Collected Papers: Volume I 1955-1966

Collected Papers Volume I 1955–1966

Kostant at family event, June 2002

Bertram Kostant

Collected Papers Volume I 1955–1966

Anthony Joseph Michèle Vergne Shrawan Kumar Editors

123

Bertram Kostant Massachusetts Institute of Technology Department of Mathematics Cambridge, MA 02139 USA [email protected]

ISBN 978-0-387-09582-0 e-ISBN 978-0-387-09583-7 DOI 10.1007/b94535 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926994 Mathematics Subject Classification (2000): 01A75, 32-06, 14-06 © Bertram Kostant 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer soft-ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Bertram Kostant is to be counted as one of the remarkable mathematicians of the second half of the twentieth century through his fundamental and varied contributions to many aspects of Lie theory, a subject which itself pervades almost the whole of mathematics. His work is marked by a rare simplicity and characteristic elegance, making it eminently readable, wonderfully enjoyable, and easily understandable even to a true novice. It is often through his work that one may understand many sophisticated developments of modern mathematics. The mathematical work of Bertram Kostant has spanned well over fifty years in which he has published at an almost constant rhythm over 100 papers of which more than a few have become a cornerstone of rich and fruitful theories. Here, in briefly listing some of his most important contributions and their eventual ramifications we can give only a small glimpse into the depth and nature of his legacy. In one of his earliest works Kostant introduced a partition function describing weight multiplicities. Significantly, it provided a tool which allowed Jantzen to compute certain fundamental determinants concerning weight spaces in Verma modules in the much more difficult parabolic case. Kostant studied extensively the principal three-dimensional subalgebra of a semisimple Lie algebra, extracting from its adjoint action the Betti numbers of the corresponding Lie group. In addition, he thereby obtained a remarkable linearization of the fundamental invariants. Geometrically this realizes an affine slice to the regular coadjoint orbits, a result subsequently broadened notably by Luna, Slowody, and Premet, and now of greatly renewed interest. Kostant followed this work by a further fundamental paper describing harmonic polynomials in the symmetric algebra. This gave a separation of variables theorem and a rather precise description of the nilpotent cone. These results have been key building blocks in many important results in representation theory, including the theory of primitive ideals. Kostant discovered a remarkable connection between the regular nilpotent element and the Coxeter element in the Weyl group. This was subsequently brought to a classification of nilpotent orbits by Carter and Elkington and then elevated to a map from nilpotent orbits to conjugacy classes in the Weyl group by Kazhdan and Lusztig. Kostant himself has returned to the many beautiful themes which may be developed from these early papers. They include an extensive study of Whittaker

vi

Preface

modules, the mathematical underpinnings of the Toda lattice which he generalized well beyond the dreams of physicists, and quantum cohomology. Building upon Bott’s geometric results, Kostant computed the Lie algebra cohomology of certain nilpotent Lie algebras arising as nilradicals of parabolic subalgebras, studying at the same time their relationship to the cohomology of the flag variety and its generalizations. This recovered the Bernstein–Gelfand– Gelfand resolution of simple finite-dimensional modules admitting far-reaching generalizations to Kac–Moody and later to Borcherds algebras. Here he inadvertently rediscovered the Amitsur–Levitski identity, so important in algebras with polynomial identity, giving it a far simpler proof and placing the result in the more general context of semisimple Lie algebras. Kostant was one of the first to realize that a construction of Kirillov to classify unitary representations of nilpotent Lie groups implied that any coadjoint orbit admitted the structure of a symplectic variety, that is, the phase space of classical mechanics. He quickly seized upon the passage to quantum mechanics, developed by physicists, as a process of “geometric quantization.” The Kostant line bundle detects which symplectic manifolds are quantizable and this result is fundamental in Hamiltonian geometry. When applied to the coadjoint orbits, quantization is designed to produce most unitary representations of the corresponding real Lie group. With Auslander he classified the unitary representations for real class 1 simply connected solvable Lie groups, laying the groundwork for the more complete theory of Pukanszky. The general case, though even more resistant, has come under the intensive study of many mathematicians, including notably Duflo, Rossmann, Vergne and Vogan. In more algebraic terms geometric quantization dramatically laid open the path to the classification of primitive ideals initiated by Dixmier and Gabriel. Further deep manifestations of geometric quantization are exemplified by the Duflo isomorphism, the Kashiwara–Vergne conjecture, and the Drinfeld associator themselves having been unified and extended following notably the powerful techniques of Kontsevitch inspired by Feynman diagrams and knot theory. Stepping outside the purely algebraic framework Kostant discovered a farreaching generalization of the Golden–Thompson rule in a convexity theorem subsequently generalized by Atiyah–Bott and Guillemin–Sternberg. Going beyond zero characteristic, he introduced the “Kostant form” on the enveloping algebra which has played a major role in modular representation theory. Moreover its generalization for quantum groups and the subsequent development of the quantum Frobenius map due to Lusztig allowed Littelmann to complete the Lakshmibai– Seshadri programme of describing standard monomial bases.

Preface

vii

In a veritable “tour de force” Kostant calculated certain fundamental determinants showing them to have linear factors from which their zeros could be determined. This led to a criterion for unitarity which has so far gone unsurpassed. Subsequently, many other such factorizations were achieved, particularly that of Shapovalov and Janzten noted above; but also of Gorelik–Lanzmann which determined exactly when a Verma module annihilator for a reductive super Lie algebra was generated by its intersection with the centre. Kostant was one of the first to firmly lay the foundations of supermanifolds and super Lie groups. He also studied super Lie algebras providing a fundamental structure theorem for their enveloping algebras. He initiated the study of the tensor product of an infinite-dimensional representation with finite-dimensional ones, leading in the hands of Jantzen to the translation principle. The latter, combined with Kostant’s description of the nilpotent cone, led to the Beilinson–Bernstein equivalence of categories which heralded a new enlightenment in Verma module and Harish-Chandra theory. Kostant described (in sl(4)) a non-trivial example of a unitary highest weight module, the theory of which was subsequently completed by Enright, Howe and Wallach (and independently by Jakobsen). In the theory of characteristics developed by Guillemin, Quillen and Sternberg, a result of Kostant proves in the finite-dimensional case the involutivity of characteristics ultimately resolved in full generality by Gabber. Among some of his important collaborative works, we mention his paper with Hochschild and Rosenberg on the differential forms on regular affine algebras, his paper with Rallis on separation of variables for symmetric spaces, his work with Kumar on the cohomology and K-theory of flag varieties associated to Kac–Moody groups, his determination with R. Brylinski of invariant symplectic structures and a uniform construction of minimal representations, and his work with Wallach on Gelfand–Zeitlin theory from the point of view of the construction of maximal Poisson commutative subalgebras particularly by “shift of argument” coming from classical mechanics. Even in the fifth and sixth decades of his career, Kostant has continued to produce results of astonishing beauty and significance. We cite here his work on the Toda lattice and the quantum cohomology of the flag variety, his re-examination of the Clifford algebra deformation of “wedge n” with its beautiful realization of the module whose highest weight is the sum of the fundamental weights, his introduction of the cubic Dirac operator, his generalization of the Borel–Bott–Weil theorem with its connection to Euler number multiplets of representations, his work on the set of abelian ideals in the nilradical of a Borel (wonderfully classified by

viii

Preface

certain elements in the affine Weyl group by Dale Peterson), his longstanding love affair with the icosahedron, his work with Wallach mentioned above and most recently an attempt to pin down the generators of the centralizer of a maximal compact subalgebra. After receiving his Ph.D. from the University of Chicago in 1954, Bertram Kostant began his academic career as an Assistant Professor in 1956 at the University of California, Berkeley rose to full professor in 1962 and shortly after moved to MIT. He taught at MIT until retiring in 1993. Kostant was elected to the National Academy of Sciences U.S.A. in 1978 and has received many other honours and prizes, including election as a Sackler Institute Fellow at Tel-Aviv University in 1982, a medal from the Coll`ege de France in 1983, the Steele prize of the American Mathematical Society in 1990, and several honorary doctorates. Kostant maintains his physical fitness after the strains of University life by a weight lifting regime well beyond the capabilities of many younger colleagues. This may be one secret to his scientific longevity. Weizmann Institute of Science, Rehovot, Israel University of North Carolina at Chapel Hill, US ´ Ecole Polytechnique, Palaiseau, France April, 2008

Anthony Joseph Shrawan Kumar Mich`ele Vergne Editors

Contents: Volume I Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Contents: Volume I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Collected Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix [1]

Holonomy and the Lie Algebra of Infinitesimal Motions of a Riemannian Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

[2] On the Conjugacy of Real Cartan Subalgebras I . . . . . . . . . . . . . . . . . . . . . . 16 [3] On the Conjugacy of Real Cartan Subalgebras II . . . . . . . . . . . . . . . . . . . . . 20 [4]

On Invariant Skew-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

[5]

On Differential Geometry and Homogeneous Spaces I . . . . . . . . . . . . . . . . 32

[6]

On Differential Geometry and Homogeneous Spaces II . . . . . . . . . . . . . . . 36

[7]

On Holonomy and Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

[8]

A Theorem of Frobenius, a Theorem of Amitsur–Levitski and Cohomology Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

[9]

A Characterization of the Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 92

[10]

A Formula for the Multiplicity of a Weight . . . . . . . . . . . . . . . . . . . . . . . . . 109

[11]

The Principal Three Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

[12]

A Characterization of Invariant Affine Connections . . . . . . . . . . . . . . . . . . 190

[13]

Lie Algebra Cohomology and the Generalized Borel–Weil Theorem . . 206

[14]

(with G. Hochschild and A. Rosenberg), Differential Forms on Regular Affine Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

[15]

(with G. Hochschild), Differential Forms and Lie Algebra Cohomology for Algebraic Linear Groups . . . . . . . . . . . . . . . . . . . . . . . . 291

[16]

Lie Group Representations on Polynomial Rings . . . . . . . . . . . . . . . . . . . . 309

[17]

Lie Group Representations on Polynomial Rings . . . . . . . . . . . . . . . . . . . . 318

x

Contents: Volume I

[18]

Lie Algebra Cohomology and Generalized Schubert Cells . . . . . . . . . . . .396

[19]

Eigenvalues of a Laplacian and Commutative Lie Subalgebras . . . . . . . . 469

[20]

Orbits, Symplectic Structures and Representation Theory . . . . . . . . . . . . 482

[21]

Groups Over Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

Kostant’s Comments on Papers in Volume I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Collected Papers The list of papers below represent the collected papers of Bertram Kostant from 1955 to 2009. The asterisk denotes the 21 papers only in this volume. *1. Holonomy and the Lie Algebra of Infinitesimal Motions of a Riemannian Manifold, Trans. Amer. Math. Soc., 80 (1955), 528–542. *2, *3 On the Conjugacy of Real Cartan Subalgebras I, Proc. Natl. Acad. Sci., 41 (1955), No. 11, 967–970; *3. On the Conjugacy of Real Cartan Subalgebras II, 1–8, 1955. *4. On Invariant Skew-Tensors, Proc. Natl. Acad. Sci., 42 (1956), No. 3, 148–151. *5. On Differential Geometry and Homogeneous Spaces I, Proc. Natl. Acad. Sci., 42 (1956), No. 5, 258–261. *6. On Differential Geometry and Homogeneous Spaces II, Proc. Natl. Acad. Sci., 42 (1956), No. 6, 354–357. *7. On Holonomy and Homogeneous Spaces, Nagoya Math. Jour., 12 (1957), 31–54. *8. A Theorem of Frobenius, a Theorem of Amitsur–Levitski and Cohomology Theory, J. Math. and Mech., 7 (1958), No. 2, Indiana University, 237–264. *9. A Characterization of the Classical Groups, Duke Math. Jour., 25 (1958), No. 1, 107–124. *10. A Formula for the Multiplicity of a Weight, Trans. Amer. Math. Soc., 93 (1959), No. 1, 53–73. *11. The Principal Three Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group, Amer. Jour. Math., 81 (1959), 973–1032. *12. A Characterization of Invariant Affine Connections, Nagoya Math. Jour., 16 (1960), 35–50. *13. Lie Algebra Cohomology and the Generalized Borel–Weil Theorem, Ann. of Math., 74 (1961), No. 2, 329–387. *14. (with G. Hochschild and A. Rosenberg), Differential Forms on Regular Affine Algebras, Trans. Amer. Math. Soc., 102 (1962), No. 3, 383–408. *15. (with G. Hochschild), Differential Forms and Lie Algebra Cohomology for Algebraic Linear Groups, Illinois Jour. Math., 6 (1962), 264–281. *16. Lie Group Representations on Polynomial Rings, Bull. Amer. Math. Soc., 69 (1963), No. 1, 518–526.

xii

Collected Papers

*17. Lie Group Representations on Polynomial Rings, Amer. J. Math., 85 (1963), 327–404. *18. Lie Algebra Cohomology and Generalized Schubert Cells, Ann. of Math., 77 (1963), No. 1, 72–144. *19. Eigenvalues of a Laplacian and Commutative Lie Subalgebras, Topology, 13 (1965), 147–159. *20. Orbits, Symplectic Structures and Representation Theory, Proc. U.S.-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965, p. 71. *21. Groups Over Z, Proc. Symposia in Pure Math., 9 (1966), 90–98. 22. (with L. Auslander), Quantization and Representations of Solvable Lie Groups, Bull. Amer. Math. Soc., 73 (1967), No. 5, 692–695. 23. (with Stephen Rallis), 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 Representations, Invited Address, Bull. Amer. Math. Soc., 75 (1969), 627–642. 26. On Certain Unitary Representations which arise from a Quantization Theory, Lecture Notes in Math., Battelle Seattle Rencontres, Springer-Verlag, 6 (1970), 237–254. 27. Orbits and Quantization Theory, Proc. Int. Congress of Mathematicians, Nice, 1970, 395–400. 28. Quantization and Unitary Representations, Lecture Notes in Math., Vol. 170, Springer-Verlag, 1970, 87–207. 29. (with Stephen Rallis), Orbits and Representations Associated with Symmetric Spaces, Amer. J. Math., 93 (1971), 753–809. 30. (with L. Auslander), Polarization and Unitary Representations of Solvable Lie Groups, Invent. Math., 14 (1971), 255–354. 31. Line Bundles and the Prequantized Schr¨odinger Eguation, Coll. Group Theoretical Methods in Physics, Centre de Physique Th´eorique, Marseille, 1972, 81–85. 32. On Convexity, the Weyl Group and the Iwasawa Decomposition, Ann. Sci. Ecole Norm. Sup., 6 (1973), No. 4, 413–455. 33. Symplectic Spinors, Rome Symposia, XIV (1974), 139–152. 34. Verma Modules and the Existence of Quasi-Invariant Differential Operators, Lecture Notes in Math., Vol. 466, Springer-Verlag, 1974, 101–129.

