considering several types of polarizations is illustrated for P'reface xxi

instance by work of Jeffrey-Weitsman and others on the moduli spaces of representations of the fundamental group of a surface. The paper [3~] with Auslander was a major achievement of geometric quantization, in which a complete description of the unitary dual of a solv• able Lie group of type I was obtained. The ideas of geometric quantization were really necessary there, since there are non simply-connected orbits, for which, by Kostant's theory, there will be several associated quantum line bundles. Furthermore Auslander and Kostant solved the problem of characterizing the solvable groups which are of type I: G is of type I if and only any A E g* is integral and its orbit is locally closed in g*. In [33] and [37] Kostant refines geometric quantization by the intro• duction of the half-form bundles. Kostant is thus able to explain the shift by ! .fi between the actual eigenvalues of the harmonic oscillator and those predicted by the WKB method. This "metaplectic correction" in geometric quantization is described in Chapter 10 of Woodhouse's book Geometric Q'uantization, Oxford Univ. Press, 1991. In [33] and [37] Kostant further developed the symplectic spinor bundle and applied it to geometric quan• tization. As symplectic spinors contain various half-form bundles, they can be used to define, pair and differentiate half-forms. Kostant's ideas were developed further by Guillemin and Sternberg. They form a very ele• gant framework for the kernels appearing in H6rmander's Fourier integral operators and for the Maslov index. Although the constructions of the Hilbert space in geometric quantiza• tion require the choice of a polarization, part of the philosophy of Kostant is that, at least in the good cases which occur in representation theory, the Hilbert space should be independent of the polarization. In the mid 70s Kostant found a mechanism for actually proving in many cases this independence of polarization. The isomorphism (which makes crucial use of the half-form bundles) is obtained by means of a pairing called the BKS pairing (after Blattner-Kostant-Sternberg), and which is defined geometri• cally. A nice example of a BKS pairing, which is already present in [3~], gives the standard isomorphism between the Fock representation and the Heisenberg representation. The book Geometric Asymptotics by Guillemin and Sternberg contains a discussion of joint work by Blattner, Kostant and Sternberg. Holomorphic methods of geometric quantization are discussed in the article of J-L. Brylinski and McLaughlin in this volume. The papers by Emmrich-Weinstein and by Flato-Sternheimer discuss another approach to quantization, usually called deformation quantization. It is very interesting to see that connections between symplectic geom• etry (more precisely, Poisson brackets) and secondary characteristic classes xxii Preface

are investigated in the article by Gelfand and Smirnov in this volume. Motivated by the study of moment mappings undertaken in [28], Kostant described the moment map for the adjoint action of a maximal torus T on an adjoint orbit 0 of a compact Lie group G (note that there is a G-equivariant isomorphism between 9 and g*, so an adjoint orbit may be viewed as a coadjoint orbit). The moment map 7r is then given by the projection of an element of 0 to its component in t. Kostant proved his famous convexity theorem, which says that the image of 7r is the convex hull of the finite set 0 n t. For the case of SU(n), this recovers a theorem of Horne on the diagonal part of hermitian matrices. This convexity theorem was generalized in the 80s by Atiyah-Bott and by Guillemin-Sternberg to any Hamiltonian action of a torus on a compact symplectic manifold. Atiyah and Bott showed that the theorem can also be deduced from the work of Duistermaat and Heckman on asymptotic ex• pansions of oscillatory integrals. Guillemin and Sternberg gave interesting asymptotic results on the repartition of weights in a pencil Vn 'A of finite• dimensional representations, which converges weakly to the push-forward to t ~ t* by the moment map of the Liouville measure of the coadjoint orbit of A. This is a sort of quantitative measure-theoretic version of the Kostant convexity theorem. Kac and D. Peterson showed that the Kostant convexity theorem still holds true for coadjoint orbits in Kac-Moody groups, and recent work of Bloch, Flaschka and Ratiu shows that the theorem extends to the group of area preserving diffeomorphisms of an annulus. The paper [32] contains a second (more sophisticated) convexity the• orem; this is roughly speaking a non-linear analog of the first theorem, in which the linear projection t ~ t is replaced by the projection G ~ A to the factor A in the Iwasawa decomposition G = KAN. This second theo• rem is much harder to prove; symplectic proofs for it have been obtained only in the complex case by Weinstein and by Ginzburg. Recent work of Lu and Ratiu suggests that this theorem should be analyzed in the setting of the "quantum category", using q-groups, q-manifolds, etc .... A third theorem in [32] is a beautiful statement about the 3 sides of a geodesic triangle in a symmetric space of negative curvature, which says that if u is one side of the triangle (viewed as an element of p), and if v is the vector sum of the other two sides, then v lies in the convex hull of the W-orbit of u. This recovers the Goldsen-Thompson inequality tr eX+Y :$ tr eXeY for x and y hermitian matrices. In the mid 70s Marsden-Weinstein used the moment map in their im• portant theory of Hamiltonian reduction. The connection of Marsden• Weinstein reduction with geometric quantization was studied by Guillemin Preface xxiii and Sternberg in the 80s. The article by Guillemin in this volume describes the relation of Marsden-Weinstein reduction with the Riemann-Roch the• orem. Another major preoccupation of Kostant in the 70s is the application of Lie theory to the integrability of Hamiltonian systems. In [45] Kazhdan, Kostant and Sternberg studied the Calogero system (which is described by the potential E~j (qi - qj)-2). This system had been shown to be completely integrable by rather complicated techniques. It is shown in [45] that the Calogero system can be obtained by Hamiltonian reduction from a linear Hamiltonian system, which not only easily implies complete integrability, but also enabled them to write down the trajectories in more or less closed form. One very intriguing aspect of [45] is that it contradicts the then accepted wisdom that reduction simplifies the mechanics, since the opposite is true for the Calogero system. The same feature was later found by Atiyah and Bott to apply to the moduli space of flat bundles on a compact oriented surface, which is obtained by Hamiltonian reduction from the space of connections. Another important feature of [45] is a connection between Howe pairs and Marsden-Weinstein reduction. In [46][47] Kostant derived the complete integrability of the Toda sys• tem (which is described by the potential Ei eq,-qi-l), from a general result which is known nowadays as the Kostant-Symes lemma. In fact, Kostant described a generalized Toda type system associated to any simple Lie al• gebra, with potential Ei eO" the sum running over the simple roots (the classical Toda system corresponds to .G[(n». Kostant identifies the phase space of the Toda system with a set of Jacobi matrices, which in turn identifies with a coadjoint orbit of a Borel subalgebra, to which he applies the Kostant-Symes theorem. To state this important criterion, let g be a finite-dimensional Lie algebra, and let a and b be Lie subalgebras such that g = a EI1 b as vector spaces. Then the Kostant-Symes lemma states

Theorem. Let oX E g* be a linear form which vanishes on [a, a] and on [b, b]. For any function f on g*, denote by 1>. the function on a* given by f>..(l) = f(l + oX) for l E a*. Then if f and g are G-invariant functions on g*, the corresponding functions 1>.. and g>.. on a* Poisson commute (equiv• alently,their restrictions to any coadjoint orbit in a* Poisson commute).