Collected Papers

xiii

35. On the Existence and Irreducibility of Certain Series of Representations, in: Lie Groups and Their Representations, edited by I. M. Gelfand, Summer School Conf. Budapest, 1971, Halsted and Wiley Press, 1975, 231–331. 36. On The Tensor Product of a Finite and an Infinite Dimensional Representation, J. Funct. Anal., 20 (1975), No. 4, 257-285. 37. On the Definition of Quantization, G´eometrie symplectique et physique math´ematique, Coll. 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 (1975), No. 5, 937–938. 39. On MacDonalds η-Function Formula, the Laplacian and Generalized Exponents, Adv. in Math., 20 (1976), No. 2, 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., Vol. 570, Springer-Verlag, 1977, 177–306. 42. Quantization and Representation Theory, Proc. Oxford Conference on Group Theory and Physics, Oxford, 1977, 287–316. 43. On Whittaker Vectors and Representation Theory, Inventiones Math., 48 (1978), 101–184. 44. (with D. Kazhdan and S. Sternberg), Hamiltonian Group Actions and Dynamical Systems of Calogero Type, Communications Pure and Applied Math., 31 (1978), No. 4, 483–507. 45. Harmonic Analysis on Graded (or Super) Lie Groups, Group Theoretical Methods in Physics, Sixth International Colloquium, Lecture Notes in Physics, Springer-Verlag, 1979, 47–50. 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, Differential Geometric Methods in Math. Physics, Lecture Notes in Mathematics, Vol. 905, Springer-Verlag, 1980, 12–28. 48. (with S. Sternberg), Symplectic Projective Orbits, New Directions in Applied Mathematics, Springer-Verlag, 1980, 81–84. 49. A Lie Algebra Generalization of the Amitsur–Levitski Theorem, Adv. in Math., 40 (1981), No. 2, 155–175. 50. Coadjoint Orbits and a New Symbol Calculus for Line Bundles, Differential and Geometric Methods in Theoretical Physics, Trieste 1980, World Scientific Publishing, 1981, 66–68.

xiv

Collected Papers

51. The Coxeter Element and the Structure of the Exceptional Lie Groups, Coll. Lectures of the AMS, 1983. 52. The McKay Correspondence, the Coxeter Element and Representation Theory, Soci´et´e Math. de France, Ast´erisque, hors s´erie, 1985, 209–255. 53. (with Shrawan Kumar), The Nil Hecke Ring and Cohomology of G/P for a Kac–Moody Group G, Proc. Natl. Acad. Sci., 83 (1986), 1543–1545. 54. (with Shrawan Kumar), The Nil Hecke Ring and Cohomology of G/P for a Kac–Moody Group G, Adv. in Math., 62 (1986), 187–237. 55. (with S. Sternberg), Symplectic Reduction, BRS Cohomology and Infinite Dimensional Clifford Algebras, Annals of Physics, 176 (1987), 49–113. 56. (with S. Kumar), T-Equivariant K-Theory of Generalized Flag Varieties, Proc. Natl. Acad. Sci., 84 (1987), 4351–4354. 57. (with V. Guillemin and S. Sternberg), Douglass Solution to the Plateau Problem, Proc. Natl. Acad. Sci., 85 (1988), 3277–3278. 58. The Principle of Triality and a Distinguished Unitary Representation of SO(4,4), in: Differential Geometrical Methods in Theoretical Physics, Editors K. Bleuler and M. Werner, Kluwer Academic Publishers, 1988, 65–108. 59. (with S. Sternberg), The Schwartzian derivative and the conformal geometry of the Lorentz hyperboloid, in: Quantum Theories and Geometry, Editors M. Cahen and M. Flato, Kluwer Academic Publishers, 1988, 113–125. 60. (with S. Kumar), T-Equivariant K-Theory of Generalized Flag Varieties, J. Diff. Geom., 32 (1990), 549–603. 61. A Formula of Gauss–Kummer and the Trace of Certain Intertwining Operators, in: The Orbit Method in Representation Theory, Editors Duflo, Pedersen and Vergne, Prog. Math., Vol. 82, Birkh¨auser, Boston, 1990, 99–134. 62. The Vanishing of Scalar Curvature and the Minimal Unitary Representation of SO(4,4), in: Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, in Honor of J. Dixmier, Editors A. Connes et al, Prog. Math., Vol. 92, Birkh¨auser, Boston, 1990, 85–124. 63. (with Siddhartha Sahi), The Capelli Identity, Tube Domains and a Generalized Laplace Transform, Adv. in Math., 87 (1990), 71–92. 64. (with Ranee Brylinski), The Variety of all Invariant Symplectic Structures on a Homogeneous Space, Symplectic Geometry and Mathematical Physics, Prog. Math., Vol. 99, Editors Donato et al., Birkh¨auser, Boston, 1991, 80–113. 65. (with Shrawan Kumar), A Geometric Realization of Minimal k-Type of HarishChandra Modules for Semisimple Groups, in: Kazhdan–Lusztig Theory and Related Topics, Editor V.V. Deodhar, Contemporary Math., Vol. 139, AMS, Providence, 1992, 249–264.

Collected Papers

xv

66. (with Ranee Brylinski), Nilpotent Orbits, Normality and Hamiltonian Group Actions, Bull. Amer. Math. Soc., 26 (1992), 269–275. 67. (with Siddhartha Sahi), Jordan Algebras and Capelli Identities, Invent. Math., 112 (1993), 657–664. 68. (with Ranee Brylinski), Nilpotent Orbits, Normality and Hamiltonian Group Actions 2, J. Amer. Math. Soc., 7 (1994), 269–298. 69. (with Ranee Brylinski), Minimal representations of E 6 , E 7 and E 8 , Unitarity and the Generalized Capelli Identity, Proc. Natl. Acad. Sci., 91 (1994), 2469– 2472. 70. (with F. Chung and S. Sternberg), Groups and the Buckyball, Lie Theory and Geometry: In Honor of Bertram Kostant, Editors J.-L. Brylinski et al., Prog. Math., Vol. 123, 1994, Birkh¨auser, Boston, 97–126. 71. (with Ranee Brylinski), Differential Operators on Conical Lagrangian Manifolds, in: Lie Theory and Geometry: In Honor of Bertram Kostant, Editors J.-L. Brylinski et al., Prog. Math., Vol. 123, Birkh¨auser, Boston, 1994, 65–96. 72. (with Ranee Brylinski), Minimal Representations, Geometric Quantization, and Unitarity, Proc. Natl. Acad. Sci., 91 (1994), 6026–6029. 73. Structure of the Truncated Icosahedron (such as Fullerene or Viral Coatings) and a 60-element Conjugacy Class in PSl(2,11), Proc. Natl. Acad. Sci., 91 (1994), 11714–11717. 74. Immanant Inequalities and 0-weight Spaces, J. Amer. Math. Soc., 8 (1995), 181–186. 75. (with Ranee Brylinski), Lagrangian Models of Minimal Representations of E 6 , E 7 and E 8 , in: Functional Analysis on the Eve of the 21st Century: In Honor of I. M. Gelfand, Prog. Math., Vol. 131, Birkh¨auser, Boston, 1995, 13–63. 76. Structure of the Truncated Icosahedron (e.g., Fullerene or C60) and a 60element Conjugacy Class in PSl(2,11), Selecta Mathematica, New Series, 1 (1995), Birkh¨auser, 163–195. 77. The Graph of the Truncated Icosahedron and the Last Letter of Galois, Notices of the American Math. Soc., Sept. 1995, 959–968. 78. Flag Manifold Quantum Cohomology, the Toda Lattice, and the Representation with Highest Weight ρ, Selecta Mathematica, New Series, 2 (1996), 43–91. 79. Clifford Algebra Analogue of the Hopf–Koszul–Samelson Theorem, the ρDecomposition, C(g) = End Vρ ⊗ C(P), and the g-Module Structure of ∧g, Adv. in Math., 125 (1997), 275–350. 80. Quantum Cohomology of the Flag Manifold as an Algebra of Rational Functions on a Unipotent Algebraic Group, Deformation Theory and Symplectic

xvi

Collected Papers Geometry, Editors D. Sternheimer et al., Kluwer Academic Publishers, 1997, 157–175.

81. The Set of Abelian Ideals of a Borel Subalgebra, Cartan Decompositions, and Discrete Series Representations, IMRN, 5 (1998), 225–252. 82. On ∧g for a semisimple Lie algebra g, as an equivariant module over the symmetric algebra S(g), in: Analysis on Homogeneous Spaces and Representations of Lie Groups, Okayama–Kyoto, Editors Toshiyuki Kobayashi et al., Advanced Studies in Pure Mathematics, Vol. 26, Mathematical Society of Japan, 1998, pp. 127–142. 83. (with B. Gross, P. Ramond and S. Sternberg), The Weyl character formula, the half-spin representations, and equal rank subgroups, Proc. Natl. Acad. Sci., 95 (1998), 8441–8442. 84. A Cubic Dirac Operator and the Emergence of Euler Number Multiplets of Representations for Equal Rank Subgroups, Duke Math. Jour., 100 (1999), No. 3, 447–501. 85. A Generalization of the Bott–Borel–Weil Theorem, Lett. Math. Phys., 52 (2000), 61–78. 86. On Laguerre Polynomials, Bessel Functions, Hankel Transform and a Series in the Unitary Dual of the Simply-Connected Covering Group of Sl(2,R), Representation Theory, AMS Electronic Jour. of Representation Theory, 4 (2000), 181–224. 87. The Weyl algebra and the structure of all Lie superalgebras of Riemannian type, Transformation Groups, 6 (2001), Birkh¨auser, 215–226. 88. Dirac Cohomology for the cubic Dirac operator, in: Studies in Memory of Issai Schur, Vol. 210, Prog. Math., 2003, Birkh¨auser, Boston, pp. 69–93. 89. (with Peter Michor), The generalized Cayley transform from an algebraic group to its Lie algebra, in: The Orbit Method in Geometry and Physics, In Honor of A. A. Kirillov, Editors C. Duval et al., Vol. 213, Prog. Math., 2003, Birkh¨auser, Boston, pp. 259–296. 90. A branching law for subgroups fixed by an involution and a non-compact analogue of the Borel–Weil theorem, in: Noncommutative Harmonic Analysis, In Honor of Jacques Carmona, Editors P. Delorme and M. Vergne, Vol. 220, Prog. Math., 2004, Birkh¨auser, Boston, pp. 291–353. 91. Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra, Invent. Math. 158 (2004), 181–226. 92. Minimal coadjoint orbits and symplectic induction, in: The Breadth of Symplectic and Poisson Geometry, Festschrift in Honor of Alan Weinstein, Editors

Collected Papers

93.

94.

95.

96. 97. 98. 99.

100. 101. 102. 103. 104.

105. 106.

xvii

J.E. Marsden and T.S. Ratiu, Vol. 232, Prog. Math., Birkh¨auser, Boston, 2005, pp. 391–422. The Coxeter element and the Branching Law for Finite Subgroups of SU(2), in: The Coxeter Legacy, Reflections and Projections, Editors Chandler Davis and E.W. Ellers, Vol. 48, Fields Institute Communications, AMS, 2005, pp. 63–70. (with Nolan Wallach), Gelfand–Zeitlin theory from the perspective of classical mechanics, I, in: Studies in Lie theory, Anthony Joseph Festschrift, Editors J. Bernstein, V. Hinich, and A. Melnikov, Vol. 243, Prog. Math., Birkh¨auser, Boston, 2006, pp. 319–364. (with Nolan Wallach), Gelfand–Zeitlin theory from the perspective of classical mechanics, II, in: The Unity of Mathematics, in Honor of I. M. Gelfand, Editors P. Etingof, V. Retakh, and I.M. Singer, Vol. 244, Prog. Math., Birkh¨auser, Boston, 2006, pp. 387-420. On the Centralizer of K in U (g), in Honor of E. Vinberg, J. Algebra 31 (2007), no. 3, 12051, 252–267. On maximal Poisson commutative subalgebras of S(g) and the generalized Hessenberg subvariety for any reductive Lie algebra g, in preparation. On the Cascade of Orthogonal Roots: Cent U (n) and Representation Theory, in preparation. Root Systems for Levi Factors and Borel–de Siebenthal Theory, arXiv:0711.2809; to appear in Progress in Mathematics, Symmetry and Spaces, in Honor of Gerry Schwarz, Campbell et al. (eds.), Birkh¨auser, 2009. On the two-dimensional projection of root systems of any reductive Lie algebra, generalizing Gossett’s projections, in preparation. Abelian ideals of dimension equal to rank and the generalized Amitsur– Levitski Theorem, in preparation. On Some of the Mathematics in Garrett Lisi’s “Theory of Everything,” in preparation. On Dale Peterson’s 2rank Abelian ideal theorem, symmetric spaces, and W. Schmid’s Discrete series construction, in preparation. (with Nolan Wallach), On a Generalization of a Theorem of Ranee Brylinski, In Honor of Varadarajan, to appear in Contemporary Mathematics, AMS, 2009. Fomenko–Mischenko Theory, Hessenberg Varieties, and Polarizations, arXiv:0812.4630, to appear in Letters in Mathematical Physics, 2009. (with Nolan Wallach), Exotic finite subgroups of E 8 and Springer’s regular elements of the Weyl group, in Oberwolfach Report No. 15/2009, Conference on Enveloping Algebras and Geometric Representation Theory, 39–42.

“This page left intentionally blank.”

Acknowledgments

We thank the publishers and copyright holders of Bertram Kostant’s papers in Volume I of his Collected Papers for their kind permission to reprint his articles. Only those articles in Volume I, noted by an asterisk in the list of Collected Papers and requiring the publisher’s permission, have been included below. All publishers’ permissions will be given in future volumes of the Collected Papers. American Mathematical Society 1. Holonomy and the Lie Algebra of Infinitesimal Motions of a Riemannian Manifold, Transactions of the American Mathematical Society, 80 (1955), 528-–542. 10. A Formula for the Multiplicity of a Weight, Transactions of the American Mathematical Society, 93 (1959), No. 1, 53—73. 14. (with G. Hochschild and A. Rosenberg), Differential Forms on Regular Affine Algebras, Transactions of the American Mathematical Society, 102 (1962), No. 3, 383—408. 16. Lie Group Representations on Polynomial Rings, Bulletin of the American Mathematical Society, 69 (1963), No. 1, 518-–526. 21. Groups Over Z, Proceedings Symposia in Pure Mathematics, 9 (1966), 90—98. Annals of Mathematics 13. Lie Algebra Cohomology and the Generalized Borel–Weil Theorem, Annals of Mathematics, 74 (1961), No. 2, 329—387. 18. Lie Algebra Cohomology and Generalized Schubert Cells,Annals of Mathematics, 77 (1963), No. 1, 72—144. Duke University Press 9. A Characterization of the Classical Groups, Duke Mathematics Journal, 25 (1958), No. 1, 107–124. Elsevier 19. Eigenvalues of a Laplacian and Commutative Lie Subalgebras, Topology, 13 (1965), 147–159. Illinois Journal of Mathematics 15. (with G. Hochschild), Differential Forms and Lie Algebra Cohomology for Algebraic Linear Groups, Illinois Journal of Math, 6 (1962), 264–281.

xx

Acknowledgments

Indiana University Mathematics Journal 8. A Theorem of Frobenius, a Theorem of Amitsur–Levitski and Cohomology Theory, Journal of Mathematics and Mechanics, 7 (1958), No. 2, Indiana University, 237–264. The Johns Hopkins University Press 11. The Principal Three Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group, American Journal of Mathematics, 81 (1959), 973–1032. 17. Lie Group Representations on Polynomial Rings, American Journal of Mathematics, 85 (1963), 327–404. Nagoya Mathematical Journal 7. On Holonomy and Homogeneous Spaces, Nagoya Mathematical Journal, 12 (1957), 31–54. 12. A Characterization of Invariant Affine Connections, Nagoya Mathematical Journal, 16 (1960), 35–50. Nippon Hyoronsha 20. Orbits, Symplectic Structures and Representation Theory, Proceedings U.S.-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965, p. 71.

Reprinted from the TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 80, No. 2, pp. 528-542 November, 1955

HOLONOMY AND THE LIE ALGEBRA OF INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD BY

BERTRAM KOSTANT

Introduction •. 1. Let M be a differentiable manifold of class Coo. All tensor fields discussed below are assumed to be of class Coo. Let X be a vector field on M. If X vanishes at a point oEM then X induces, in a natural way, an endomorphism ax of the tangent space Vo at o. In fact if yE Vo and Y is any vector field whose value at 0 is y, then define axy= [X, Y]o. It is not hard to see that [X, Y]o does not depend on Y so long as the value of Yat 0 is y. Now assume M has an affine connection or as we shall do in this paper, assume M is Riemannian and that it possesses the corresponding affine connection. One may now associate to X an endomorphism ax of Vo for any point oEM which agrees with the above definition and which heuristically indicates how X "winds around 0" by defining for vE Vo, axv= - v.X, where V. is the symbol of covariant differentiation with respect to v. X is called an infinitesmal motion if the Lie derivative of the metric tensor with respect to X is equal to zero. This is equivalent to the statement that ax is skew-symmetric (as an endomorphism of Vo where the latter is provided with the inner product generated by the metric tensor) for all oEM. §1 contains definitions and formulae which will be used in the remaining sections. In §2 we consider g, the space of all infinitesimal motions on M. g is a finite dimensional Lie algebra under the usual bracket operation for vector fields. Each element X Eg is uniquely determined over M by the value x of X and ax at any single point oEM. These known facts and a statement giving [X, Y]o and a[X,YJ for X, YEg (and consequently giving the structure of g) in terms of x, y, ax, ay and the curvature tensor at 0, generalizing that given by E. Cartan when M is symmetric, are contained in Theorem 2.3 of this section. In §3 we consider ~o the Lie algebra of the restricted homogeneous holonomy group at oEM. (~o is a Lie algebra of skew-symmetric endomorphisms of V o.) We are interested in the question as to when axE~o for XEg. As Euclidean space clearly illustrates, this is not true in general. When x = 0, Lichnerowicz [12] has shown that it is true if ~o acts irreducibly on Vo (M is thereby called irreducible as this is independent of 0) and the Ricci tensor does not vanish. It is true under these conditions whether or not x=o. It is the main result of this section that this is true (axE~o) whenever M is compact. Received by the editors

M~rch

20, 1955.