In the application, a is a Borel subalgebra and b is the Lie algebra of a maximal compact subgroup. Kostant then derives the complete integra• bility of the Toda system and integrates it explicitly in terms of the matrix coefficients of the fundamental representations of g. The Kostant-Symes theorem was applied by Adler, van Moerberke, 1. xxiv Preface

Gelfand and others to show the complete integrability of many Hamiltonian systems. Kostant was a pioneer in the 70s in the theory of supermanifolds and of super Lie groups. His theory is exposed in [41]. Motivated by Berezin's (et ai.) work and by Kac's classification of the simple Lie superalgebras, the theory was built up on Kostant's work in the 60s on cocommutative Hopf algebras. The Lecture Notes [41] gives the analogs in the supercate• gory of the Frobenius theorem, of the tangent and cotangent spaces, of de Rham theorem and homogeneous spaces. Symplectic structures (of even type) are also studied, and the super analogs of Hamiltonian mechanics, coadjoint orbits (of even linear functionals) and quantum line bundles are also treated. A very nice feature is that the Cartan-Chevalley construction of the Clifford algebra from the exterior algebra now appeared as a case of geometric quantization in the super category. In this fashion the Clifford algebra appears as an "odd" version of the algebra of differential operators on a manifold. This analogy was to be pushed further by Witten and Get• zler in their local proof of the Atiyah-Singer index theorem. It is interesting to note that the theory of BV quantization involves odd symplectic forms. Kostant's theory of supermanifolds remains one of the most important approaches to the field; other theories were developed later by B. DeWitt and also by Manin for algebraic supermanifolds. Needless to say, superman• ifolds and super Lie groups have been of great importance in theoretical physics since the 1970s. The paper [34] was motivated by a question by 1. Segal; it explains why the space of solutions / of the wave equation 6./ = 0 on compactified Minkowski space is conformally invariant, even though the operator 6. is not invariant under the conformal group SO(4,2). Kostant introduced in [34] the notion of a quasi-invariant differential operator on a flag manifold and classified all these operators. They are parametrized by singular vec• tors in a certain Verma module, and each operator defines a subrepresenta• tion of a principal series representation. [34] should be further studied, and should have applications to differential operators on automorphic forms. During the period where Kostant was making huge strides in geomet• ric quantization, symplectic geometry and Hamiltonian systems, he was also extremely active in pioneering new methods in representation theory. Kostant undertook his famous work on spherical principal series [35], for which he was awarded the Steele Prize of the AMS for an influential pa• per in 1990. The citation, in part, reads "he used algebraic methods to solve completely a problem that has resisted analytic methods, and as a consequence found a simple and powerful construction for new series of representations". We will describe informally the results in [35] using the Preface xxv language of Harish-Chandra modules. A Harish-Chandra module X for a group G is called spherical if the space XK of K-invariant vectors is non• zero. If X is spherical and irreducible, then X K is one-dimensional. The spherical Harish-Chandra modules X(..\) are parametrized by a character ..\ of a maximally split abelian subalgebra n. It was known since the 1950s that the X(..\) are unitary for..\ purely imaginary, and Bruhat showed in his thesis that X(..\) is irreducible for ..\ regular (which means that ..\ has trivial stabilizer in the so-called little Weyl group W). However the essential gen• eral problem to determine when X(..\) is irreducible was open; this is the problem that was completely solved in [35]. In particular Kostant showed that X(..\) is irreducible whenever ..\ is purely imaginary, and more gener• ally when ..\ is in an explicit critical strip. He also showed that the unique irreducible quotient Z("\) of X(..\) admits an invariant Hermitian form if and only if there exists some w E W such that w· ..\ = -'X. Kostant's description of the hermitian form is quite explicit and enables one to prove its positivity by analytic continuation. The methods of Kostant in [35] grew out of the theory of harmonics in [29]. For each K-type 'Y occurring in the harmonics, he constructs a polynomial valued matrix p-r, and was able to compute its determinant. This led to Shapovalov's work on Verma modules. The irreducibility and unitarity problems were expressed in terms of the p-r. A further wonderful property of these matrices is that the multiplicity of a K-type in an irre• ducible spherical representation is the rank of a certain matrix R-r which is a product of two matrices of type p-r. This work of Kostant has had many applications to the theory of sym• metric spaces. It led Helgason to the precise description of the values of ..\ for which the so-called Poisson transform, from a space of hyperfunction sections of a certain line bundle on G / P, to a space of functions on G / K, is injective. This was important in the proof of the Helgason conjecture, due to Helgason in the rank one case, and to Kashiwara, Minemura, Okamoto, Oshima and Tanaka in the general case. Another ground-breaking paper from this period is [36], where Kostant studies the infinitesimal characters which can occur in the tensor product of an infinite-dimensional representation with one of finite dimension. This was the starting point for the theory of translation functors, which was developed very successfully - by Bernstein-Gelfand-Gelfand and by Jantzen in the category 0; - by Knapp, Stein and Zuckerman for Harish-Chandra modules; - by Borho and Jantzen for primitive ideals. These translation techniques have become a fundamental technique in representation theory. In their localization theory Beilinson and Bernstein xxvi Preface

show that the geometric counterpart of a translation functor is the opera• tion of twisting a V~-module over the flag variety by a line bundle. In the paper [44] Kostant solves the question of which representations of a semisimple Lie group G admit a Whittaker model (for a given character X of a maximal nilpotent subalgebra n), or equivalently which representations occur in the Whittaker model, which is the representation of g induced from that character. If V is the dual of an irreducible Harish-Chandra module, then V admits a Whittaker model if and only if its annihilator is a minimal primitive ideal; in that case, the space of Whittaker vectors has dimension equal to the order of the Weyl group. This has applications in the theory of automorphic functions. The paper [44] contains a number of other impor• tant results. For instance, it is shown that higher n-cohomology vanishes for representations which admit a Whittaker model. [44] gives structure theorems for the structure of the n-invariants of U(b) and a splitting the• orem for the enveloping algebra of b. Around that time Kostant obtained results on the cascade of orthog• onal roots and the polynomial ring structure of the center of U(n). These results are referred to in the paper by A. Joseph in Jour. of Algebra, 48 (1977). In the early 70s, following up on his work with Rallis, Kostant came up with his correspondence between nilpotent elements e in Pc and nilpotent elements z E g. The correspondence is as follows: Given e E Pc which is nilpotent, it was proved in [29] that there is a TDS Oc of gc, containing e and stable under (J. Kostant further observed that, after conjugating e by an element of Kc if necessary, one may assume that the TDS is stable under complex conjugation. Thus we obtain a real TDS 0 of g. Then take z to be a nilpotent element of o. Conversely, given a nilpotent z E g, by Jacobson-Morozov z is contained in a real T DB 0 C g. After conjugating o by an element of G if necessary, we may assume that 0 is stabilized by (J. Then e is taken to be a nilpotent element of Oc n Pc. Kostant was able to show that in nearly all, but not in all cases, these give two inverse bijections between Kc-conjugacy classes of nilpotent elements in Pc and G-conjugacy classes of nilpotent elements in g. Kostant's work was not published, but he gave numerous lectures on this correspondence in the early 70s. Several years later Sekiguchi proved the correspondence in all cases; he also established the correspondence in a more general setting in• volving two commuting involutions of a real semisimple Lie algebra; see Sekiguchi's paper in Jour. Math. Soc. Japan. 39 (1987). This is why the correspondence is widely referred to as the Kostant-Bekiguchi correspon• dence. The correspondence is a crucial tool in important recent work of Schmid and Vilonen on the characteristic cycles and associated varieties of Preface xxvii