528 B. Kostant, Collected Papers, DOI 10.1007/b94535_1, © Bertram Kostant 2009

1

INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD

In §4 we consider

529

90 the Lie algebra of skew-symmetric endomorph isms of

Vo generated by all ax for XEg. One knows that in general 90Cn(~0) where n(~o) is the normalizer of ~o in the Lie algebra, ao, of all skew-symmetric endomorphisms of Vo. In this section we take up the reverse question of §3,

namely, when does ~oC90. We show that this holds whenever g is sufficiently big, i.e. when g is transitive in the sense that for any vector vE Vo at any point oEM there exists XEg such that x=v. This is of course true whenever Mis any open submanifold of a homogeneous space. It is then the principal result of this section that ~o = 90 when M is a compact homogeneous space. 1. Preliminaries.. 1. Let M be a Riemannian manifold of class C... By this we mean that not only is M a differentiable manifold of class C.. but that the metric tensor is likewise of class C... Similarly, without any mention to the contrary all tensor fields considered in this paper are assumed to be defined on all of M and are of class Cao. We adopt the following notations: (a) 0 is an arbitrary point of M, ((1) Vo is the tangent space at 0, h) x, Y, Z are vector fields on M, and unless specified otherwise x, y, z are, respectively, the values of these vector fields at 0, (5) S is an arbitrary tensor field and s its value at 0, (e) u, v, are arbitrary vectors . . 2. For any oEM we define for the present V!= Vo and V;;-l= V: (the dual of Vo) and where'Y is the k-tuple ('Yl, 'Y2, ... ,'YIo), 'Yi = ± 1, i = 1, 2, . n, define V;; = V;;1 ® ... ® 1';;10. Then the direct sum

is of course the mixed tensor algebra at oEM. If 'G is the algebra of all mixed tensor fields on M let us consider any linear transformation Do from 'G into 'G( Vo) which satisfies (1) D o (SlS2) =slDo(S2) +D o(Sl)S2, (2) Do preserves tensor type, (3) Do(SC) =Do(S)c (where C designates any specific contraction). We shall call such a linear transformation a 'G-differentiation at oEM. An example of a 'G-differentiation at 0, with which we shall be concerned, is given by covariant differentiation Vv of 'G by a vector vE Vo. Where i is a function V vi is just vi, i.e. differentiation of i by the vector v. However, any 'G-differentiation at 0 defines a differentiation, at oEM, of the space of functions of class C"" on M and hence there exists a vector v E Vo such that Do- Vv vanishes on all such functions. If we then apply Do- Vv to a vector field X we see easily that (Do- Vv)X depends only on x so that Do- Vv defines an endomorphism a on Vo. Similarly, it defines an endo-

2

530

BERTRAM KOSTANT

[November

morphism on Vo* which by (3) is simply -a* (where a* designates the transpose of a on V o*). Moreover, letting al =a and a_I = -a* for the present, it follows that Do- \l v applied to any tensor field S of type "I = ("II, "12, ... ,'Yk) is equal to k

a"y =

L: 1 ® 1 ® ... ® a"Yi ®

... ® 1 (a"y. the ith term, k = number of terms)

;=1

applied to s. Conversely, given any endomorphism a on Vo we may consider the endomorphism a-y of Vo defined in just this way. Furthermore if S is any tensor field decomposing into the sum S= S-y according to type and we define v(a)S= L:"Y a-ys"Y we see that v(a) is just a 'G-differentiation at o. Thus we may associate to every endomorphism a of Vo a 'G-differentiation v(a) at o. In the future we shall drop the symbol p in p(a) and simply understand that any endomorphism a of V by an extension of its domain may be applied to any tensor field in the manner discussed above. It follows then that any 'G-differentiation Do at oEM may be uniquely written as

L:-y

Do =

\l~

+a

for some vector vE Vo and some endomorphism a of Vo and that every such sum is a 'G-differentiation. We see then that a 'G-differentiation is determined once we know how it acts on functions and vector fields . .3. If X is a vector field we shall let \lx be the derivation of 'G defined by setting \lxS equal to \l",S at any point oEM. I ts action on functions already given, the definition of covariant differentiation on M is uniquely determined in that it satisfies for any two vector fields X and Y the equations (1.3.1) (1.3.2)

= (\l ~X, Y) + (X, \l ~Y), [X, Y] = \lxY - \lyX

v(X, Y)

where ( , ) designates the inner product in the tangent space at any point given by the metric tensor at the point and [X, Y] is just the Poisson bracket of the vector fields X and Y. (1.3.1) expresses the fact that the metric tensor has covariant derivative zero and (1.3.2) the fact that the torsion tensor is zero . .4. Now if X and Yare any two vector fields, then it is easy to see that [\lx, \lY] -

\l[X,Y]

defines, at each point oEM, a 'G-differentiation. Moreover this 'G-differentiation vanishes on functions and thus is given by an endomorphism of Vo. Depending only on x and y this endomorphism, t(x, y), is well known to be the contraction of the curvature tensor, t, at 0 on the last two indices by x and y. Letting T be the curvature tensor field and contracting in the same way with X and Y we thus have (1.4.1)

[\lx, \lY] -

\l [X,Y] = T(X, Y)

3

1955]

INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD

531

as a field of tangent space endomorphisms. Besides being skew-symmetric in u and v the endomorphism t(u, v) satisfies the first Bianchi identity (1.4.2)

t(u, v)w

+ t(v, w)u + t(w, u)v = o.

.5. We shall call an endomorphism a of Vo symmetric (resp. skew-symmetric) if a satisfies (au, v) = (u, av) (resp. (au, v) = - (u, av)) for all u, vE V o , i.e. if the second order covariant tensor obtained from a by lowering its upper index is symmetric (resp. skew-symmetric). It is well known that t(u, v) is skew-symmetric. The final relation involving the curvature tensor which we shall make use of in this paper is the second Bianchi identity, (1.5.1)

[V .. T](v, w)

+ [VvT](w, u) + [VwT](u, v)

= o.

The contraction again being on the last two indices. The above identities are proved, for example, in [13]. 2. A structure theorem .. 1. Let X be any vector field. We may associate to X a field of tangent space endomorphisms Ax, i.e. a tensor field of one upper index and one lower index, in the following way: For vE Vo we define the value ax of Ax at 0 by (2.1.1) Consider the derivation Vx+Ax of 'G. Clearly

('Ix

+ Ax)J = Xf

where f is a function.

('Ix

+ Ax)Y =

VxY

+ AxY

VxY - VyX = [X, Y] by (1.3.2). However if Lx is the derivation of 'G corresponding to Lie differentiation with respect to X, then at each point Lx defines a 'G-differentiation. Moreover, Lxf = Xf for f a function and

LxY = [X, Y] for Ya vector field. Thus by §1.2 (2.1.2)

Lx= Vx+Ax.

I t is well known that for X and Y (2.1.3)

[Lx, Ly] = L[x,y).

4

532

BERTRAM KOSTANT

[November

In fact this follows easily since [Lx, Ly] -L[x,y) defines a 'G-differentiation at each point which vanishes on functions and by the Jacobi identity vanishes on vector fields and hence is zero. The relation (2.1.3) enables us to give an expression for A [X ,Y) in terms of Ax and A y . Indeed by (2.1.2) and (2.1.3) ['ilx

+ Ax,

'ily

+ Ay]

= 'il [X,Y)

=

'ilx(Ay) ,

+ A [X,Y)

or

But ['ilx, Ay]

['ily, Ax] = 'ily(Ax)

and since ['ilx, 'ilY] -

'il [X,Y) = T(X, Y)

we have (2.1.4)

T(X, Y)

+ 'ilx(Ay)

-

'ily(Ax)

+ [Ax, Ay] =

A [X,y).

.2. A vector field X is called an infinitesimal motion or Killing vector if the Lie derivative of the metric tensor with respect to X is equal to O. This is equivalent to saying that ax is skew-symmetric at all points oEM, or as we shall say Ax is skew-symmetric. Since it is trivial to show that if Ax is skew-symmetric then 'il v(Ax) is likewise skew-symmetric it follows from (2.1.4) that if X and Yare irifinitesimal motions then so is [X, Y]. Thus if g is the space of all infinitesimal motions on M, g becomes a Lie algebra under the Poisson bracket. g, as is well known, and which will follow later, is finite dimensional; however, finite dimensionality is not understood here to be a requirement for the definition of a Lie algebra. It is a known fact (see [13, p. 45]) that if X is a vector field then the endomorphism 'il v(Ax) - t(x, 'II)

of Vo is the value at 0 of the Lie derivative, by X, of the affine connection contracted by 'liE Vo. Since if XEg the Lie derivative of the affine connection is zero it follows of course that for XEg the above expression vanishes. For completeness we shall give a proof of this which does not involve a computation of the Lie derivative of the affine connection. We state it in the following form. LEMMA

2.2. Let X be a vector field. Then at any point oEM, 'ilv(Ax)

5

1955]

INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD

=t(x, v) for all

533

vE Vo if and only if \7v(Ax) is skew-symmetric for all vE Vo.

Proof. It is of course trivial that if the equality holds then \7 v(Ax) is skew-symmetric for all vE Vo. Let X be arbitrary. We first show that the vector \7 y(Ax)z - t(x, y)z

as a bilinear function of y and z is symmetric in y and z. In fact embed y and z in the respective vector fields Y and Z. Now \7y(Ax)Z

= [\7y, Ax]Z = ,\JyAxZ - Ax ,\7yZ

= - \7y \7 zX - Ax \7yZ.

Then \7y(Ax)Z -

\7z(Ax)Y

= [\7z, \7Y]X + Ax[Z, Y] (by (1.3.2)) = ([ \7 z, \7Y] - \7 [z .y])X = T(Z, Y)X (by (1.4.1)).

But by (1.4.2) t(z, y)x

= t(x, y)z - t(x, z)y.

Thus we have the symmetry in y and z expressed in the relation

= \7.(A,,)y - t(x, z)y.

\7 y(A,,)z - t(x, y)z

Let cp(v, u, w)

= (\7 v(Ax)u, w) - (t(x, v)u, w).

Then is symmetric in u and v. If we now make the assumption that \7 v(Ax) is skew-symmetric for all vE V o , then (v, u, w) is skew-symmetric in u and w. But any trilinear form symmetric in two variables and skew-

symmetric in two others is necessarily identically zero. Thus \7 v(Ax)

= t(x, v).

Q.E.D .

.3. If we consider the question as to how much information is yielded about XEg by knowledge of x and ax, we are led, heuristically at any rate, to thinking of X decomposing into a sum of an infinitesimal translation at 0 in the direction of x and an infinitesimal rotation about 0 given in the tangent space of 0 by ax. Indeed if we consider for the moment the case where M is Euclidean n-space with the usual metric, and we let 0 be the origin, we know that g decomposes into the direct sum g = a+ V where a is the subalgebra of g composed of all elements in g vanishing at 0 and V is the commutative subalgebra of all infinitesimal translations. Now if we let no be the Lie algebra of all skew-symmetric endomorphisms of Vo and we define go = no+ Vo and introduce a bracket operation into go by setting

6

534

BERTRAM KOSTANT

[November

[aI, a2] = ala2 - a2al E ao, [a, v] = a(v) E Yo, [v, a] = - a(v) E Yo, [Vb V2] = 0 for a, aI, a2Eao; V, VI, V2E Yo, then go is a Lie algebra and g is isomorphic to go under the mapping which sends X into ax+x. This mapping of course sends a into ao and V into Yo. More generally it was shown by E. Cartan that if M is a simply connected Riemannian symmetric space and 0 is any point of M, then g is isomorphic to go = fo+ Vo where fo is the subalgebra of all skew-symmetric endomorph isms a of Vo which leave invariant the curvature tensor at oEM (infinitesimally, i.e., aT=O). The bracket relation in go is introduced in the same way as before with respect to [fo, fo], [fo, Vo], [Vo, fo], but for [Vo, Vo] we now have more generally [Yo, Vo]efo where for

VI,

V2 E Yo.

The isomorphism between g and go is again established by mapping X into ax+x. We return now to the case where M is perfectly general. Let ao Qe all skew-symmetric endomorph isms of Yo. Let Qo=a o+ Yo. We introduce into go a bracket operation where [aI, a2] = ala2 - a2al, [a, V] = a(v) , [V, a] = - a(v) ,

[VI,

V2] = t(V2,

VI)

for a, aI, a2Ea o and V, VI, V2E Yo. go is not in general a Lie algebra under this bracket (the Jacobi identity fails in the general case) but of course subalgebras of go may be Lie algebras. We have seen that in the symmetric case g is isomorphic to a sub algebra go of go under the mapping X-+ax+x. The following theorem states that this is true in general. THEOREM

2.3. Let oEM. Let Qo=ao+ Vo be the algebra defined in §2.3. Let

g be the Lie algebra of infinitesimal motions on M. Let 90 :g-+go be the mapping

defined by 80 (X)

= Xo = ax + x

and go =90 (g); then 90 is an isomorphism of g onto go. Proof. Implicit in the statement of the theorem is that, first of all, 90 is one-one. This is equivalent to the statement that an infinitesimal motion X is uniquely determined over all of M by the value of Ax and X at anyone point. To see this, let M and let r-+o(r) be a differentiable curve, 1~r~2, connecting 01 with 02. Then ifax(r) and x(r) are respectively the values of Ax and X respectively at oCr), and vCr) is the tangent vector to the curve at oCr), then it follows from (2.1.1) and Lemma 2.2 that the pair x(r) and arCr) satisfy the linear differential equations

7

1955]

INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD

'l,,(r)x(r)

(2.3.1)

=

ax(r)v(r),

-

'l,,(r)ax(r) = t(x(r),

(2.3.2)

535

vCr»~

so that a knowledge of them at anyone point of the curve determines their value at any other. To show ()o is a homomorphism, let X, YEg and let Xo=()oX, Y"=(),,Y. Now by (2.1.4) T(X, Y)

+

'lx(Ay) - 'ly(Ax)

+

[Ax, Ay] = A [X,y).

But since X and Yare infinitesimal motions we have by Lemma 2.2 that 'lx(Ay) = T(Y, X), 'ly(Ax) = T(X, Y). Thus (2.3.3)

A[x,Y) = T(Y, X)

+ [Ax, Ay].

Moreover by (1.3.2) [X, Y] = 'lxY - 'lyX or by (2.1.1) [X, Y] = AxY - AyX. Hence, by evaluating [X, Y] and Arx,y) at Oo[X, Y]

=

0

we see that Q.E.D.

[XO, Yo].

Note. With respect to the curve r~o(r) mentioned in the proof of Theorem 2.3 and by means of the linear differential equations (2.3.1) and (2.3.2) we obtain, by solving these equations, a linear one-one mapping from gOl to g.,J' If now 01=02=0 it is clear that the elements of go are pointwise fixed. In fact it is not hard to see that go is characterized as the set of those elements in go which remain invariant under all mappings of go into itself obtained in this way for all closed differentiable curves passing through o. For any XoEgo, Xo=a+x, we define Ix. to be the 'G-differentiation 'l ",+a at o. The question as to whether the Jacobi identity is satisfied for three elements X o , Yo and ZoEgo is, by using the second Bianchi identity (1.5.1), seen to be equivalent to the question as to whether (lx.T)(y, z)

+ (ly.T)(z,

x)

+ (lz.T)(x,

y)

vanishes. (The contraction is again on the last two indices.) In case Xo=()oX for some vector field X, then Ix. is by (2.1.2) just the Lie derivative Lx evaluated at o. If therefore XEg it follows that Ix T=O since the curvature tensor is invariant under motions. 3. Holonomy and infinitesimal motions.. 1. Let 1/10 be the homogeneous holonomy group at oEM. 1/10 is a group of orthogonal endomorphisms of the 0

8

536

BERTRAM KOSTANT

[November

tangent space V o, orthogonal with respect to the inner product in Vo. We recall, by definition, that an orthogonal endomorphism a belongs to 1/10 if the endomorphism, a: Vo~ V o, is induced by transporting (by parallelism) Vo completely around a closed piecewise differentiable curve on M. It returns to its initial position only after being subjected to the rotation a. The restricted homogeneous holonomy group (To at 0 is a normal subgroup of 1/10 and is obtained in the same way as 1/10 except that the curves are restricted to be homotopic to zero. It is a known result [3] that (To as a group of orthogonal endomorph isms is compact and connected and is in fact the connected component of the identity in 1/10 • . 2. Let ~o be the Lie algebra of (To. ~o is of course a subalgebra of no. Consider, in no, the inner product, (3.2.1)

B is clearly negative definite. Let 110 be the orthocomplement of ~o in ao with respect to B. Let fo be the set of XEg such that x vanishes and let fo =8 fo. fo is a subalgebra of g; in fact fo is a subalgebra of no. If Ko is the group of all motions of M which keep 0 fixed then its Lie algebra, represented as vector fields on M, is a subalgebra of fO so that it is natural to call fO the isotropy algebra at 0 and fo the linear isotropy algebra at o. It is a simple fact that if a E fo then a lies in the normalizer of ~o. It is not much harder to show this is true for any ax where XEg. A more general statement is derived as an easy consequence of the next lemma; it is more general in that it deals with 1/10 instead of (To or ~o. For any XEg we consider the decomposition of ax, 0

(3.2.2)

ax = bx

+ ex,

where bxE~o and exE11o. Let Ex (resp. Bx) be the field of tangent space endomorph isms which at any point 0 takes the value ex (resp. bx). LEMMA

3.2. The field Ex is covariant constant.

Proof. We use in the proof two basic facts about holonomy; (1) if oCr), rl ~r ~r2' is a differentiable curve then ~OCrl) as a subspace of noC'l). is carried

into ~0(r2) under parallel transport along the curve, and (2) at any point 0 and for any u, vE Vo , t(u, v) E~o, see, for example, [1]. But Jrom (1) it follows that the orthocomplement 11oh) of ~o(rl) is carried into 11o(r,) under parallel transport along the curve. Thus to prove the lemma it is sufficient to show the covariant derivative of Ax by any vector vE Vo lies in ~o. Now by Lemma 2.2, v.(Ax) = t(x, v) E £1 0 •

Thus we have shown V.(Ex) =0. Q.E.D.

9

1955J

INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD

537

.3. We are interested in the question as to when axE~o, i.e. when ex =0, or equivalently, Ex=O. It follows from a result of Lichnerowicz [12] that this is the case when the Ricci tensor is not zero and ~o acts irreducibly on Vo. In fact that author uses his result to prove that axE~o when XEf o under those assumptions. For completeness we repeat his argument (slightly modified) here. Assume Ex~O. Since ~o acts irreducibly it follows that Mis pseudo-Kahlerian. The result in [12] alluded to above is that if Mis pseudo-Kahlerian and the Ricci tensor is not zero then ~o has a nonzero center. But if cE~o is in the center then c, ex, and the identity endomorphism generate a three-dimensional commutative algebra in the centralizer of ~o. Since the latter acts irreducibly this contradicts Schur's lemma. Of course we know that in general axEE~o, as Euclidean space clearly illustrates. We shall show, however, that axE~o whenever M is compact. THEOREM 3.3. Let M be a compact Riemannian manifold, X an infinitesimal motion on M, and ax the endomorphism of the tangent space Vo at an arbitrary point 0 E M defined by axv = - \I vX for any v E Vo. Then ax E ~o, the Lie algebra of the restricted homogeneous holonomy group at o.

Proof. We use the notation of §3.2. We shall show Ex=O. Let Y be a vector field defined by

Y = ExX. That is, Ex is applied to X at every point. For any vE Vo \IvY

=

(by Lemma 3.2).

Ex\lvX

Thus ayv = exax'IJ

and consequently Ay

=

ExAx.

Assume for the present that Mis orienfable. It is just the statement of Green's theorem, for example see [13, p. 31] that if Z is any vector field on Mthen

f

tr Azd'IJ = 0

where dv is the volume element on M associated with the metric. Thus

f

tr Aydv =

But tr ExBx=O at every point

f 0

tr

[E~ + ExBx]dv

=

o.

by definition of llo. Thus

10

538

BERTRAM KOSTANT

f However tr

tr

Ei is constant on 2

tr ex

E~dv

[November

= O.

M since Ex is covariant constant. Thus

=

B(ex, ex) = O.

But since B is negative definite, ex = O. In case M is not orientable then M has a two sheeted compact locally isometric covering space M which is orientable. The vector field X when lifted to M determines an infinitesmal motion X. The above argument shows that ex=O and hence obviously ex=O. Q.E.D. 4. Transitivity.. 1. Let g* be a subalgebra of g. Let V(g:) be the subspace of Vo defined by

{x E Vol

V(g:) =

X E g*}.

We shall say that g* is transitive on M if V(g:) = Vo for every oEM. .2. Let G be a connected Lie group and H a compact subgroup. Let M be the homogeneous space G/H (left coset space) when the latter is provided with a metric tensor which is kept invariant by the action of G on G/H. In this case we see that g contains a subalgebra g* which is transitive on M. In fact the right invariant Lie algebra of G maps homomorphically into g under the mapping which associates to everyone-parameter group get) of G the vector field XEg on M where

XJ(o)

=

lim J(g(t)o) - J(o) HI t

for any Coo function,J(o), on M. The image, g*, under this mapping is transitive on M. More generally then if M is an open sub manifold of such a homogeneous space, then g contains a sub algebra which is transitive on M . .3. For any point oEM consider the sub algebra rOCg, see §3.2. It is not hard to see that fo is reductive in g (the adjoint representation of fO on g is completely reducible). In fact g is isomorphic to goCao+ Vo and fo is isomorphic to foCa o (Theorem 2.3). Let

10

=

{ax I X E g}.

Of course foCIoCa o. ao is a Lie algebra under the usual bracket operation for endomorph isms. Furthermore any sub algebra, as for example f o , is reductive in ao• Moreover, by Theorem 2.3, 10 is invariant under the adjoint representation of fo on ao • Let jo be the orthogonal complement (with respect to the bilinear form B) of fo in 10 , Thus

11

1955]

INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD

539

where [fo, fo] C fo, [fo, io] C io. Define

\lo = {X0 E go I ax E io}. Then it follows that

go = fo

+ \lo,

where [fo, \lo]cpo . .4. The mapping of \lo into V(go) which sendsXo into x is one-one onto and defines an equivalence between the representation of fo on Po and that on V(go) (V(go) is an invariant subspace of Vo under the action of fo as endomorphisms). Since the latter is reductive the action of fo in \lo is reductive and hence the adjoint representation of fo on go is reductive. Now if M is a simply connected Riemannian symmetric space then it is a known result of E. Cartan that ax=O if BoX=XoE\lo, that is, io=O or Io=fo. Such a vector field is an infinitesimal transvection at oEM. It is also known that g is transitive on M. Conversely, if we assume that g is transitive on M and that io = fo for all 0 E M, then M must be Riemannian symmetric since if XEg then we know that LxT=O, i.e. "the curvature tensor is invariant under motions." Thus since Lx= \7x+Ax

(4.4.1)

\7",T = - axT.

However, since V(go) = Vo and ax =0 when BoX E \lo it follows that \7vT=O for all vE Vo. Thus the curvature tensor is covariant constant which is another way of saying that M is symmetric. More generally we see that M is symmetric if g is transitive and axT=O for all XEg and all oEM. Thus a distinction between a symmetric space and a non symmetric space on which g is transitive is that in the latter case there always exists X for which axT=;t'O whereas in the former axT=O for all XEg. N ow if M is symmetric and simply connected it is a result of E. Cartan that

(4.4.2)

~o C

Io( =fo)

C n(~o)

where n(~o) is the normalizer of ~o. N ow for the general case let 1)0 be the Lie subalgebra of ao generated by Io (in the case above 1)0 = Io since Io is already a Lie algebra). The next theorem extends the result expressed by (4.4.2) to the case where g is only assumed transitive on M and 1)0 replaces 10• In fact it is not necessary to deal with the full Lie algebra of infinitesimal motions and in fact Corollary 4.5 illustrates why the following theorem should be formulated in terms of a transitive subalgebra of g. THEOREM 4.4. Assume g* is a Lie algebra of infinitesimal motions on M which is transitive (see §4.1.). For any point oEM let 1): be the Lie algebra of

12

540

BERTRAM KOSTANT

[November

skt;w-symmetric endomorphisms of Vo generated by all ax for XEg*. Then if ~o is the Lie algebra of the restricted holonomy group,

*c

~o C 90

where

n(~o)

is the normalizer of

n(~o)

~o.

Proof. It was already mentioned in §3.2 that axEn(~o) for any XEg, a fact which follows from Lemma 3.2 since the latter implies [~o, ex] =0. Thus 9:Cn(~o). In order to prove ~oc9: we first need a lemma. LEMMA 4.4. Let S be a tensor field in an arbitrary Riemannian manifold M which is invariant under motions, i.e., LxS = 0 for all X E g. Let Wo be any subspace of 'G( Vo) which is invariant under ax for all X E g and such that sE Wo. Then for any Xl, X 2 , • • • , XkEg the vatue of the field \lXI

\Ix• ... \lx,p

at oEM is contained in Wo.

Proof of Lemma 4.4. If YEg and X is arbitrary then it follows easily from (2.1.2) and Lemma 2.2 that as operators on 'G [L y , \Ix] and consequently for any X 2 ,

••• ,

=

\7[y.x]'

Xk k

[L y , \7x.· .. \7Xk] =

L: \7x• ...

\7 [Y.Xil

•••

\7Xk'

i~2

Thus (4.4.3)

Ie

L y \7x.··· \7Xk = \7x.··· \lxkLy

+ L:

\7x.· .. \7[Y.X;l ... \7Xk'

i~2

Now assume Y=X I and Xl, X 2 • • • XkEgo. Assume also that the lemma is true for k-l. If then we apply (4.4.3) to S we see that LXI \7x• ... \7XkS evaluated at 0 is contained in Woo But Lx! = \lx1+A xl . Thus \7x I \7x • ... \7XkS

= LXI \7x• ... \7XkS - AXI \7X2 ... \7XkS ,

But \Ix• ... \7 Xk S at 0 lies in Wo and since Wo is invariant under ax! we see that \7Xl \7x• ... \lXkS at oEM lies also in Wo. But we already know that sEWo. Q.E.D. Proof of Theorem 4.4 continued. Assume g* is transitive on M. We apply Lemma 4.4 in the case where S = T, the curvature tensor. Let Wo be the subspace Vo ® ® ® consisting of all s such that for all u, v E Vo,

v: v: v:

s(u, v) E

90*

where s(u, v) means the contraction of s on the last two indices by the vectors u,v.

13

1955]

INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD

541

To show that Wo is invariant under ax we observe that for any sE Vo ® v:® V:® v: and any u, vE Vo (axs) (u, v)

= [ax, s(u, v)] - s(axu, v) - s(u, axv)

so that if sEWo it follows that axsEWo. Thus to' show that (\7 Xl \7 X 2 • • • \7 xkT) (u, v) lies in ~: for all Xl, X 2 • • XkEg it follows from Lemma 4.4 that all we have to show is that t(u, v)E~: for all u, vE Vo. It is at this point that we first use transitivity. Given any u, vE Vo, by the transitivity of g*, there exists X, YEg* such that x=u, y=v. Now by (2.3.3) [Ax, Ay] - A [X,Yl

= T(X, V).

Thus t(u, v) E~: for all u, vE Vo. We assert it follows from a theorem of Chevalley and Nijenhuis, see [8], that the Lie algebra generated by all endomorphisms of Vo of the form (\7Xl \7X2 ... \7xkT)(u, v)

where X.Eg, u, vE Vo, is ilo • In fact if we form this algebra at each point oEM where the vector fields Xi are arbitrary then it follows from that theorem that if the dimension of the algebra is constant over M then at each point oEM the algebra is ilo • What is asserted here is that since g* is transitive, we need only consider vector fields belonging to g* and that the dimension of the resulting algebra is constant. The first of these assertions follows from the fact that if Xl, X 2 , • • • , XnEg* is such that Xl, X2, ... , Xn form a basis of Vo, then any vector field Y equals L~-lfiX; in a neighborhood of 0 wheref;, i = 1, 2, ... , n, are suitable chosen functions. The second follows from the fact that the transitivity of g* yields the existence of two neighborhoods Vo and V; of 0 such that for each PEV; there exists an isometric mapping of Vo onto a neighborhood of P in which 0 maps into p. Consequently the dimension of the algebra is constant over M. Q.E.D . .5. As mentioned before if the Ricci tensor of M does not vanish and ilo acts irreducibly (i.e. M is irreducible) then axEilo for all X Eg, or as proved before (Theorem 3.3) this conclusion holds if M is compact. Consequently, we have, combining these facts with Theorem 4.4, THEOREM 4.5. Let M be a Riemannian manifold and g* any Lie algebra of infinitesimal motions which is transitive on M (see §4.1). Now if either one of the following conditions hold: (a) M is compact or (b) M is irreducible and the Ricci tensor does not vanish, then

14

542

BERTRAM KOSTANT

go*

= ~o

where ~o is the Lie algebra of the restricted homogeneous holonomy group at oEM and is the Lie algebra of endomorphisms of Vo generated by all ax (see § 2.1) for XEg*.

g:

COROLLARY 4.5. Let G be any connected Lie group and K a compact subgroup. Let M be the homogeneous space Gj K (left coset space) when the latter is provided with a metric tensor which is kept invariant by the action of G on GjK. Now if either one of the following conditions holds: (1) G is compact or (2) M is irreducible and the Ricci tensor does not vanish, then

go*

= ~o

g:

where is the Lie algebra of endomorphisms of Vo generated by all ax where X runs through the Lie algebra of infinitesimal motions on M arising from the Lie algebra of G (see §4.2) and ~o is the Lie algebra of the restricted homogeneous holonomy group at o. REFERENCES

1. W. Ambrose and 1. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. vol. 75 (1953) pp. 428--443. 2. A. Borel and A. Lichnerowicz, Espaces riemannien et hermitiens symetrique, C. R. Acad. Sci. Paris vol. 234 (1952) pp. 2332-2334. 3. - - - , Groupes d'holonomie des varietes riemanniennes, C. R. Acad. Sci. Paris vol. 234 (1952) pp. 1835-1837. 4. E. Cartan, Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math. France vol. 54 (1926) pp. 214-264; vol. 55 (1927) pp. 114-134. 5. - - - , La geometrie des groupes de transformations, J. Math. Pures Appl. vol. 6 (1927) pp.1-119. 6. R. Hermann, Sur les isometries infinitesimales et Ie group d'holonomie d'un espace de Riemann, C. R. Acad. Sci. Paris vol. 239 (1954) pp. 1178-1180. 7. - - - , Sur les automorphismes infinitesimaux d'une G-structure, C. R. Acad. Sci. Paris vol. 239 (1954) pp. 1760-1761. 8. A. Nijenhuis, On the holonomy group of linear connections, Indagationes Math. vol. 15 (1953) pp. 233-249; vol. 16 (1954) pp. 17-25. 9. K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. vol. 76 (1954) pp. 33-65. 10. - - - , Sur les transformations affine d'une variete riemannienne, C. R. Acad. Sci. Paris vol. 237 (1953) pp. 1308-1310. 11. - - - , Applications de l'etude de transformation affine aux espaces homogeneous reimanniens, C. R. Acad. Sci. Paris vol. 237 (1953) pp. 1386-1387. 12. A. Lichnerowicz, Espaces homogenes kahleriens, Colloque International de Geometrie Differentielle, Strasbourg, 1953, pp. 171-184. 13. K. Yano and S. Bochner, Curvature and Bette numbers, Annals of Mathematics Studies, no. 32, Princeton, 1953. INSTITUTE FOR ADVANCED STUDY, PRINCETON,

N. J.

15

Reprinted from the Proceedings of the NATIONAL ACADEMY OF Vol. 41, No. 11, pp. 967-970. November, 1955

SCIENCES,

ON THE CONJUGACY OF REAL CARTAN SUBALGEBRAS.

I

By BERTRAM KOSTANT DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO

Communicated by Saunders Mac Lane, September 2,1955

Among the questions which have been raised concerning the structure of a connected semisimple Lie group are those relating to conjugacy of its Cartan subgroups. In case the group is either compact or complex, it is a well-known fact (and indeed a fundamental one) that all Cartan subgroups are conjugate. It is also known that this is not true in general. The interest in the general case is heightened as a result of statements of Harish-Chandra l relating the" classes of conjugate Cartan subgroups and the various' series' of unitary representations which occur in the Plancherel formula." It is clear that one may reduce the problem to a consideration of the conjugate classes of Cartan sub algebras (C.S.'s) of a real, simple Lie algebra (conjugate, under the action of the adj oint group). 2 Moreover, in view of the above, one may restrict the Lie algebra to be a real noncompact form of All BII CII DII G2 , F 4 , E 6, E 7 , and E 8 • It is the purpose of this note to list a series of general theorems enabling us to "determine" the conjugate classes in everyone of the real forms mentioned above. By "determine" is meant, among other things, giving (1) the number of conjugate classes, (2) the number of classes for which the" vector part" (or" toroidal part") of a C.S. in that class has a given dimension, (3) a characterization of each conjugacy class with respect to the full group of automorphisms. (This may be done in all but a few cases by giving the centralizers of the "vector and toroidal parts" of a C.S. in that class. A more convenient type of characterization may be given in the C3.se of all but one classical algebra.) In a succeeding note, tables will be given listing enough of this information so that every C.S. (up to conjugacy by the full group of automorphisms) in all the real forms mentioned above may be identified. A list of all W-subspaces of the simple complex algebras up to C-conjugacy (see below) will be included also. Proofs and further elaboration will appear elsewhere. Let go be a real, semisimple Lie algebra. Let Go be the adjoint group. Let fo be the subalgebra of go corresponding to a maximal compact subgroup Ko ~ Go.

B. Kostant, Collected Papers, DOI 10.1007/b94535_2, © Bertram Kostant 2009

16

968

MATHEMATICS: B. KOSTANT

PROC.