Harish-Chandra modules. Another unpublished result of Kostant in this period is the determi• nation of the outer derivations of the nilradical n of a Borel algebra; this result is quoted and generalized in G. Leger and E. Luks "Cohomology of the nilradical of a Borel subalgebra of a semisimple Lie algebra H*(n, n)", Trans. AMS, 195 (1974). In [39] Kostant gave a beautiful representation-theoretic version of the MacDonald-Kac formula from the early 1970s, which expresses 1Jdim(o) as a sum over the weight lattice. His formula involves the Coxeter conjugacy class a which was so important already in [9]. To write it down, let P+ be the set of dominant weights; for >. E P+, let X.\ denote the character of the corresponding finite-dimensional representation V.\, and let c(>.) be the eigenvalue of the normalized Casimir operator. Then we have:

1]dim(g) = E n(a)· dim(V.\). qC(.\) • .\EP+

This and other results of [39] were presented in Bourbaki Seminar no. 483 by Demazure. For related "super" developments, see the paper of Kac and Wakimoto in this volume. Some results of [39] are used by Howe in his article appearing in this volume. Fegan and Millman recognized that the above formula for 1Jdim(o) is really a statement about the heat kernel on K; see their paper in Amer. Math. Monthly 93 (1986).

80s

Kostant continued in the 80s to work on geometric quantization. The paper [50] contains the notion of shifted cotangent bundle, which is very impor• tant in geometric quantization as a symplectic manifold X equipped with a smooth submersion X -+ M with lagrangian fibres is necessarily a shifted cotangent bundle of M. Kostant also develops in [50] a new, sharpened, symbol calculus which allows him to recover the correct coadjoint orbit (as opposed to a nilpotent orbit) attached to an ideal in the enveloping alge• bra. These results of Kostant and related results are also discussed in W. Borho-J.-L. Brylinski, Bull. Soc. Math. Fr. (1990). The 80s saw many applications of geometric quantization to the rep• resentation theory of reductive Lie groups. Many of these applications originated in the coherent cohomology of holomorphic vector bundles over some open domains in flag bundles, and were motivated by the geometric quantization program of Kostant and by the Langlands conjecture. These xxviii Preface

applications were developed by Schmid and by Wolf already in the 60s and 70s, and there were important works in the 70s and 80s by them and by Guillemin, Rawnsley, Sternberg and many others. To a large extent, the theory of analytic localization of representations, developed in the 80s by Hecht, Milicic, Schmid and Taylor, is very close to geometric quantization. We will refer the reader to the articles by Kashiwara-Schmid and by Wolf in this volume for further information. In the papers [60J [64J Kostant gave a beautiful geometric construction of the minimal representation of SOC 4,4). This representation H is realized as the kernel of a Laplace type operator acting on the space of sections of some line bundle over S3 x S3. Kostant constructs a hermitian pairing on H via a Radon transform which is closely related to triality. He gives a beautiful description of the restriction of H to the maximal compact subgroupK = SU(2)xSU(2)xSU(2)xSU(2): they form a pencil. Kostant proves that the annihilator of H in the enveloping algebra is the Joseph ideal. Thus H is a so-called minimal representation. The papers [60J and [64J have had enormous influence on the study of minimal representations. The paper [64J also contains the marvelous fact that in dimension 6, and only in dimension 6, is the vanishing of the scalar curvature a con• formally invariant condition. It also discusses a theorem of Biedrzycki (a student of Kostant) on constructing solutions of the Einstein equations on compactified Minkowski space by embedding it into S3 x S3. In the book [BI], Kostant develops the idea that both the gravitational field and the electromagnetic field on compactified Minkowski space M can be obtained from an embedding M "--> X, where X is equipped with a conformal structure of signature (3,3). The gravitational field is obtained from the induced Lorentzian metric on M, and the electromagnetic field on M is obtained from the connection on the normal bundle induced by the Levi-Civita connection on X. There are three different models of confor• mally flat completions of 1R3,3, which are related by triality. In each model there is a natural embedding of M. This sort of Kaluza-Klein picture of gravitation and electromagnetism is called the "Kostant universe" in the book Variations on a theme of Kepler by V. Guillemin and S. Sternberg. The paper [56J of Kostant and Sternberg is a mathematical treatment of the theory BRS quantization (named after 13ecchi, Rouet and Stora), which is of great importance in mathematical physics (in particular, in gauge theory and in string theory). The first result of [56J is a homological construction of the space of functions on the Marsden-Weinstein reduction B of a symplectic manifold M with respect to a Hamiltonian action of a lie group G. The space F(B) of functions on B is the degree 0 cohomology group of the complex Ag* 0Ag0F(M), equipped with the BRS differential Preface xxix