N. A. S.

Let Po be the orthogonal complement to fo in go with respect to the Cartan-Killing bilinear form B. Then go = fo + Po. Let mo be a maximal commutative subalgebra of go contained in po. The group Ko leaves Po invariant. Let W. be the group of transformations of mo induced by the subgroup of Ko which leaves mo invariant. W. is a finite group. Let ~o be a Cartan subalgebra. Let ~o-, the "vector part" (~o+, the "toroidal part"), be the subspace {X E ~o I the eigenvalues of adX are real (pure imaginary) }. Then ~o = ~o+ ~o-. We will call a Cartan subalgebra ~o standard if ~o + ~ fo and ~o c:; mo cpo. Theorems, essentially equivalent to our first four theorems, have been proved independently (unpublished) by A. Borel. THEOREM 1. Every Cartan subalgebra is conjugate to a standard Cartan subalgebra. Moreover, two standard Cartan subalgebras are themselves conjugate if and only if their" veCtor parts" are conjugate with respect to W •. 2 Let g be the complexification of go. Let ~ be the complexification of a Cartan subalgebra of go containing mo. Let ~6 be the real subspace of ~ spanned by the roots. Then ~ = ~6 + i~6 and go n ~6 = mo. We may regard the Weyl group W or the Cartan group C associated with g as operating in ~6. We will call a subspace of ~6 a W-subspace if it is the eigenspace belonging to the eigenvalue -1 of an element of order 2 in W. LEMMA 1. 1 ~ ~o is a W-subspace if and only if it has a basis of orthogonal (with respect to B) roots. Where 11- means the orthogonal complement of r with respect to B in ~o, we have THEOREM 2. n c mo is the vector part of a standard C.S. of go if and only if n is of the form mo n 11-, where ( c:; mo and 1is a W-subspace. THEOREM 3. The group W. is obtained by the transformations induced on mo by the subgroup of W leaving mo fixed. Moreover, two W-subspaces of mo are conjugate with respect to W if and only if they are conjugate with respect to W •. We are thus reduced to the study of how W conjugates the W-subspaces. Assume that g is simple. We begin with Theorem 4, which tells how W acts on onedimensional W-subspaces. THEOREM 4. The orbits of W acting on the set of roots are the subsets of all roots having the same length. For a subspace 1 ~ ~6, let g[l] denote the complex subalgebra of g generated by r and those root vectors whose roots are in L Let f¥' denote the hyperplane in ~6 orthogonal to the root I{J. For any root a we will say that a root (:J is related to a if (a) B(a, (:J) = 0; (b) R(a, a) = B({:J, (:J); (c) for any root I{J, B(a, I{J) = 0, I{J ~ ±(:J implies B({:J, I{J) = O. If we now say that a root is related to itself and its negative, we have THEOREM 5. The relation above is an equivalence relation partitioning the roots into equivalence classes. Clearly C preserves these classes. If we order the roots in anyone of the usual ways and define n¥, to be the number of positive roots in the equivalence class of I{J, we have THEOREM 6. g [f¥'] = (n¥, - l)AI ffi g¥" where g¥, is simple or simple ffiD I (the latter case occurs only in A,).3 Also, the n¥, -1 copies of Al are in/act g[(I{J/)], where l{Ji runs through positive roots other than I{J which are related to I{J.

+

17

VOL.

41, 1955

MATHEMATICS: B. KOSTANT

969

The key factor in our method of determining the class of W-conjugate W-subspaces 4 is the simplicity of [g." g.,] in the above theorem; it enables us to repeat the use of Theorem 4, that is, in view of Lemma 1 and LEMMA 2. For any X Eg6, let Llx be the set of roots ({J such that B«({J, X) = 0; then, if U E W leaves X invariant, u is generated by the reflections through f.,for ({J E Ll x . We need LEMMA 3. Let Wa be the (commutative) subgroup of W generated by reflections through ffJ for all {3 related to a. If ({J is not related to a or orthogonal to a, then the set ± Wa({J is a union of equivalence classes. LEMMA 4. If I is a W-subspace and a E I, then, if {3 is related to a, either {3 E Ior {3 d.L. For a W-subspace I, let n ~ I be the subspace generated by all a E I such that I contains all {3 related to a. Both nand n.L n 1 are W-subspaces. We will call r complete if n = I, incomplete if n ~ 1, and totally incomplete if n = O. /j Because B2 = C2, our definitions make B2 a degenerate case (its W-subspaces up to W-conjugacy are obviously 0, «({J), (tY), and g6, where ({J and tY are roots and B(1/;, tY) = 2B«({J, ({J)). The following theorems are our main results: THEOREM 7. Assume that g is classical but not B 2; then two W-subspaces II and 12 are conjugate with respect to the Cartan group if and only if dim h = dim h,

dim nl

=

dim n2.

They are conjugate with respect to W if and only if these conditions hold, with the following exceptions: g = D 4 , dim I = 2, dim n = 0 (three conjugate classes), and g = D 2k , k ~ 3, dim I = k, dim n = 0 (two conjugate classes). For the exceptional Lie algebras two W-subspaces are W-conjugate if and only if they are C-conjugate. For any subspace 1 S. g6, let Ll [I] be the set of roots in L We now have THEOREM 8. If g is any simple complex Lie algebra and hand 12 are two W-subspaces, then II and 12 are C-conjugate if and only if there exist (one to one, onto) maps i and j (i: Ll [h] -- Ll [12], j: Ll [h.L] -- Ll [12.L]) which preserve the structure (length, addition, negatives), with the sole exception g = D 2k , k ~ 3, dim II = k, dim nl = 0, dim f2 = k, dim n2 = 2. In this case maps i andj exist but hand 12 are not C-conjugate. Returning to a real form go of g and a C.S. go S. go, we first observe that igo+ and go - are complementary orthogonal subspaces of some g6 ~ g. Consequently, we may speak of Ll [igo+] and Ll [go -]. We have, finally, THEOREM 9. For a real, simple Lie algebra go, the structure of Ll [igo+], together with that of Ll[go-], uniquely determines the conjugacy class of any C.S. go under the full group of automorphisms of go, with the sole exception g = D 2k , k ~ 3; go is the unique real form for which dim rno = 2k. For this case, corresponding to the pair of nonconjugate W-subspaces hand 12 of Theorem 8, there exist two distinct conjugate classes of C.S.'s both determining, for a representative C.S. go of either class, identical structuresfor Ll[igoT] and for Ll[go-]. We may substitute the adjoint group for the full automorphism group in the above statement, altering only the exceptional case by letting k ~ 2. Corresponding now to the exceptional W -subspaces of Theorems 7 and 8 there are three distinct conjugacy classes of C.S.'s all determining, for a representative C.S. go, identical structures for Ll [igo+] and for Ll [l)o-].

18

970

MATHEMATICS: B. KOSTANT

PROC. N. A. S.

* Based on research supported in part under Contract N60ri-02053, monitored by the Office of Naval Research. I Harish-Chandra, "Plancherel Formula for the 2 X 2 Real Unimodular Group," these PROCEEDINGS, 38, 337-342, 1952. • In general, the word "conjugate" will be used to mean transformable with respect to the action of the group under consideration. When speaking of conjugacy of C.S.'s, this group will be understood to be the adjoint group unless the full group of automorphisms is specified. , DI , the one-dimensional complex Lie algebra, is not semisimple, and D2 = AI E9 AI is not simple. , We say that II
P - J

j odd

both cases. D4(i)

(+)

j

(4 - j)2 + 12

q=4[~]

Dq)y

(See above for

(-)

; even;

; even

definition); [D4(2) = D4(A)]

16-(4-j)2

j

=

1.2.3.4

([~] (j -

«p - q)A1EB

O..sp..s;

P)Al)y EB

1

[1 -/1] D4 WEB 3

([~] (4 -

;+1

j even

j)Al)t

j even ............. .....

..................

..s p ..s;. q =

((I-[f])

1

; odd

°

(p - I)AIEB

............... .......

((1-

(j - p) EB

([f] Dl}}t

; odd

(1-[f])D1EB

[f]A3}y ; odd --

[f])AI5-2PI

--

..............

. ........

I

; ; odd

~ '"

~

On the Conjugacy of Real Cartan Subalgebras II Table IV. Real Forms

Dim

Dim

Dim

to

Po

rno

{go[~~J, go[~tJ}

No. of

Total

Classes for

No. of

fixed p

Classes

B2(AI EB AI)

6

4

1

{OIB2(AI EB Ad}; (At(l), At(O)}

1

2

B2(DI EB Ad

4

6

2

{O, B2(DI EB Ad}; (At(l), At(1)};

1

4

{At (1), At (l)}; {B2(DI EB All, O}

(p

= 0, 2) 2

(p G2(AI EB All

6

8

2

1)

1

{O, G2(AI EB All}; (At(l), At (I)}; (At (1), AtO)}; {G2(AI EB All, O}

=

(p

4

= 0,2) 2

(p F4(B4)

F4(AI EB C3)

(+)

(-)

36

16

24

28

1

2

(O, F4(B4)}; (AIO), B3(0)}

(O, F4(AI EB C3)}; (AIO), B3(3)}; (AI(l), C3(3)}; (2AIO),2AI(I)}; (B2(2), B2(2)}; (B3(3), Al (I)}; (C3(3). AI(1)}; (F4(AI EB C3), O}

E6(DI EB Ds)

46

32

2

=

{O, E6(DI EB Ds}; (AI(1), As(I)};

(p

1)

1

2

1

8

=

0,4)

2 (p=I,2,3)

1

3

(2AI (1), DI (0) EB A3(0)}; E6(F4)

52

26

2

(2DI(1), D4(0)}

1

1

E6(AI EB As)

38

40

4

to,

1

5

1

5

E6(AI EB As)}; (AIO), As(3)};

(2AI(l), A3(2) EB DdO)}; (3AI(I), Al (1) EB 2DI (O)}; (D4(4), 2DI (O)} E6(C4)

36

42

6

(2DI(1), D4(4)}; (2DI(I) EB AI(1), 3AI(I)}; (DI(1) EB A3(3), 2AI(I)}; (As(5), Al (1)}; {E6(C4),0};

E7(DI EB E6)

79

54

3

(O, E7(DI EB E6)}; (AIO), D6(2)};

1

4

1

5

(2AI(I), Adl) EB D4(0)}; (3AI (1), D4(0)} E7(AI EB D6)

69

64

4

(O, E7(AI EB D6)}; AI(I), D6(A)}; (2AI (1), Al (0) EB D4(2)}; {3AI (1), Al (1) EB 3AI (O)}; (D4(4), 3AI (O)}.

26

7

B. Kostant

8

Table IV (continued). Real Forms

E7(A7)

Dim

Dim

Dim

to

Po

mo

63

70

7

{lJo [~;;-], lJo [~t]}

{O, E7(A7»); {AI (I). D6(6)}; {2AI(I), Adl) EB D4(4)}; {3AI (I), D4(4)}; {3AJ (1), 4AI (I)};

No. of

Total

Classes for

No. of

fixed p

Classes

1

10

(p=O,I,2, 5,6,7)

{4AI (I), 3AI (I)}; {D4(4), 3AI (I)}; {AI(1) EB D4(4), 2AI(I)}; {D6(6), Al (I)}; {E7(A7), O}. E8(E7 EB All

136

112

4

to, E8(E7 EB All};

2 (p = 3,4)

1

5

{AI (1), E7(DI EB E6)};

{2AJ (1), D6(2)}; {3AI(I), AI(l) EB D4(0)}; {D4(4), D4(0)} E8(D8)

120

128

8

to, E8(D8)};

{AI(1), E7(A7)};

1

{2AI (1), D6(6)};

(p = 0,1,2,

(3AI (1), Al (1) EB D4(4)};

3,5,6,7,8)

10

{4AI (1), 4AI (I)}; {D4(4), D4(4)}; {D4(4) EB Al (1). 3AI (I)}; {D6(6), 2AI(I)}; {E7(A7), AI(1)};

2 (p = 4)

{E8(D8), O}

Note. Thanks to the editorial review by Fokko du Clos (February, 2005), the present Tables III and IV correct misprints and minor notation that were contained in the 1955 version.

27

Reprinted from the Proceedings of the NATIONAL ACADEMY OF SCIENCES, Vol. 42, No.3, pp. 148-151. March, 1956.

ON INVARIANT SKEW-TENSORS By BERTRAM KOSTANT DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY

Communicated by S. Le/schetz, January 16, 1956

Throughout this note V will denote a real n-dimensional vector space with a positive-definite inner product, its value for u, v E V being denoted by (u, v). Let 1\ PV be the space of contravariant skew-tensors over V of degree p. The inner product in V induces a positive-definite inner product in 1\ PV, where (UI 1\ ... 1\ Up, VI 1\ ... 1\ Vp ) = (lip!) . determinant (Ui' Vj). Let f be a Lie algebra of a compact group. A representation of f on V will be denoted by the triple { u, f, V}. It will be always understood that the representation is by skew-symmetric operators (skew-symmetric with respect to the given inner product), the operator corresponding to X E f being denoted by u(X). Given {u, f, V}, we will let {up, f, 1\ PV} be the corresponding representation of P

f on 1\ PV, where, as one knows, UP(X)VI 1\ ... 1\ Vp

=

:E VI 1\

... 1\ U(X)Vi 1\ ...

i = I

Note that { uo, f, 1\ °V} is the zero representation of f. Now let W p = {s E I\Pvl uP(X)s = 0 for all X E The elements of W p are called the pth-order skew-invariants of {u, f, V}. Let dp be the dimension of W p • 1\ Vp.

fl.

n

We will call P v(t)

:E dpt P the Poincare polynomial of {u, f, V}.

=

P = 0

We consider for the present the special class of representations which one associates to compact homogeneous Riemannian symmetric manifolds (hereafter called symmetric representations). That is, let M = GIK be such a manifold, G being the connected component of the identity in the full group of motions of M. We consider the representation { u, f, V}, where V is the tangent space at the point of M corresponding to the coset K (with the inner product given by the metric tensor), f is the Lie algebra of K, and u is the representation of f on V induced by the obvious action of K on V. All symmetric representations have been studied and classified by ~. Cartan. He has shown that they are all of the form ad f restricted to V (ad fl V), where g = l V is a LiE algebra satisfying (a) [f, fl £ f, (b) [V, Vl = f, and (c) [l, Vl £ V, where ad fl V is faithful and skew-symmetric with respect to a positive-definite inner product on V. Conversely, every such representation is a symmetric representation. Moreover, M may be chosen so that K is connected and hence as one knows the Poincare polynomial of M is precisely P v(t). We shall call {u, f, V} an S-representation if it is irreducible and symmetric. One knows all symmetric representations when one knows the S-representations. I The S-representations form a fairly extensive class (they are, of course, all faithful). Important for our purposes is a "small" subclass whose elements we call T-representations. We say that {u, f, V} is a T-representation if it is an S-representation and f is simple (it may be the zero- or one-dimensional Lie algebra). These representations will be characterized among all representations by Theorem 1. The T-representations are of two types: T I , the adjoint representations of the simple (non-Abelian) compact Lie algebras, and T 2 , the nonadjoint representations. The latter are listed below; they are specified by giving a corresponding compact

+

148

B. Kostant, Collected Papers, DOI 10.1007/b94535_4, © Bertram Kostant 2009

28

MATHEMATICS: B. KOSTANT

VOL. 42, 1956

149

homogeneous Riemannian symmetric space: Type T 2: (a) SU(m)/SO(m), m = 3,5, ... ; (b) SU(2m)/Sp(m), m = 3, ... ; (c) SO(m + 1)/SO(m), m = 1,2,5, ... ; (d) F4/Spin (9); (e) E 6/F 4; (f) E6/Sp (4); (g) E 7/SU (8); (h) Es/Spin(16).2 The representations {u, f, V} corresponding to SO(n + 1)/SO(n), n = 1, ... , we shall call complete. They are complete in the sense that f is isomorphic to the Lie algebra of all skew-symmetric operators on V, and u gives the isomorphism. That is, {u, f, V} is the "natural" representation of f. Note that they are all S-representations, and in fact, except for n = 3, 4, all are T 2-representations. When n = 4, {u, f, V} is non-To We shall call the complete representation the trivial representation 1 when n = 1 and the circular representation when n = 2. N ow let { u, f, V} be an arbitrary representation. It is not difficult to give necessary and sufficient conditions that d 2 = o. Our first main result hinges upon d4 • We wish to point out that the proof, outlined below together with that of Theorem 1', does not employ the classification (but only the definition) of T-representations. We have THEOREM 1. Let {u, f, V} be any faithful irreducible representation. Then d4 = oif and only if it is aT-representation. Remark 1; Assuming the condition of Theorem 1 ({ u, f, V} is a T-representation), one has (a) d a = 0 if and only if it is type T 2, (b) d 2 = 0 if and only if it is noncircular, and (c) d1 = 0 if and only if it is nontrivial. Given the arbitrary representations {UI, f l, V,},i = 1,2, ... , q, we define the·composite to be the representation q

q

i=l

i = 1 q

L: Vi = L: P,,(t)

Ui(Xi)Vi, where X

E

{u, itl f" itl Vi} defined by setting uCtl Xi)

fi' Vi

E

Vi.

One can show without difficulty that

= II P".(t). i = 1

"

We now have, more generally, THEOREM 1'. Let {u, f, V} be an arbitrary faithful representation. Then d. = 0 if and only if it is a composite of T-representations {Uj, fl' Vi}' i = 1, 2, ... , q, such q

that the coeJficient of t4 in II P ". V 2 such that T0 1 (x) = 02(X) T for all XE K. In the above sense 0 and 1

02 are disjoint if and only if the only intertwining operator is the one mapping

V 1 into zero. For the sake of convenience we shall assume in this section that 9 acts effectively on G/ K.

(This is not as restrictive as the assumption that G acts

effectively on G I K.) LEMMA

5. 2.

Assume ad:p and ad! are disjoint.

Then

(1) the complement 1J, to f, is unique in that it is the one and only linear complement which is invariant under adK, (2) if g' is any ideal in 9 then g' = Il'

n f + Il' n 1J,

s

(3) if go is the center of 9 and 9 = Ljlli is the direct sum where the gj, i> 0, i=O

s

are the simple ideals in Il then 1J = Lj 1Jj is a direct sum where lJj = 1J ngj and j=O

the subspaces 1Jj are identical with the subspaces 1Ji(B) in some order. 6)

This means that if X

E

9,

X=

0 implies X = O.

57

49

ON HOLONOMY AND HOMOGENEOUS SPACES

Let

Proof.

0 ~!I

be any invariant subspace for adK.

Let ada be the

representation of K obtained by restricting adK to o. Now assume ada is irre· ducible. Then ada is equivalent to one of the irreducible components of either adr or adp but not both. Without loss of generality assume ada is not equivalent

to any irrebucible component of adp• mapping X

~

Xp of

0

This defines an intertwining operator T for the

into p.

representations ada and adp• must have Xp = 0 for all X joint it follows that u ~ p. o ~ f or

0 ~

p.

Now for any X E 0 consider the linear

But since these representations are disjoint we

E 0

so that

0 ~

Similarly if ada and adr are dis-

f.

Thus we see that if ada is irreducible then either

More generally then, if ada is not necessarily irreducible, by

decomposing it into irreducible summands, we see that ada may be uniquely written as the direct sum

0 = 01

+ 02

where

01 = 0

n I and

02 = 0

n p.

Now if pI is a linear complement to f which is invariant under adK then it is clear that adp' is equivalent to adp and hence :p = pl. !I'

= !I' n f + !I' n p.

It should be noted that if !I'

~

Also from above,

0 then !I' n p ~ 0 since otherwise

!I' ~ I which contradicts our assumption that !I acts effectively on G/ K. s

It follows immediately then that p =

2J Pi

is direct sum and that Pi ~ 0 for

i::::O

i>

o.

Furthermore the pj are invariant under ¢o(Bl.

On the other hand if p'

is invariant under ¢o(B) then obviously !I' = [pI, .)J'J +:p' = I' + p' where I' == il' n I is an ideal in

Q.

Hence the

.)Ji

are irreducibly invariant under ¢o(B). But since

Po == Po(B) by Lemma 4. 4B the subspaces

.)Jj

must be identical with the subspaces

Pi( B) in some order.

5.3.

Q.E.D.

Now both the condition that adpK has inequivalent representations

and the condition adr and adp are disjoint are conditions on K as a subgroup of G.

We will say that a compact subgroup K is detached when both these

conditions are satisfied.

In other words K is a detached subgroup of G when

the irreducible components of adp occur not only with multiplicity one in adp but with multiplicity one in adg.

One sees easily that if K is a detached sub-

group of G then any compact subgroup K' of G which contains K is also detached. We now observe LEMMA

5. 3.

Let T be a maximal torus in G.

group.

58

Then T is a detached sub-

50

BERTRAM KoST ANT

Proof.

Let f be the Lie algebra of T. Then as one knows we may find a r

complement u such that u =

~ lJj

is direct sum where r is half the number of

i=l

roots of the complexification of g,

is two· dimensional for each i and

lJi

irreducibly invariant under the action of adT.

lJj

is

Furthermore the subspaces

lJi

are in one· one correspondence with the set of all pairs consisting of a root and its negative. This correspondence is such that the representation of T on the complexification of

is a direct sum of the character representations of T

lJj

defined by the root and its negative. Since two different roots define inequivalent character representations of T it follows that the representations of T on the various planes

lJj

are inequivalent. Furthermore since adt is just a multiple of

the identity representation, T is therefore detached.

Q.E.D.

Now one knows that X( G IK) ~ 0 if and only if K contains a maximal torus of G. Hence by Lemma 5. 3 and the remark preceeding it K is detached whenever XCGI K)

~

Combining the results of Corollary 4.6 and Lemma 5.2

O.

or more directly Theorem 4. 6 and Lemma 5.2 we obtain the following theorem. THEOREM

5.3.

Assume XCG/K)

~

O-cand that g acts effectively on GIK.G)

Then with respect to any invariant metric, GI K is irreducible if and only if is simple. More generally,

9

is necessarily semi·simple and if

•

9 = ~gj

9

is the decompo-

i=l

•

sition of 9 as the direct sum of its simple ideals then p = ~Pj is direct where J -1

.):lJ =

p n gj and the subspaces pj are identical with the subspaces

.):li( C)

in some

order. One should note that we may substitute the condition K is detached for the condition X(G/K) 5. 4.

~

0 in the previous theorem.

Of course it is quite possible for a compact Riemannian manifold M

to admit as a transitive, effective group of motions both a simple Lie group and a non-simple Lie group. An example of this is when M is itself a compact non-abelian simple Lie group G which is provided with a left and right invariant metric.

Then G and G x G modulo a finite group both act transitively and

effectively as motions on G. if X(M)

~O.

The following corollary says this cannot happen

The proof follows immediately from Theorem 5.3.

COROLLARY

5. 4.

A simple and a non-simple compact Lie group cannot both

59

51

ON HOLONOMY AND HOMOGENEOUS SPACES

act transitively and e..flectively as a group of motions on a compact Riemannian manifold M if X(M) ~ O. 5.5.

For homogeneous spaces G / K where X( G / K)

~

0 the relation between

invariant metrics and affine connections they induce becomes quite direct. We have COROLLARY

5.5. Assume X( G / K)

~

0 and g is simple. Then two permissible

me tries C I and C 2 induce the same affine connection if and only if there exists a scalar A such that ACI Proof.

=

C2 •

Let T be defined as in Lemma 4.2. Now if CI is strongly equiva·

lent to C2 then by Lemma 4.2, T reduces to a scalar since Theorem 5.3 implies

Q.E.D.

rpo(C I ) is irreducible. The converse is trivial.

6.1. Invariant vector fields and the dominance of natural metrics. from the case where X(G/K)

~

Aside

0 the condition that adpK has inequivalent

representations seems to be fairly restrictive. One would like a less restrictive condition in order to imply that S leaves the spaces Pj(C) and Pi(B) invariant. In the following theorem we give a more likely condition which implies at least that S leaves the spaces MB) invariant.

Thus according to Theorem 4.3,

C < B for every permissible metric C when this condition is satisfied.

First

we would like to point out that the example in § 3. 2 illustrates that B < C does not always hold.

In fact in that example we had reducibility with respect to

C and irreducibility with respect to B. We now give an example where C < B does not hold. Here we shall have reducibility with respect to Band irreduci· bility with respect to C. We have seen (see § 4. 3) that C < B implies that Po( C) = Po(B).

The following example also illustrates that the strict inequality

in the relation dim Po( C) example 0 = dim Po( C)

~ lJo( B)

of Lemma 4.3 may hold.

That is, in our

< dim Po(B) = 1.

Let G = U(2), the 2 x'2 unitary group.

Then the decomposition of g as

the direct sum of its center and its simple ideals becomes g = go + lit where go is one· dimensional with basal element Xo and gl is 3·dimensional with basal elements XI. X 2 , Xa. [X2 , Xa]

= Xl,

The basis of gl may be chosen so that [XI. X 2 ] = X a,

[X" Xl] = X 2 •

Now let K

=

e so that lJ

= It

Now it is clear that we may choose B so that

Xi, i = 0, 1, 2, 3 forms an orthonormal set.

60

It is obvious also that go = Po(B)

52

BERTRAM KosT ANT

Now since K = e, any metric C is permissible and hence every positive definite (with respect to

B)

operator S may be obtained.

Now given C then by Lemma 4.3 either dim lJo( C) = 1 or O. that in case dim lJo( C)

0, G is irreducible.

=

We assert

Indeed if G were reducible then

dimension-wise the only possible decomposition is lJ = lJI(C) + lJ2(C) where dim lJl(C)

= dim lJ2(C) = 2.

are subalgebras of g.

On the other hand by Lemma 3.4A, lJl(C) and MC)

But since G is compact lJl(C) and MC) must be com-

mutative. Since g contains a non-zero center this easily implies the existence of a three dimensional commutative subalgebra of g.

This, however, is a contra-

diction. It suffices then to exhibit a metric C for which lJo(C)

= O.

and q( B) are defined as in § 4. 3 then as we have seen there q(B) q(B) = gl,

to show lJo(C)

=

Now if q(C) ~

q( C ) . Since

0 it suffices to choose C so that q(C) contains a vector

not in gl. Let S be an operator on g defined as follows: SXo = Xo + Xa, S-?Cl = 2 X!, SX2

= X 2 , SXa = 2 Xa + Xo.

It is easy to verify that S is positive definite with

respect to B and hence defines C.

Y

= ax,X2 E

q(C).

We observe that S-lXa = Xa - Xo.

Now let

Then Y == (S-lDx,S+ DXl - S-lD sx) Xz =Xo•

Since Xo is not in III we conclude that g is irreducible. Now the condition that we shall impose on g, roughly speaking, produces the opposite situation from the one above.

The condition is that the space of

vectors in lJ which are point-wise fixed by adpK* should be at most one dimensiona1.7l

We shall show that this always implies C

chosen properly.

< B for all C when

lJ is

In order to prove this we have to appeal to the results in

[3J. However, these results were obtained under the additional hypothesis that Il acts effectively on GIK.

Hence we shall have to make this assumption.

However, we wish to point out that the results of [3J, especially Theorem 4, can be modified so as to yield the following theorem without the assumption of effectiveness. Now we choose lJ so that the ideal g( lJ) = lJ + [lJ, lJJ in g is equal to g. 7)

K* is the identity component of K.

61

In

53

ON HOt-ONOMY AND HOMOGENEOUS SPACES

[3J we called V pervasive in g if g(V) = g.

When g acts effectively on G/ K we

showed (Theorem 4 and Corollary 5) that a necessary and sufficient condition that V be pervasive in g is that B can be extended to an invariant (under adG) bilinear form B* on g such that P is orthogonal to f.

In such a case B* is

necessarily unique and non-singular on g. As mentioned in [3J even though the unique B* is non-singular and of course positive definite on P it is not necessarily true that B* is positive definite on g. We now have THEOREM

6. 1.

Assume g acts effectively on G / K.

Let B* be a non-singular

invariant (under adG) bilinear form on g which is non-singular on f and which is positive definite on the ortho-complement V to 1.

Let B be the restriction of

B* to V. Now if the identity representation of ad:pK* on P occurs with a multiplicity of at most one then C < B for any permissible metric C. Proo/. Now let PI

~

P be a subspace invariant under l/Jo(B). We must !?how

then that PI is invariant under l/Jo(C) where C is any permissible metric.

By

Theorems 2.3 and 3.3 it suffices then to show that VI is invariant under S. Now let V2 be the orthocomplement to VI relative to B. We may now apply the results of [3J (in particular Theorem 5). Let Qi = Vi + [1:1i, PiJ where i

= 1,

2.

Then gl, !J2 are ideals in !J where !J =!JI EB g2 and fl' f2 are ideals in f where f = fl EEl f2 and fl = gl n f and f2 = Q2 n f.

and f 2 •

Let KI and K2 be groups corresponding to fl

Now by our hypothesis either KI or K2 must have the property that

ad:pKi leaves no non-zero vector fixed in Pi.

Since otherwise 2 linearly inde-

pendent vectors would be invariant under ad:pK*.

Assume without loss that

i = 1. Thus !he subspace V2 is characterized as the set of fixed vectors for ad:p K I • Since S commutes with ad:pK* it follows that V2 and hence 1:11 is left invariant by S. Now it is obvious that if the identity representation of ad:pK* occurs at all on P then there exists a non-singular invariant vector field on G/ K*. Hence we have COROLLARY

6.1.

Let B arise from a B* in the manner of Theorem 6.1.

Then if there exists no non-singular invariant vector field on G / K and K is connected, C < B for all permissible metrics C.

62

54

BERTRAM KOSTAN1'

BIBLIOGRAPHY

[1] B. Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian

manifold, Trans. Amer. Math. Soc., vol. 80 (1955), pp. 528-542. [2] B. Kostant, On differential geometry and homogeneous spaces, I. Proc. Nat. Acad.

Sci., vol. 42 (1956), pp. 258-261. [3] B. Kostant, On differential geometry and homogeneous spaces, II. Proc. Nat. Acad. Sci .. vol. 42 (1956), pp. 354-357. [4] K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. Jour. Math., vol. 76 (1954), pp. 33-65. [5] A. Lichnerowicz, Espaces homogenes riemannien et reductibilite, Comptes Rendus,

vol. 242 (1956), pp. 1410-1413. [6J A. Lichnerowicz, Sur la reductibilite des espaces homogimes riemanniens. Comptes

Rendus, vol. 243 (1956), pp. 64Q-642. [7J Y. Matsushima and J. Hano. Some studies on Kaehlerian homogeneous spaces, Nagoya Math. J., vol. 11 (1957), pp. 77-92.

Berkeley, California Added in proof. [8] A. Lichnerowicz, Transformations affine et holonomie, Comptes Rendus, Vol. pp. 1868-1870.

63

2~4

(1957),

Reprinted from the JOURNAL OF MATHEMATICS AND MECHANICS Vol. 7, No. 2, March, 1958, pp. 237-264

A Theorem of Frobenius, a Theorem of Amitsur-Levitski and Cohomology Theory * BERTRAM KOSTANT Communicated by N.

JACOBSON

1. Introduction. 1. Let U be a complex n-dimensional vector space and G = G( U) the group of all non-singular linear transformations on U. If U =

r

@ru

is the space of r-tensors over U, then it is a well known theorem that the group G and the symmetric group S = Sr both admit natural representations on ru and that the algebras generated by the two image groups are each other's commutators. This theorem plays a large role in WEYL'S treatment of the classical groups as given in [5]. One of the most important, yet lesser known, applications is in essence, that it leads to an identification of the cohomology group Hr(g) with an explicit subspace of the group algebra I of S. Here g = g(U) is the Lie algebra of all linear transformations on U so that Hr(g) is the same as the rth cohomology group (with complex coefficients) of the n X n unitary group. WEYL uses this identification to obtain a proof of the fact that (1

+ t)(1 + t)

... (1

+ en-I)

(1.1.1)

is the Poincare polynomial of the n X n unitary group. 2. Our investigation originated with the desire to go further than this in the following respect. We wanted to find explicitly those operators on ru made to correspond to the primitive elements of Hr(g) by the identification above (the subspace of I corresponding to Hr(g) is faithfully represented on rU). As it turned out, results completely unexpected were obtained. This paper contains only a partial account of these results. If r is even, there is no non-zero primitive class in Hr(g). If r is odd, say r = 2p - 1, where 1 ;;;:; P ;;;:; n, then the space reg) of primitive elements is one-dimensional. Hence in such a case there is a single operator D on r U to be sought. Consider the usual direct sum decomposition U = U a of U, indexed by the partitions, Ct., of r having n or less parts. An early result (obtained

r

La

r

* This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49(638)-79. 237 Journal of Mathematics and Mechanics, Vol. 7, No.2 (1958).