D. Then Kostant and Sternberg make the crucial observation that since a = gEeg* is an orthogonal vector space, one can put on Ag*®l\g a structure of associative algebra, the Clifford algebra of o. Then I\g* ® I\g ® F( M) becomes a Poisson superalgebra. A wonderful result of [56] is that the differential D is given by Poisson bracket with a canonical element e of this Poisson algebra. The total degree in the complex is also given by a Poisson bracket. Then [56] studies the conditions under which {8,8} is a cOinstant (or equivalently, D has square zero), which are quite delicate in the infinite-dimensional case, and which is a case of importance in physics. One new feature of the infinite-dimensional case is that there are many different types of Spin representations. The article of Ding and Frenkel in this volume contains the construction of analogs of the Spin representation for quantum groups. Finally [56] discusses so-called BRS quantization, which involves ten• sOiring with the spin representation of the Clifford algebra of o. BRS coho• mology and its applications in conformal field theory are discussed in the paper of Lian and Zuckerman in this volume. The related theory of vertex operator algebras is the topic of the article by Huang and LepowskYi this theory was developed by 1. Frenkel, Lepowsky and Meurman in connection with the monster group . BRS quantization is further developed in C. Duval, J. Elhadad, G. M. Tuynman, "Comm. Math. Phys. 1126 (1990). The paper [59], with Guillemin and Sternberg, recasts the solution of the Plateau problem by J. Douglas in terms of the action of Diff+ (S1) on the space of vector-valued functions on S1. There are some interesting formulas in [59] which deserve further study. The paper [61] with Sternberg gives a geometric interpretation of the classical Schwartzian derivative of a function in terms of the structure of the torus S1 x S1 under the diagonal action of Diff+ (S1 ). There are two orbits: an open orbit C, diffeomorphic to an open cylinder, and a closed orbit D (the diagonal of 8 1 x 8 1), diffeomorphic to 8 1• There is a natural (up to constant) PSL(2, lR)-invariant measure J.l on C, since C identifies with a coadjoint orbit of P8L(2,lR). Given 7 E Diff+(81), it admits a Radon-Nikodym derivative IT> such that J.l7 = IT· J.l, and IT extends to smooth function on 8 1 x 8 1 which vanishes on D and has vanishing normal derivative along D. Then the Schwartzian derivative of 7 is the second normal derivative of IT along D. This easily gives the cocycle property of the Schwartzian derivative. This work has intriguing connections with string theory and conformal field theory. Kostant proved in [62] an equality expressing, for an arbitrary semisim• pIe Lie group G, the trace of the product of an intertwining operator and a xxx Preface certain bounded operator as a sum over the K-types. The trace is equal to the inverse of the well-known c-function. The resulting equality specializes, in the case of SO(n, 1), to an identity of Kummer expressing certain values of hypergeometric functions in terms of the r function. [62] contains a dis• cussion of the relation of the Kunze-Stein operators with the p"Y matrices of [35], as well as a theorem on the traceability of intertwining operators. Continuing his work on the Amitsur-Levitski theorem in [49], Kostant related the whole subject of that identity to the theory of transgression in the Leray spectral sequence for the fibration G -+ EG -+ BG. There are a number of further unpublished results of Kostant in this direction having to do with a Clifford algebra analog of the Hopf-Koszul theorem which says that the cohomology ring of a Lie algebra is an exterior algebra. An important result, obtained by Kostant around 1980, gives quadratic equations for the closure of the G-orbit of a highest weight vector v in a finite-dimensional irreducible representation VA of a simple Lie group G. Specifically, a non-zero vector u belongs to the orbit G . v if and only if the vector u®u of VA ® VA belongs to the irreducible component V2'A of VA ® VA. Kostant then writes down explicit quadratic equations as follows: if C is the Casimir operator, and y the scalar by which C operates on VA' then u belongs to G· v if and only if C· (u ® u) = y. (u ® u). This a far-reaching generalization of the Plucker equations which characterize decomposable vectors inside I\PCn . In terms of projective geometry, Kostant gives explicit quadratic equations for any flag manifold G / P inside a projective space IP'(VA). This theorem of Kostant is recalled and used in the thesis of D. Garfin• kle, MIT, 1982. It was generalized to affine Kac-Moody groups by Date, Jimbo, Kashiwara and Miwa; the quadratic equations become the Hirota bilinear wave equations for the T-function, which characterize the solu• tions to the KdV type hierarchies. These equations play a role in the Kac-Peterson theory of Kac-Moody groups. The results of Bernstein-Gelfand-Gelfand and of Demazure on the co• homology and K-theory of flag manifolds were generalized to the case of an arbitrary Kac-Moody group G in the two papers [55] [58] in collaboration with S. Kumar. The Kac-Moody groups considered in [55] [58] are those constructed by Kac and Peterson. A main object introduced in [55] is the nil Heeke ring R, which is a subring of the smash-product of the group ring of W with the field of rational functions on a Cartan subalgebra ~. [58] contains a structure theorem for the ring R. The cohomology of the flag manifold G / B is described combinatorially in terms of the ring Rand its "dual" A, which is defined explicitly in terms of the Coxeter group W acting on ~. In the paper [58] Kostant and Kumar introduce another ring Preface xxxi

Y inside the smash-product of Z[W] with the field of rational functions on the maximal torus T, and its dual W. They prove that W identifies with the equivariant K-theory Kr(GIB). They can then describe explic• itly various operations on equivariant K-theory analogous to those studied by Demazure in the finite-dimensional case, and describe the equivariant K-theory of the Bott-Samelson-Demazure varieties. Kostant and Kumar prove that the K-theory K(GIB) itself is a quotient of Kr(GIB) which corresponds to a specific specialization of A. The work of Kostant and Kumar has had a vast influence on the study of flag manifolds and Schubert varieties, including the finite-dimensional case. Kumar used [58] in his study of characters of affine Kac-Moody groups; similar results were obtained independently by O. Mathieu. A. Arabia has studied the T -equivariant cohomology of G I B and of Schubert varieties. Kumar has used the nil Hecke ring in developing criteria for smoothness and rational smoothness of Schubert varieties. The paper of Kumar in this volume represents his further work, and contains references to other important works by Deodhar, Carrell, D. Peterson, Polo, M. Dyer, and others. It should also be noted that equivariant K-theory and equivariant co• homology have become a major instrument of geometric quantization and of representation theory. We will mention the work of Borho, J-L. Brylinski and MacPherson on primitive ideals, the work of Kazhdan-Lusztig and of Ginzburg on representations of affine Hecke algebra and of p-adic groups, the work of Berline-Vergne, of Rossmann and of Duflo-Heckman-Vergne on the Kirillov conjecture and the work of Duflo-Vergne (and also Kumar) on equivariant cohomology, invariant distributions and characters of represen• tations, for which we will refer the reader to the article of Duflo and Vergne in this volume. In the early 80s Kostant found a beautiful representation-theoretic framework for the McKay correspondence between finite subgroups of 8L(2, q (up to conjugacy) and simply-laced complex simple Lie algebras. Recall that for a finite subgroup r of 8L(2, C), one can parametrize the irreducible representations 7I"i of r by the vertices or simple roots ai of the extended Dynkin diagram of a simple Lie algebra g, in such a way that if Cij is the affine Cartan matrix, the tensor product by the tautological two-dimensional representation 71" of r is given by the formula