B. Kostant, Collected Papers, DOI 10.1007/b94535_8, © Bertram Kostant 2009

64

238

BERTRAM KOSTANT

through the theory of self-associated partitions) showed that D vanishes on all the spaces U a , save one, U, , and that D2 is a non-zero scalar on U, . Here E is the simplest of all self-associated partitions of r. Its graph is given in §4.8. (In this paper this result is Theorem 5.4 asserting that D = B(h~) where h~ I: I is given by (5.4.1». On the other hand, there was a natural candidate, B(h,), (see §4.6) for the operator D. The operator B(h,) is the one used by WEYL to obtain the Poincare polynomial. The fact that B(h,) is indeed D (as we later proved) then yielded each of the following three, seemingly unrelated, known (but unknown to us at the time) theorems. Conversely, each of these theorems, as we shall see, implies B(h,) = D. I. The theorem of AMITSUR-LEVITSKI on the minimal standard identity satisfied by the general matrix algebra over any field. (See [1].) II. The formula of DYNKIN on the invariant cocycles of 9 representing its primitive cohomology classes. (See [7], p. 141.) III. A theorem of FROBENIUS applied to the characters of the alternating group n. This is an immediate consequence of Lemma 4.8 and (4.7.2). Hence the FRoBENlUs theorem implies and is implied by

Theorem 4.8. Let r = 2s - 1 and let rT £ 5''Y be the cycle (1, 2, ... , r). Then if B is the representation of S, on r U given in §4.8,

B(L sg

l (r)TrTT- )

.. s

if and only if s

>

= 0

n.

9. Let g = g(U) be the full linear algebra of operators on U and 'g = g('U) the full linear algebra of operators on 'u. The notation 'g is consistent with that of §4.8 since, as one knows, r g may be identified with ® 'g where in fact one has (Al

®

A2

® ... ®

AT)

Xl

®

X2

® ... ® Xr

=

Alx l

®

A2x2

® ... ®

A,x,

forA. £ g;x; £ U. Now, as observed in §3.5, one has an immediate equivalence between the AMITSUR-LEVITSKI theorem and (3.4.3). We now show that the AMITSURLEVITSKI theorem is equivalent to a part of the FROBENIUS theorem (h, £ I,) by showing the equivalence of Theorems 4.8 and (3.4.3). This equivalence will

82

256

BERTRAM KOSTANT

be an immediate consequence of the following generalization of a result of r. SCHUR. Lemma 4.9 reduces to the result of SCHUR when Al = A2 = ... = A, • Lemma 4.9. Let PES, and let the cycle decomposition of p-1 be given by p-l = (il ... i,J(i',+1 ... i,,+!,) ... (idk+l ... i,) where i) , i2 , ... , i, is a permutation of 1, 2, ... , rand d k AI , A2 , ... , A, E 9 = g(U). Then

trB(p)·A l

®

A2

® ... ®

L:~:~ 0; . Let

=

A,

(4.9.1)

= tr (Ai, ... Ai!) tr (A i !,+, ... Ai!,+I) ... tr (Aid.+> ... Ai.)' (It should be noted that the trace taken on the right side of this equation is for operators

on U.) Proof: Let U* be the dual space to U. To prove the lemma it suffices by linearity to prove it when all the Ai are of the form x ® w. Here x E U and WE U* and x ® W is that operator on U defined by x ® w(y) = (w, y)x for any y E U. Note that Xl ® WI 'X 2 ® W2 = (WI' X2) Xl ® W2 and tr X ® W = (x, w). Now '(U*) is naturally identified with ("U)* (and hence can be written 'U*) where (Xl

® ... ®

X,

,WI

® ... ® W,)

= 11 (Xi' Wi)' i=1

Let ej E U, i = 1,2, ... , n, be a basis of U and let OJ N ow if G is any operator on 'U then

L:

tr G =

(G(e;,

® ... ® e;.),

0;,

U* be the dual basis.

E

® ... ®

O;n)

l~;i:in

Now assume Ai = Xi Ar ·B(p) (G(e;,

® Wi, i =

® ... ® e;.),

1,2, ... ,r. Then if G

= Al ® A2 ® ... ®

,

0;,

® ... ®

0;,) =

IT (Xi'

O;)(e;p-I(i) ,Wi)

i=I

,

IT (Xi'

O;,)(e;, ,WpCi»'

i=l

But now if we sum this last expression (m' terms) over alII that , tr G = IT (Xi' Wpco)·

~

ii

~

m it follows (4.9.2)

i=l

Now to prove the lemma it suffices only to rearrange the factors in (4.9.2) so that it takes the form of (4.9.1). Indeed, (4.9.2) may be rearranged so that tr C = «Xi, ,Wi!)

IT (Xi. , Wi._,»)(Xi!,

+,

p=2

83

,Wi!, +I.>

IT (Xi!, p=2

+. ,

Wi!, +0_,»)

ON THEOREMS OF FROBENIUS AND AMITSUR-LEVITSKI

257

But now from the first paragraph in this proof the terms in the parenthesis are respectively (tr Ai •.. A ia ,),

(tr A ia ,+ •

•••

Ail,H,)' ... ,(tr A idk +,

•••

AiJ.

Q.E.D.

Now in Lemma 4.9 let p = rur- 1 where u- 1 = (1,2, ... ,r).

It follows then from Lemma 4.9 that trB(rur- 1)·A 1 A2 .•• AT

=

tr A r (l)A r (2)

•••

Ar(d

and hence tr B( L: sg (T)rur-1). Al A2 .•. AT = tr [AIA2 ... AT]'

(4.9.3)

TES

We can now go back and forth between Theorem 4.8 and (3.4.3). That is, (4.9.3) implies B(h,) = 0 if and only if tr [AJA2 ... AT] = 0 for all Al , A2 , ... , AT I: g. Setting 8 = n + 1 so that r = 2n + 1, Theorem 4.8 implies tr [A 1 A 2 ... A 2n + 1 ] = 0 for all Al , A2 , ... , A 2n + 1

I:

9 and conversely.

5. Cohomology classes as operators. 1. In this section we complete the circle of ideas relating the AMITSUR-LEVITSKI theorem, the theorem of FROBENIUS and the formula of DYNKIN. We return to the cohomology theory of Lie algebras specializing to the case where g = g(U), the Lie algebra of all operators on U. First, by use of part of the FROBENIUS theorem (h, I: I,) we shall exhibit a natural relationship between the primitive r-cohomology classes of 9 (r odd) and the operator B(h,). As a consequence of this we obtain a proof of the formula of DYNKIN. a This then completes the proof of the equivalence of I, II and III of §1.2. Also as a consequence we sharpen a result of WEYL. (See §5.5 here.) Most important (in the author's opinion), it is demonstrated that the space of tensors U, acts as a focal point where the three theorems of §1.2 come together. Finally the circle of dependencies among these theorems is broken into by giving a direct proof of the formula of DYNKIN. This proof is based on the operatortheoretic significance of cup and Pontrjagin products. 2. Let U and TU, 9 = g(U) and 9 be as in §4.8. Also, as in §4.8, let B designate the representation of S = ST on TU. Now let BO be the representation of S on Tg obtained in the same way as B except that 9 is substituted for U. We will be especially interested in the subspace go. erg of anti-symmetric r-tensors over g. That is, go. is the range of the projection BO(eo.) where 1 eO. = I L: sg (u)u r T

a

CT£S

aWe wish to observe that in [7], p. 141, DYNKIN isolates a particular class in PT(g). Since our concern is with cohomology over fields, no such distinction is made here.

84

258

BERTRAM KOSTANT

is the identity element of the one-dimensional ideal I., . Here W = w(r) = (WI, W2 , '" , W r ) £ Q is the partition given by WI = W2 = ... = Wr = l. Now recall that rg is also the full algebra of operators on ru. It is easily verified that the representations Band BO of S are related in the following way. For any G £ rg

Indeed it suffices only to verify this for all G of the form Al @ A 2 @ ... @ Ar , A; £ g. In such a case it is obvious. Now then

and hence the space g., of anti-symmetric elements in rg as operators on ru can be characterized as the set of all operators on ru which commute with all B(u) for even u and anti-commute with all B(u) for odd u. Now consider the mapping 7]r : Gr(g)

-7

g., (d

given by

Let the non-singular symmetric bilinear form (G I (GI

,

G2)

,

G2 ) on r g be defined by

= tr GI G2

(5.2.1)

It is clear that BO(u) is orthogonal with respect to this inner product. This implies that BO(e.,) is symmetric with respect to it and consequently the bilinear form (5.2.1) is necessarily non-singular on g., . We define r{ : g.,

-7

Gr(g)

by (7]r(C) , u) = (G, 7]r(U»

where G £ g., and u £ Gr(g). It is clear that both "'r and ."r are onto isomorphisms. If B; , A; £ g, i, j = 1, 2, ... , r, then it is a simple fact that r

(AI @ ...

® Ar

,BI

® ... ® Br) = IT tr AiB, ,,-1

(Also note that Lemma 4.9 reduces to this when ."r(A;) = a, £ Gr(g), 7J,(BO(e",)(A I

® ... ®

Ar»

p = e.)

•

It follows then that if

= ~ a l A ... A ar • r.

(5.2.2)

3. Let G = G(U) and rG = GrU) be the groups of non-singular operators,

85

259

ON THEOREMS OF FROBENIUS AND AMITSUR-LEVITSKI

respectively, on U and ru. Let defined by 7r(A)

7r :

G

--t

TG be the representation of G on ru

=

A

® ... ® A.

'-------,r---'

r Now the representation r 9 defined by

7r

of G on ru induces the representation

7r

0

of G on (5.3.1)

for G £ r g. Since all 7r(A) commute with all. B(u) it follows trivially that g., is invariant under all 7r°(A). Let 7r~(A) be the restriction of 7r°(A) to g" . Recall now the representation Or of 9 on Gr(g) considered in §2.3. We now observe that the isomorphism TJr defines an equivalence between the representation Or and the differential representation of 9 defined by 7r ~ • It follows then that TJr maps Jr(g) onto the invariants of 7r~ • But it follows from (5.3.1) that the space of invariants of 7r ~ is exactly the set of all operators on g., which commute with all 7r(A). We now apply the already classical result asserting that the set of all operators on ru which commute with all 7r(A) is the set B(l) (see [5], Theorems 3.5.D and 4.4.D). Thus if 1/Ir is the restriction of TJ, to (g) it follows that the range of 1/Ir is g., n B(l). On the other hand, by the definition of H we see that g., n B(l) = B(H). Furthermore, since 7r~(A) is orthogonal with respect to the bilinear form (5.2.1) it follows that this bilinear form is nonsingular on B(H). Hence if 1/Ir is the restriction of TJr to B(H) it follows that 1/Ir maps B(H) isomorphically onto reg). Summarizing we have proved

r

Theorem 5.3. Let g.,(r) be the space of anti-symmetric r-tensors over the Lie algebra of all operators 9 on U. Let TJr : Gr(g) TJr : g., (r)

--t

--t

g., (r)

Gr(g)

be the maps defined as in §5.2. Then if (1) H = Hr is the subspace of the group algebra I = I, of the symmetric group S = Sr defined as in §4.6, (2) B is the representation of I on the space ru of r-tensors over U given in §4.8, (3) Jr(g) C Gr(g) and reg) C Gr(g) are, as in §5.3, respectively, the spaces of invariant r-cycles and r-cocycles and, finally, (4) 1/1, and 1/1' are, respectively, the restrictions of TJr to J r and TJr to B(H) we have

tl!T : Jr(g)

--t

B(H)

tl!r : B(H)

--t

r(g)

and, furthermore, 1/Ir and 1/Ir are both onto isomorphisms.

The shift to homology and cohomology is now immediate. Since J.(g) and

86

260

BERTRAM KOSTANT

reg) u

--t

U

are respectively isomorphic to Hr(g) and Hr(g) under the mappings and a --t a, 1/;r and 1/;r induce onto isomorphisms ~r

:

Hr(g)

--t

B(H)

~r :

B(H)

--t

Hr(g).

The maps ~r and ~r neatly convert the study of the more or less intangible spaces Hr(g), Hr(g) to a study of the very concrete space B(H). This is particularly significant in the light of the FROBENIUS theorem which gives an almost complete picture as to how B(H) acts as operators on ru, or more particularly on the subspaces U{J of tensors (quantics of type (3) corresponding to the selfassociated partitions {3 of r. We recall that the elements {h{J}, (3 £ no , form a basis of H and that Corollary 4.6 implies that B(h{J), which leaves U{J invariant, vanishes on all U a , where a =1= (3. That this is especially relevant for the traditional study of either H*(g) (or H*(g)) as an exterior algebra over its space of primitive elements will be borne out in the remaining sections where we see how the map ~r(resp. ~r) relates the primitive elements and their Pontrjagin products (resp. cup products) with the particular elements B(h{J), (3 £ no , of B(H). 4. Let U' CUbe a subspace of dimension m and let U" be such that U = U' + U" is a direct sum. Now let g' = g(U') and let

O. 2.5. Now we recall that the Lie algebra of SL(n, C) is written as A n - 1 so that the Lie algebra of Gl(n, C) is A n _) EB Dl where D) is the one-dimensional complex Lie algebra. Also the Lie algebras of SO(n, C) is Dn/2 in case n is even and B(n-ll/2 in case n is odd. Finally the Lie algebra of Sp(n, C) is C" ; the latter is, of course, a set of 2n X 2n matrices. Of course the notion of natural representation carries over to these classical Lie algebras. In the case of Lie algebras, however, the question of establishing whether a representation is natural or not becomes simpler. This is so mainly because the Cartan-Weyl theory of weights which is employed is more directly applicable to Lie algebras. The following theorem gives a simple criterion (an orthogonality condition on the weights) when given a representation of a Lie algebra g, that (1) g is isomorphic to a classica~ Lie algebra and (2) the representation is then equivalent to a natural one. (Ei

,cp) (cp,

Ej)

=

ninj

~

THEOREM 2.5. Let 71" be a faithfuL irreducible representation of a complex Lie algebra g on V. Let n = dim V. (1) Then (a) g is isomorphic to A n - 1 EB D) and (b) with respect to this isomorphism 71" is equivalent to the natural representation of A n - 1 EB D) if and only if there exists a weight II £ .:i(7I") which is orthogonal, with respect to B .. , to any Il £ .:i(7I") for which Il ~ II. (2) Then (a) g is isomorphic to B (,,-1)/2 in case n is odd, or either Dn/2 or C,,/2 in case n is even and (b) with respect to this isomorphism 71" is equivalent to the natural representation in any of these cases if and only if there exists a weight

96

112 11 I:

BERTRAM KOSTANT

~(11")

any

p.

I:

such that - 11 I: ~(11") and such that for which p. ¢ ± 11.

p

is orthogonal, with respect to B". , to

~(11")

Proof. It is simple fact (see e.g. [7; 52] translation) since 11" is faithful and irreducible that 9 is reductive. In the notation of §2.1 let :L!-l g' = 13 be the direct sum decomposition of 13 into its simple ideals. Write a = gO so that 9 = :L!=o g'. Then l) = :L!-o l)8 where l)8 = g' n l). Of course l)' is a Cartan subalgebra of g'. The notation of §§2.3-2.5 established relative to l) and 9 will carry the same meaning here relative to l)' and g' when it appears with the superscript s. Furthermore if X I: l) (respectively WI: l)*) then X' (respectively w') will designate its component in l)' (respectively l)*') so that X = :L!-o X· (respectively W = :L!=o w'). N ow since 11" is irreducible and faithful, we observe that gO = a is at most one-dimensional and that p.~ = p.~ for any P.l , P.2 I: ~(11"). Thus if X I: ~(11") is the highest weight of 11" and p. is any other weight, then X - p. I: it~. In fact we may write X - p. = :L~-l P;oi; where Oil , ... , Oi r are the simple positive roots and PI , ... , p; are non-negative integers. (See [13], Theoreme 1, p. 17--04). Now we recall the well-known fact (see e.g. [8; 376]) that if p. I: ~(11") and cp I: ~, then 2(p., cp)/(cp, cp) = p - q where p and q are the largest non-negative jcp I: ~(11") when - p ~ j ~ q. In particular p. cp I: ~(11") integers for which p. if (p., cp) < 0 and p. - cp I: ~(11") when (p., cp) > o. Let L". be the operator on l)* defined by (L"Wl , W2) .. = (WI, W2). Since there is only one invariant bilinear form, up to a scalar multiple on a simple Lie algebra (see [11; Theoremes 11.1 and 11.2]), it follows for all s that the l)*' are eigenspaces for L". Moreover, if c. is the corresponding eigenvalue, c. > 0 for s ~ 1, and Co = o. Since any cp I: ~ is contained in il)~' for some s, it follows that 2(p., cp) / (cp, cp) = 2(p., cp),,/(cp, cp) .. so that we may replace B". for B in the criterion above determining whether p. + cp or p. - cp is contained in ~(11"). Consider the decomposition X = :L!-o X' of the highest weight X. The condition that 11" is faithful is equivalent to the condition that X' ¢ 0 for all s. To see this we first observe a ¢ 0 immediately implies that X° ¢ O. Now if X' = 0 for s ~ 1, then from the criterion above 1I"(E ,,)vA = 0 whenever cp I: ~. = ~ n il)*', E" is the corresponding root vector and VA I: V is the weight vector for X. Indeed since p = q and X ± cp cannot both be weights, p = q = O. Hence 1I"(g")vA = o. Since g' is an ideal in 9 and 11" is irreducible, this implies 1I"(g') = O. But this contradicts the faithfulness of 11". Now if p I: ~(11") is orthogonal to all other elements of ~(11"), this is obviously true of UP for any u I: W. Hence we may assume P I: D. We now show that P = X. Since (il)*", '/,1)*') = 0 for s ¢ t, it follows that

+

+

k

1

k

(p, X) .. = :L (p", X') .. = (pO, XO) .. 8=0

+ :L C,a- (p', X'). 1

Since .,,0 = XO, it then follows from Lemma 2.4 that (.,,', X') ~ 0 for any s ~ 0, and (.,,', X') = 0 implies .,,' = O. Hence (.", X) .. > O. But by our hypothesis

97

A CHARACTERIZATION OF THE CLASSICAL GROUPS

113

this can only happen if A = II. Now for any Jl t Ll(7r) there exists U t W such that UJl t D. By a similar argument then it follows that UJl = A. Thus we see that one obtains all occurring non-zero weights by applying W to A. Moreover, since n,,(7r) = n.,,(7r), they all occur with multiplicity one. Let A = AI, A2 , ... , Ap be the elements of Ll(7r). Since Ai = UAI for some u, it follows that (Ai, A;),.. = 0 for i r!' j. But then, being mutually orthogonal, they are linearly independent. Hence P :::; r where r is the rank (dim f)) of g. On the other hand, were X t f) orthogonal to all Ai , it would follow that 7r(X) = O. But that contradicts faithfulness. Thus the Ai span if)*, and hence we conclude r = p. We have of course assumed condition (1) of the theorem. We observe in this case that 3 r!' O. Indeed, in general D n - D = if~ and hence if g were semi-simple, D n - D = O. On the other hand - D is a cochamber so that there exists A' t - D n Ll(7r). But then A r!' A'SO that (A, A') .. = O. But also - A' t D so that CA, -A') .. =

t

8=1

1. (A',

_A'B) = O.