71" ® 7I"i = L (2 - C ij)· 7I"j. j

On the other hand the Dynkin diagram in question can be recovered from r by means of algebraic geometry, as was observed by M. Artin in xxxii Preface

the 70s. The quotient algebraic variety Y = ((:2/r is a normal algebraic surface with an isolated singular point 0, and there is a minimal resolution of singularities 71" : Y -+ Y. The "exceptional fibre" 71"-1(0) is a union of rational curves D i , which correspond to the vertices of the Dynkin diagram, in such a way that the Cartan matrix is Cij = -Di . D j , where Di n Dj denoted an intersection number in Y. Kostant's work in [53] goes a long way toward unifying these two points of view. To explain some of the results, choose the parametrization of the 7I"i in such a way that 71"0 is the trivial representation. It is known from algebraic geometry that the generating function of the graded algebra of r- invariants in S«((:2) is equal to the quotient ( 1 ;/h b)' where h is the I-ta I-t Coxeter number and a, b are integers such that a+ b = h+ 2 and a· b = 2 ·Irl. Kostant studies the generating function Pi(t) for the multiplicities of any 7I"i in the symmetric powers sn«((:2). Kostant proves that

where Zi(t) is a polynomial which is determined explicitly as follows: Let CT be a Coxeter element in Wand let D. be the set of roots of g. Then there are l orbits of CT in D., each of which has h elements. For a distinguished choice of CT, the D.i are naturally parametrized by the simple roots (ti (for i > 0). On the other hand there is the height function h on D.. Let

0 is not a branch point one has

Zi(t) = L tn(t/». t/>E4>;

For the branch point of the D and E diagrams, the formula is even simpler. Kostant in fact was always fascinated by the exceptional Lie algebras and groups. This led him to study some amazing embeddings of finite simple groups into exceptional groups, which are the topic of the paper [63]. To give an example, Kostant shows that the 248-dimensional Lie algebra Es decomposes as the direct sum of 31 Cartan subalgebras, which he considers as the 31 points of the projective plane over lFs. The symmetry group PSL(3, 5) of this finite geometry then embeds into the Lie group Es, and this allows Kostant to derive beautiful formulas for the bracket in the Lie algebra in terms of Gauss sums. In a similar fashion, Kostant shows that the 13 points of the projective plane over lFa embed PSL(3,3) into Preface xxxiii

F4 , and the 7 points of the projective plane over 1F2 embed PSL(3, 2) into G2 • The paper [63] also contains the conjecture that if G is a simple compact Lie group, with Coxeter number, for which 2h+ 1 is prime, then the group SL(2,2h + 1) embeds into G; this conjecture is verified in [63] in many cases. The last open case of the conjecture, that of E8 , was solved recently by Griess and Cohen.

90s

Kostant and Sahi found in the papers [66] and [70] a remarkable general• ization of the classical Capelli identity in the context of symmetric spaces. They give a new interpretation of that identity based on Jordan algebras. For a symmetric space G / K with corresponding Cartan decomposition g := t E9 \" it is known that the algebra of G-invariant differential oper• ators on G / K is isomorphic to the ring S( a) w, where a C \' is a maximal abelian subspace. Now let V be an irreducible finite-dimensional repre• sentation of G which admits a K-invariant vector v. Then G/K embeds into V; if G / K is of co dimension 1 in V, then its dilates span an open set in V, hence any operator in R defines a differential operator on V of degree 0. This co dimension 1 condition is not frequently satisfied, but it certainly holds true (by the Tits-Koecher theory) whenever V has a Jor• dan algebra structure, v is the identity of the Jordan algebra, and G is the "norm-preserving group". Assuming this to be the case, let p E S(V) be the Jordan norm, and let op be the corresponding constant coefficient differential operator. Then p . op is a G-invariant differential operator on V of degree 0. Kostant and Sahi identify this operator with that induced by an explicit element {j of S(a)w. They are then able to compute the eigenvalues of p . op acting on the various K-types in S(V). The original Capelli identity is recovered from the Jordan algebra of n x n-matrices. The identities of [66] [70] have found many applications in the theory of minimal representations, and are now being used by I. Gelfand and others in their work on free algebras. Another approach to proving the Capelli identity and similar identities is given by Sahi in his contribution to this volume. In [67], R. Brylinski and Kostant determine completely the structure of the set Symp(M) of all G-invariant symplectic structures on any homoge• neous space M of G, for G a complex semisimple Lie group. By a theorem of Kostant in [28], any G-homogeneous symplectic manifold is a covering of a coadjoint orbit 0 = G· x. The question reduces to the case where 0 is nilpotent. Some of the main results are: xxxiv Preface

- Symp(M) is naturally isomorphic to Ox = On gX; - Ox is Zariski dense in gX and its complement is a finite union of hyperplanes; - Ox has a group structure, and is isomorphic (up to a finite cover) to the centralizer of G in Diff( M). - for X principal nilpotent, if N(GX) is the normalizer of GX, then

dimN(GX) = 2· dimGx.

In [69] (announced in [68]), R. Brylinski and Kostant discover new phenomena involving nilpotent orbits by combining techniques of algebraic and symplectic geometry. The basic object is a G-homogeneous covering M of a complex nilpotent orbit a so that M is both a complex algebraic variety and a symplectic manifold with a Hamiltonian G-action. Here G is a complex simply-connected semisimple algebraic group with Lie algebra g. The ring R = E9pcoRIP] of regular functions on M is then a graded Poisson algebra where (with respect to the natural G-invariant grading) the Poisson bracket has degree -2 and the Hamiltonian action defines an inclusion g C R[2]. One of the main discoveries is that R "sees" a maximal (algebraic) symmetry group G' which contains the action of G. More precisely, they prove that the space of functions g' = R[2] is a (in fact, the unique maximal) finite-dimensional semisimple Lie subalgebra of R containing g. Let G' be the simply-connected Lie group corresponding to g'. They show that there exists a unique nilpotent orbit A' of G' and a G'-homogeneous cover M' of A' such that M' contains M as an open dense set (with boundary of codimension at least 2) and the infinitesimal Hamiltonian action of g' on M integrates to an action of G' on M'. The two orbits M and M' then have the same ring of regular functions and share many geometric properties; Brylinski and Kostant call (M, M') a shared orbit pair if g i= g'. To construct the G'-action, they consider the variety X = Spec(R). They prove X is a normal affine variety which is equal to the normalization of a in the function field of M. The action of G on X has finitely many orbits and extends to an action of G'. In fact M and M' are, respectively, the unique open dense orbits of G and G' on X. There is a unique G-fixed point 0 in X, and there is G-equivariant embedding X <-> T; X which is dimension-wise a minimal embedding of X into an affine space. Except in the metaplectic case, R[l] = 0 and X is singular; then X is the closure of a coadjoint orbit of G' if and only g' = T; X. There are a number of interesting cases where g' is strictly larger than g; these are completely classified in [69]. A beautiful example occurs when g = sZ(3, q and M is the 3-fold simply-connected cover of the 6-dimensional Preface xxxv principal nilpotent orbit. Then g' is the 14-dimensional exceptional simple Lie algebra of type G2 and M' is the minimal (non-zero) nilpotent orbit in g'. Another striking example provided much of the motivation for [68) [69): Levasseur and Smith, proving a conjecture of Vogan, discovered that the normalization of the closure of 8-dimensional nilpotent orbit M = 08 of G2 is equal to the closure of the minimal nilpotent orbit Oroin of 80(7, C); thus (08 ,Oroin) is a shared orbit pair. In this case 0 8 is the orbit Os of a short vector; Brylinski and Kostant find that for the other doubly-laced simple Lie algebras the simply-connected 2-fold cover of Os lies in a shared orbit pair. They also prove several general results, e.g., g' is a simple Lie algebra if 9 is simple. The full classification of all such pairs (g, g') is related to the well-known occurrence of 2 root lengths for a simple Lie algebra. The study of "hidden symmetries" in [69) has interesting connections with work of Souriau and others on the Kepler problem. The papers [72) [74) [75) of R. Brylinski and Kostant break new ground by showing that the machinery of geometric quantization, suitably gener• alized, works for the so-called minimal nilpotent orbit of a semisimple Lie group and produces a unitary minimal representation of GR (in particular the representation is irreducible and its annihilator is the Joseph ideal). An example is the Fock space model of the metaplectic representation. Vo• gan showed that there are strong restrictions on the K-types of minimal representations. The models of the minimal representations constructed by Brylinski and Kostant are in the spirit of the Fock model, as the ac• tion of K is very natural and explicit in their models. Other models for most of the representations of [74) [75) (of a rather different nature) had already been constructed by Vogan, Kazhdan-Savin, by Gross-Wallach (cf. their article in this volume), by Kostant in [60) [64), and by Binegar-Zierau (precise references can be found in [74)). In particular, Gross and Wallach obtain beautiful results on discrete restriction of the minimal representa• tions to subgroups. Many mathematicians and physicists have constructed very interesting examples of minimal, and more generally singular, repre• sentations. The papers [72] [74) [75) are based on far-reaching ideas from algebraic geometry and symplectic geometry, and give beautiful and explicit models for the Harish-Chandra modules where 9 acts by explicit pseudo-differential operators. They give a uniform construction of minimal representations in all cases where they exist, including the non-spherical minimal represen• tations. In the spirit of geometric quantization, the geometry is based on the lagrangian subvariety Y of the minimal orbit 0 n p* (there is no loss of generality in assuming Y not empty, since no minimal representation can exist if Y is empty). Here 9 = t+p is a complexified Cartan decomposition xxxvi Preface

corresponding to the real simple simply-connected Lie group Gilt such that 'Y is simple and the pair (g, t) is non-Hermitian. The Harish-Chandra mod• ule is to be H = r(Y, N~), where N~ is a half-form bundle over Y. The Lie algebra t acts naturally on H, and the difficulty lies in constructing operators associated to elements of p. The difficulty is that 0 admits no G-invariant polarization. However [74] gives the amazing result that the cotangent bundle T*Y and a ramified double cover of 0 share an open set M; this defines an infinitesimal action of g on M c T*Y, which is Hamil• tonian. Brylinski and Kostant go further in [74] to show that the function on M associated to any v E p decomposes as fv - gv, where the fv generate the Poisson commutative subalgebra R(Y) of R(M), and the gv generate another (transverse) Poisson commutative subalgebra. The construction of the operators associated to elements of p, given in detail (in a uniform way) [74], converts the functions fv - gv on Minto pseudo-differential differential operators on H. While fv obviously defines a multiplication operator, gv defines a pseudo-differential differential oper• ator which is equal to (E'(E' + 1))-1 Dv where Dv is an order 4 differential operator while E' is the Lie derivative with respect to the Euler vector field on Y (they prove that E' has positive spectrum on H). The proof that these operators form a Lie algebra isomorphic to g combines their results on the geometry of Y and T*Y with a skillful application of the work of Kostant and Sahi on the generalized Capelli identity; a crucial point is that a certain subspace of t is always a Jordan algebra of rank S 4 and further• more there is a monomial P in the Jordan norms of degree exactly 4. The paper [75] (in this volume) contains the beautiful construction of the new differential operators Dv. If e E p is the highest weight vector then De is the quotient of an operator corresponding to P by the linear function on Y defined bye. The explicit and uniform construction of the· minimal representations by means of geometric quantization allows Brylinski and Kostant to give a beautiful description of the invariant hermitian form. In [72] and [74] this hermitian form is written down in terms of hypergeometric functions. The methods of [72] [74] [75] open the road to a study of the minimal representations from the point of view of harmonic analysis; already in [74] there is a formula expressing the spherical function restricted to a root TDS. The methods of [72] [74] [75], which are based on geometry, have the po• tential to carryover to more general nilpotent orbits; Brylinski and Kostant have embarked on their program to do this. Kostant has also made great progress in the last few years in a vast new program of relating the group PSL(2, 11) to the polyhedron called Preface xxxvii buckminseterfullerene (or simply buckyball). This is a truncated icosahe• dron which has 60 edges and 90 vertices. Of these edges, 60 edges (referred to as pentagonal) belong to the boundary of the 12 pentagons surrounding the vertices of the icosahedron, and the remaining 30, referred to as hexag• onal, bound only hexagons. The graph r made of these vertices and edges materializes in the coatings of many viruses and in the structure of the carbon molecule G60 • The polyhedron itself was used a number of years ago by the architect Buckminster Fuller. The fact that the 60-element icosahedral group A acts as a symmetry group leads one to believe that noncommutative harmonic analysis can explain some of the chemistry of these molecules. This was already the case in work of F. Chung and S. Sternberg, who deduced the vibrational spectrum from group theory. In the paper [73] by Chung, Kostant and Sternberg, appearing in this vol• ume, the geometry of the buckyball is studied further in connection with the group PSL(2, 11); it is shown in [73] that the 12 vertices of the icosahe• dron can be identified naturally with the projective line Pl1, and the action of PSL(2, 11) extends naturally to an action on the 90 vertices of r (the reason being that the edges of the icosahedron have a beautiful interpreta• tion in terms of the cross-ratio in P l1 ). The action of PSL(2, 11) preserves the pentagonal edges but not the hexagonal edges. The paper [73] also de• scribes the embedding A ~ PSL(2, 11) and the corresponding branching rules, as well as their relation to the electronic spectrum. Then the paper discusses the Huckel model modified to include the magnetic field, and the possible rule of the double cover of A in the physics of the Fullerene. The very recent paper [76] goes much further by describing the en• tire structure of the buckyball graph r in terms of PSL(2, 11). Kostant identifies the set of vertices of r with a conjugacy class M in PSL(2,11) consisting of elements of order 11; then the two pentagonal edges having x E M as a vertex are {x,x3} and {x,x-3}, and the unique hexagonal edge having x as a vertex is {x,Px}. Here Px = x-I. (lx' X· (lx, where (lx is the unique element of order 2 of A c PSL(2,11) such that the above expression is again an element of order 2 of A. Thus the entire graph r is constructed from group theory. This should pave the way to using group representations to further study the physics and chemistry of the molecule G450. The recent paper [71] gives a beautiful generalization of a positivity theorem of J. Stembridge which shows that if an n x n- matrix A is totally positive (Le., all its minors are 2: 0), then for any partition A of n the immanant ImmA(A) is 2: O. First it is shown in [71] that the immanant can be written as

ImmA(A) = Tr PA7fA(A)PA, xxxviii Preface

where 1I"A : GL(n) -+ Aut(VA) is the representation of highest weight A, and PA is the projection of VA to the zero weight space. Then the generalization of the Stembridge theorem is the inequality

where 11" : GL(n) -+ Aut(V) is any finite-dimensional representation of GL(n), and 11" : V -+ VH is the H-equivariant projection onto the space V H of H-invariant vectors, for H a maximal torus. The proof of this theorem uses a result of A. Whitney on the structure of the semigroup of totally positive matrices, together with a deep positivity result of Lusztig concerning his canonical basis (cf. the paper of Lusztig in this volume). When we last spoke with Bert he was busy, as always, working on many projects. We have all benefited immensely, not only from reading many of Kostant's papers, but also from so many exciting lectures and inspiring conversations. We wish him the best for the next century. The editors warmly thank everyone at Birkhauser and in particular Elizabeth Hyman. We are very grateful to Ann Kostant for all her guidance, support and patience in preparing this volume. Published Works of Bertram Kostant

1. Holonomy and the Lie Algebra of Infinitesimal Motions of a Rieman• nian Manifold, Trans. Amer. Math Soc. 80 (1955), 528-542. 2. On the Conjugacy of Real Carlan Subalgebras, I, II, Proc. Acad. of Sci. 41: 11, Nov. 1955. 3. On Invariant Skew Tensors, Proc. Nat. Acad. of Sci. 42: 3, March 1956, 148-151. 4. On Differential Geometry and Homogeneous Spaces, I, II, Proc. Nat. Acad. Sci. 42 (1956), 258-261, 354-357. 5. On Holonomy and Homogeneous Spaces, Nagoya Math. J. 12(1957), 31-54. 6. A Theorem of Probenius, A Theorem of Amitsur-Levitski and Coho• mology Theory, J. of Math. and Mech. 7: 2 (1958),237-264. 7. A Characterization of the Classical Groups, Duke Math. J. 25: 1(1958), 107-124. 8. A Formula for the Multiplicity of a Weight, Trans. Amer. Math. Soc., 93: 1 (1959), 53-73. 9. The Principal Three Dimensional Sub-Group and the Betti Numbers of a Complex Simple Lie Group, Am. J. Math., Oct. 1959,973-1032. 10. A Characterization of Invariant Affine Connections, Nagoya Math. Jour. 16 (1960), 35-50. 11. Lie Algebra Cohomology and the Generalized Borel- Weil Theorem, Ann. of Math. 74: 2 (1961),329-387. 12. (with G. Hochschild and A. Rosenberg), Differential Forms on Regular Affine Algebras, Trans. Amer. Math. Soc. 102: 3 (1962),383-408. 13. (with G. Hochschild), Differential Forms and Lie Algebra Cohomology for Algebraic Linear Groups, III, Jour. of Math. 6(1962), 264-281. 14. Lie Group Representations on Polynomial Rings, Bull. Amer. Math. Soc. 69(1963), 518-526. 15. Lie Group Representations on Polynomial Rings, Am. J. Math. 85 (1963), 327-404. 16. Lie Algebra Cohomology and Generalized Schuberl Cells, Ann. of Math. 77: 1 (1963), 72-144. 17. (with A. Novikoff), A Homomorphism in Exterior Algebra, Canad. J. Math. 16 (1964), 166-168. 18. Eigenvalues of a Laplacian and Commutative Lie Subalgebras, Topol• ogy, 13 (1965), 147-159. xl Published Works of Bertram Kostant

19. Orbits, Symplectic Structures and Representation Theory, Proc. of U.S.-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965. 20. Groups Over Z, Proc. Symposia in Pure Math., 9 (1966), 90-98. 21. (with L. Auslander), Quantization and Representations of Solvable Lie Groups, Bull. Amer. Math. Soc. 73: 5 (1967),692-695. 22. Irreducibility of Principal Series and Existence and Irreducibility of the Complementary Series, Bowdoin Conf., Aug. 1968, 1-41. 23. (with Stephen Rallis), On Orbits Associated with Symmetric Spaces, Bull. Amer. Math. Soc. 75 (1969),879-883. 24. (with Stephen Rallis), Representations Associated with Symmetric Spaces, Bull. Amer. Math. Soc. 75 (1969),884-888. 25. On the Existence and Irreducibility of Certain Series of Representa• tions, pub. as Invited Address in Bull. Amer. Math. Soc., May 1969, 627-642. 26. On Certain Unitary Representations which arise from a Quantization Theory, Lecture Notes in Phys., Vol. 6, Battelle Seattle Rencontres, Springer-Verlag, 1970, 237-254. 27. Orbits and Quantization Theory, Proc. Int. Congress of Mathemati• cians, Nice, 1970, 395-400. 28. Quantization and Unitary Representations, Lecture Notes in Math. 170 (1970), Springer-Verlag, 87-207. 29. (with Stephen Rallis), Orbits and Representations Associated with Symmetric Spaces, Amer. Jour. of Math. 93 (1971), 753-809. 30. (with L. Auslander), Polarization and Unitary Representations of Solv• able Lie Groups, Inventiones Math., 1971, 255-354. 31. Line Bundles and the Prequantized Schrodinger Equation, ColI. Group Theoretical Methods in Physics, Centre de Physique Theorique, Mar• seille, June 1972, 81-85. 32. On Convexity, the Weyl Group and the Iwasawa Decomposition, Ann. Sci. Ecole Norm. Sup. 6: 4 (1973), 413- 455. 33. Symplectic Spinors, in: Symp. Math., Vol. XIV, Instituto Naz. di Alt. Mat. Roma., Academic Press, London-New York 1974, 139-152. 34. Verma Modules and the Existence of Quasi-Invariant Differential Op• erators, Lecture Notes in Math. 466(1974), Springer-Verlag, 101-129. 35. On the Existence and Irreducibility of Certain Series of Represen• tations, in: Lie Groups and Their Representations, edited by I. M. Gelfand, Summer School Conf. Budapest, 1971, Halsted Press, Wiley Press, 1975, 231-331. 36. On The Tensor Product of a Finite and an Infinite Dimensional Rep• resentation, J. Funct. Anal. 20: 4 (1975), 257-285. Published Works of Berlmm Kostant xli

37. On the Definition of Quantization, Geometrie Symplectique et Physique Mathematique, ColI. CNRS, No. 237, Paris, 1975, 187-210. 38. (with D. Sullivan), The Euler Characteristic of an Affine Space Form is Zero, Bull. Amer. Math. Soc. 81: 5 (1975). 39. On MacDonalds .,.,-Function Formula, the Laplacian and Generalized Exponents, Adv. in Math., 20: 2 (1976), New York, 179-212. 40. (with J. Tirao) On the Structure of Certain Sublalgebras of a Universal Enveloping Algebra, Trans. Amer. Math. Soc. 218(1976), 133-154. 41. Graded Manifolds, Graded Lie Theory, and Prequantization, Lecture Notes in Math. 570 (1977), Springer-Verlag, 177-306. 42. Quantization and Representation Theory, in: Representation Theory and Lie Groups, Proc. SRC/LMS Res. Sympos., Oxford, 1977, pp 287-316; London Math Soc. Lecture Note Ser 34, Cambridge Univ Press, Cambridge-New York, 1979. 43. Harmonic Analysis on Graded (or Super) Lie Groups, Group Theoreti• cal Methods in Physics, Sixth International Colloquium, Lecture Notes in Physics, Springer, 1979, 47-50. 44. On Whittaker Vectors and Representation Theory, Inventiones Math. 48 (1978), 101-184. 45. (with D. Kazhdan and S. Sternberg), Hamiltonian Group Actions and Dynamical Systems of Calogero Type, Communications Pure and Ap• plied Math., 31:4 (1978), 483-507. 46. The Solution to a Generalized Toda Lattice and Representation Theory, Adv. in Math., 34 (1979), 195-338. 47. Poisson Commutativity and Generalized Periodic Toda Lattice, Differ• ential Geometric Methods in Math. Physics, Lecture Notes in Mathe• matics 905 (1980), Springer-Verlag, 12-28. 48. (with S. Sternberg), Symplectic Projective Orbits, New Directions in Applied Mathematics, Springer-Verlag, (Cleveland, Ohio, 1980), Springer-Verlag, 1982, 81-84, 49. A Lie Algebra Generalization of the Amitsur-Levitski Theorem, Adv. in Math. 40 : 2 (1981), 155-175 :50. Coadjoint Orbits and a New Symbol Calculus for Line Bundles, Conf. on Diff. and Geometric Methods in Theoretical Physics, Trieste 1980, World Scientific Publishing, 1981, 66-68. :51. The Coxeter Element and the Structure of the Exceptional Lie Groups, ColI. Lectures of the AMS, 1983, Notes available from the AMS. 52. On Finite subgroups of SU(2), simple Lie Algebras, and the McKay Correspondence, Proc. Nat. Acad. Sci. U.S.A., 81:16 (1984), Phys. Sci., 5275-5277. xlii Published Works of Bertmm Kostant

53. The McKay Correspondence, the Coxeter Element and Representation Theory, Societe Math. de France, Asterisque, hors series, 1985, 209- 255. 54. (with S. Kumar), The Nil Hecke Ring and Cohomology of GIP for a Kac-Moody Group G*, Proc. Nat. Acad. Sci. USA. 83:6 (1986), 1543-1545. 55. (with S. Kumar), The Nil H ecke Ring and Cohomology of G / P for a Kac-Moody Group G*, Adv. in Math. 62 (1986), 187-237. 56. (with S. Sternberg), Symplectic Reduction, BRS Cohomology and In• finite Dimensional Clifford Algebras, Annals of Physics 176 (1987), 49-113. 57. (with S. Kumar), T-Equivariant K -Theory of Generalized Flag Vari• eties, Proc. Nat. Acad. Sci. U.S.A. 84:13 (1987),4351-4354. 58. (with S. Kumar), T-Equivariant K-Theory of Generalized Flag Vari• eties, Journal of Differential Geometry, 32 (1990), 549-603. 59. (with V. Guillemin and S. Sternberg), Jesse Douglas' Solution to the Plateau Problem, Proc. Nat. Acad. Sciences, May 1988, 3277-3278. 60. The Principle of Triality and a Distinguished Unitary Representation of SOC 4, 4), Differential Geometrical Methods in Theoretical Physics, edited by K. Bleuler and M. Werner, Kluwer Academic Publishers, 1988, 65-108. 61. (with S. Sternberg), The Schwartzian Derivative and the conformal geometry of the Lorentz hyperboloid, Quantum Theories and Geometry, edited by M. Cahen and M. Flato, Kluwer Academic Publishers, 1988, 113-125. 62. A Formula of Gauss-Kummer and the Trace of Certain Intertwining Operators, The Orbit Method in Representation Theory, edited by Duflo, Pedersen and Vergne, Birkhauser, 1990, 99-134. 63. A Tale of Two Conjugacy Classes, in preparation. 64. The Vanishing of Scalar Curvature and the Minimal Unitary Represen• tation of SO(4,4), Operator Algebras, Unitary Representations, En• veloping Algebras, and Invariant Theory, edited by Connes et aI, PM 92, Birkhauser-Boston,1990, 85-124. 65. (with Shrawan Kumar), A Geometric Realization of Minimal K-Types of Hansh-Chandra Modules, Kazhdan-Lusztig Theory and related Top• ics, Contemporary Math., edited by V.V. Deodhar, Vol. 139, 1992, Providence. 66. (with Siddhartha Sahi), The Capelli Identity, Tube Domains and a Generalized Laplace Transform, Adv. in Math. 87 (1990), 71-92. Published Works of Bertmm Kostant xliii

67. (with Ranee Brylinski), The Variety of all Invariant Symplectic Struc• tures on a Homogeneous Space, Symplectic Geometry and Mathemati• cal Physics, PM 99, edited by Donato et al. Birkhauser-Boston, 1991, 80-113. 68. (with Ranee Brylinski), Nilpotent Orbits, Normality and Hamiltonian Group Actions, Bull. AMS. 26 (1992), 269-275. 69. (with Ranee Brylinski), Nilpotent Orbits, Normality and Hamiltonian Group Actions, JAMS 7 (1994), 269-298. 70. (with Siddhartha Sahi), Jordan Algebras and Capelli Identities, Inven• tiones Math., 112 (1993), 657-664. 71. Immanant Inequalities and O-weight Spaces, Tto appear, JAMS.

72. (with Ranee Brylinski), Minimal representations of E6 , E7 and Es and the Generalized Capelli Identity, Proc. Natl. Acad. Sci. USA 91 (1994), 2469-2472. 73. (with F. Chung and S. Sternberg), Groups and the Buckyball, to ap• pear in: Lie Theory and Geometry, edited by J.-L. Brylinski, R. Brylin• ski, V. Guillemin, V. Kac, PM 123, Birkhauser, 1994 74. (with Ranee Brylinski), Minimal representations, geometric quantiza• tion and unitarity, Proc. Natl. Acad. Sci. USA 91 (1994), 6026-6029. 75. (with Ranee Brylinski), Differential operators on conical lagrangian manifolds, to appear in: Lie Theory and Geometry, edited by J.-L. Brylinski, R. Brylinski, V. Guillemin, V. Kac, PM 123, Birkhauser, 1994.

76. Structure of the truncated icosahedron (e.g. fullerene or C60, viral coat• ings) and a 60-element conjugacy class in PSL(2,11), 1994, Selecta Math., to appear

Books

Bl A Course in the Mathematics of General Relativity, ARK Publications, 1988