Ca

This, however, contradicts Lemma 2.4 since A' r!' 0 for all 8 ;;::: 1. On the other hand, if we assume (2) then 3 = O. This is immediate since as we have observed 11° = Jlo for any Jl t Ll(7r). But since -II t Ll(7r) , 11° = - 11° = 0, and hence 3 = 0 by the faithfulness of 7r. Now the property assumed for II in (2) is clearly invariant under the action for W. Hence as above we may assume II t D. In an argument identical with that of the previous case we may also conclude that A = II. Furthermore any Jl t Ll(7r) is of the form UA for some U t W so that n,,(7r) = 1. It also follows then that - Jl t Ll(7r) so that Ll(7r) = - Ll(7r). (This latter fact is, as one knows, equivalent to the condition that 7C" be self-contragredient; see [12] for example.) Now let AI, ... , Ap be the elements of Ll+(7r) where A = AI' Then, as above, (Ai , A;) .. = 0 for i r!' j and as before, we conclude that P = r (but in this case P is half the number of non-zero weights). We have deduced no information as yet about the zero weight. Write, as a direct sum V = VO + V' where VO is the weight space for the zero weight and Viis the subspace spanned by the weight vectors for the elements of Ll(7r). We have shown under the assumptions of (1) that dim V' = r and for (2) dim V' = 2r. On the other hand, for (1) it is immediate that VO = 0 since for any weight Jl, Jlo = AO r!' O. Hence in that case dim V = n = r. We now show that for (2), VO is at most one-dimensional. Assume VO r!' O. Then A= L~~l Piai where the Pi are positive integers (positive, by virtue of Lemma 2.4, and integral since the zero weight occurs) and the ai are the simple positive roots. Thus L:~l Pi = 8;;::: r. Now if E a , t g are the root vectors corresponding to the ai , and 'Y1 , ... ,'Y. are the roots - a; in some order, where - ai occurs with multiplicity Pi , then the vectors u, = 7C"(E'Yd,)7r(E'Yd') ... 7r(E'Yd.)VA ,

where T runs through all permutations of the integers from 1 to 8, span VO. Since VO r!' 0, at least one of the u, is not zero. We may assume this is the case

98

114

BERTRAM KOSTANT

for the identity permutation. Let 0; = - L.:;-1 'Yi and let V; = 7r(E"i+,)7r (E"i+,) ... 7r(E.,Jv),. Then Vi is a weight vector for 0; (: A+ (7r) , j = 1, 2, ... , 8. However, since Oi - 0; > 0 if i > j, the 0; are distinct. Thus (ai' 0;) ~ = 0 for i ~ j and hence the 0; are linearly independent in ~1)*. But since 8 ~ r, it follows in fact that 8 = r and that the 0; are identical with the Ai , in some order. We may assume Ai = Oi. Also since Pi = 1, we may assume 'Yi = - ai so that A; = 2::-1 ai , 1 :::; j :::; r. N ow consider U r where r is not the identity permutation. Let 1 < P :::; r be the greatest integer such that rep) ~ p. Let w = 7r(E_ a ,c.Jvp . Then w (: V'is a weight vector for the weight op - ar(p) = fJ. > o. But fJ. ~ Ai for i = 1, 2, ... , r and hence n,,(7r) = O. Thus w = 0 and hence O U r = o. It follows then that dim V = 1. We now prove under the assumptions of (1) (respectively (2» that if fJ. (: A(7r) (respectively if fJ. is a non-negative weight), then A - fJ. is a root, assuming, of course, that A ~ fJ.. We first observe that there exist positive roots f3i , i = 1,2, ... , q, such that A - fJ. = L.:~-1 f3i and such that v" = 7r),(E_ fl ,) 7r),(E-fl') ... 7r(E_ fJ ,)v), is non-zero and hence is a weight vector for fJ.. Let fJ.o = fJ., fJ.; = fJ. + L.:;-1 f3i· Then for all j, fJ.; is a weight of 7r and since fJ.i+l > fJ.i , all the fJ.i are distinct. Furthermore (fJ.i , fJ.i) or = O. This follows from distinctness in the case of (1) and distinctness together with the fact that fJ.i > fJ.o ~ 0 for i ~ 1 in the case of (2). Now it is obvious that fJ.l - fJ.o = f3 is a root. Assume we have shown fJ.j - fJ.o is a root. Then fJ.;+1 -

But for j

>

fJ.o =

(fJ.;+1 -

fJ.i)

+ (Mi

-

fJ.o) =

f3i

+ (fJ.i

-

fJ.o).

0

+

Thus f3; fJ.i - fJ.o = fJ.i+J -fJ.o is a root. Hence A - fJ. is also a root. Hence under the assumptions of (1), Ai - A; (: A for j = 2, ... r. Under the action of W this implies Ai - Ai (: A for all i ~ j. On the other hand, the Ai being linearly independent, this implies the roots obtained as differences in this way are distiI].ct. Thus the roots are at least r(r - 1) in number and hence dim g ~ r2. But dim V = n = r and since 7r is faithful, g must be isomorphic to grey), and 7r expresses an isomorphism. Hence of course, there exists an isomorphism between g and A n - 1 EB Dl such that the representation 7r corresponds to the natural representation of A n - 1 EB Dl . It may be recalled that there is an isomorphism of A n - 1 EB DJ on grey), where dim V = n, which is not equivalent (n > 2) to the natural representation. This representation is equivalent to the representation of A n - 1 EB Dl on An - 1 V which is induced by the natural representation of A n - 1 EB Dl on V. This, however, does not affect our result since it is only an isomorphism of g with A .. _l EB Dl with which we are concerned. We may restrict our consideration now to the assumptions of (2) . We have

99

A CHARACTERIZATION OF THE CLASSICAL GROUPS

115

shown A - p. £ .1 if p. is any non-negative weight distinct from A. N ow if is another non-negative weight, distinct from A and p., then (A -

p.,

A - p.')"

=

(A, A) ..

>

p.'

O.

Thus (A - p.') - (A - p.) = p. - p.' is also a root. If the zero weight occurs and we let p. or p.' equal zero, it becomes clear then that .1(11") C.1. On the other hand, given just that p., p.' £ .1(11") and that p. and p.' are linearly independent, we may choose the lexicographical order in such a way that p. and p.' are both positive (for example, by embedding p. and p.' in the order defining basis). Hence p. p.' £.1. Thus if Al , A2 , ... , Ar are the elements of .1+(11"), we see that ± Ai ± Ai , for i < j, are roots. Moreover, by the independence of the Ai we are thus assured of obtaining 2r(r - 1) roots in this manner. Also, if the zero weight occurs, then we obtain 2r more roots; namely ± Ai' Conversely, of course, if some Ai is a root, then the zero weight occurs for 11" since (Ai' Aj) .. > 0, and hence Aj - Aj = 0 is a weight. N ow consider the question as to whether we have exhausted all the possibilities for a root. If!p is a root, then since 1I"(E",) ~ 0 there must exist weights p., p.' such that p. - p.' =!p. Hence the only other possibility is that !p is of the form ± 2 Aj. Now we have seen that dim VO = 1 or 0 so that dim V = 2r + 1 or 2r according as the zero weight occurs or not. Case 1. dim V = 2r + 1. In this case ± Aj , j = 1, 2, ... , r, are roots. This excludes the possibility that ± 2Aj is a root for any j. Hence the totality of roots is given by the set ± Aj , ± Ai ± Ai for i < j. On the other hand, the linear independence of the Aj insures that these expressions for the elements of .1 in terms of the elements of .1(11") are unique. Hence the additive structure of .1 is given completely by our being able to write the roots in this way. But then, as one knows, .1 with the additive property so given, is isomorphic to root structure of Br (see e.g. [9; 126]) and hence g is isomorphic to B r . Moreover, the expression for the highest weight, Al , of 11" in terms of the roots, agrees with the expression of the highest weight of the natural representation of Br in terms of the roots of B r • Hence with respect to this isomorphism 11" is equivalent to the natural representation of Br = B en - ll /2 • Case 2. dim V = 2r. Recalling the transitive action of the Weyl group on the set {± Ai} we see that in this case it is simply a question as to whether all the ± 2Aj occur as roots or none occur. Both possibilities are realized. Indeed if the ± 2Ai are not roots, then ± Ai ± Ai for i < j defines (see argument above) the root structure of Dr , r ~ 2, (see e.g. [9; 126]) and if the ± 2Aj occur as roots, then ± Ai ± Ai , ± 2Ai , i < j, defines the root structure of Cr , r ~ 1. In either case the highest weight Al defines the natural representation. This concludes the proof. 2.6. Now we shall obtain the corollary of the introduction (Corollary 2.6) as a consequence of Theorem 2.5. The device enabling us to go from the orthogonality condition on a weight of Theorem 2.5 to the low rank condition of Corollary 2.6 is Lemma 2.3.

100

116

BERTRAM KOSTANT

COROLLARY 2.6. Let 9 be any complex Lie algebra of linear transformations acting irreducibly on the n-dimensional complex vector space V: (1) Then 9 is the Lie algebra of all linear transformations in V if and only if it contains an operator A of rank 1 such that A 2 ~ (i.e., tr A ~ 0). (2) If 9 leaves invariant (see Remark 1) the non-zero bilinear form B on V, then 9 is the Lie algebra of all linear transformations leaving B invariant if and only if 9 contains an operator A of rank 2 such that A 2 ~ 0.

°

Proof. As mentioned above, the fact that 9 acts irreducibly implies that 9 is reductive. Now, as in Lemma 2.3 identify V @ V* with the set gf(V) of all operators on V. N ow if an element A £ V @ V* is semi-simple as an operator on V (that is, A is diagonalizable) then, clearly, ad A as an operator on V @ V* is semisimple. If, furthermore, A £ g, then the restriction of ad A to 9 is semi-simple. That is, A is, by definition, a semi-simple element of g. It is well-known, however, that any semi-simple element of a reductive Lie algebra can be embedded in a Cartan subalgebra (see [9; 119]). Hence if A £ 9 is semi-simple as an operator on V, then it can be embedded in a Cartan sub algebra 1) of g. Now if A £ V @ V* is of rank 1, then A is of the form v @ v* for v £ V and v* £ V*. Clearly tr A = (v, v*). Hence if A 2 = (v, v*)A is not 0, then v is an eigenvector of A belonging to the non-zero eigenvalue, tr A. It follows immediately then that if A 2 ~ 0, A is a semi-simple operator on V. Hence A may be embedded in a Cartan sub algebra 1) of g. Let 7r designate the given representation of 9 on V. By commutativity it follows that v is a simultaneous eigenvector for any B £ 1), and hence v is a weight vector for a non-zero weight A £ .1(11} Furthermore, since [B, A] = 0, and since [B, A]

=

[B,v@v*]

=

Bv@v* +v@B*v*

=

A(B)v @ v*

+ v @ B*v*

= 0, it follows that v* is weight vector for the weight - A of the contragredient representation 7r* of g. Thus A = Vx @ v!x. On the other hand, since A has only one non-zero eigenvalue it follows that AJ (A) = for any Al £ .1C7r) for which Al ~ A. But now, by Lemma 2.3,

°

rCA) = r(vx @ v!x) =

(vx , v!x)A"

=

tr A·A".

Since A £ g, however, rCA) = A so that A = tr A· A". But then, if Al £ .1(7r) and AJ ~ A,

(Al , A) ..

=

(A; , A")"

101

A CHARACTERIZATION OF THE CLASSICAL GROUPS

=

t/A 1

= tr A =

(A~

,

117

A) ..

AI(A)

0.

Thus the conditions of (1), Theorem 2.5, are satisfied and hence (1) of Corollary 2.6 has been proved. N ow assume the condition of (2) in Corollary 2.6. Since g acts irreducibly, it follows that B must be non-singular. Moreover, again using the irreducibility, B is either symmetric or skew-symmetric. In any case'll" is self-contragredient so that among other things tr B = for any B t g. But then it follows easily that since rank A = 2 and A2 ;e 0, that A has 2 non-zero eigenvalues a and -a. Consequently A is semi-simple and hence may be embedded in a Cartan subalgebra l) of g. Let VI and V_I be the respective eigenvectors of A corresponding to the eigenvalues a and - a. As before it follows that VI and V-I are weight vectors for weights Al and A-I of'll". But .1('11") = - .1('11") and hence - Al t .1('11"). It follows then that A_I = - Al since A has rank 2 and A-I(A) = -Xl (A). Let X = AI' Now for any Btl), (A, B)" = tr AB = 2X(A) X(B). But then A" = (1/2X(A))A. On the other hand, in' t .1('11") and X' ;e ± X, then X'(A) = since A has rank 2. Using an argument identical to that above we conclude that (A', X) .. = 0, and hence the conditions of (2) in Theorem 2.5 are satisfied. The remainder of Corollary 2.6 follows immediately from the definitions of the natural representations of the classical Lie algebras, and in the case of Br , Cr and Dr , from the fact that the invariant form under such a representation is unique up to scalar multiple. 2.7. For the situation when we are dealing with the real numbers rather than the complex numbers we have the following corollary. Corollary 2.7 is actually weaker than Corollary 2.6 but it is more directly applicable for us (see §3.1).

°

°

COROLLARY 2.7. Let Vo be a real finite dimensional space and let B be a positive definite bilinear form on V. Let go be any Lie algebra of skew-symmetric (with respect to B) operators acting irreducibly on Vo. Then if go contains an operator W of rank 2, either (1) go is the Lie algebra of all skew-symmetric operators in Vo , or (2) Vo is even-dimensional, and there exists an operator J on Vo such that (a) J is skew-symmetric with respect to B, (b) r = - 1, (c) J commutes with every element of go. Moreover, if Vo is regarded as a complex vector space where i acts like J and C is the Hermitian inner product on Vo defined by

C(u, v)

=

(u, v)

+ i(u, Jv),

then go regarded as complex linear transformations on Vo is the Lie algebra of all skew-Hermitian operators in Vo .

102

118

BERTRAM KOSTANT

Proof. Let V = Vo + iVo be the complexification of Vo and g = go + ig o the complexification of go. We can of course regard g as acting on V. Case 1. Assume g acts irreducibly on V. Extend B to V (unique) as a complex bilinear form. It is clear of course that g leaves B invariant. Now W is still of rank 2 regarded as a complex operator on V. On the other hand W 2 :;c 0, since skew-symmetric operators are not nilpotent. Hence by Corollary 2.6 g is the Lie algebra of all operators on V which leaves B invariant. But then go must be the Lie algebra of all skew-symmetric operators on Vo . Case 2. V is not irreducible under g. In this case, as one knows, the ring of all operators on Vo which commute with g is isomorphic either to the complex numbers or to the quaternions. In either case we may find J, such that J2 = - 1, in the commuting ring of go. Since D = J + J* is symmetric and since D commutes with go , it follows that D = }..] for some scalar}... But tr J = 0 implies tr D = 0 and hence D = o. Thus J is skew-symmetric. Now let Vo be regarded as a complex vector space where i acts like J. It is clear that C is a Hermitian positive definite inner product and that go , with respect to C, is a Lie algebra of skew-Hermitian operators. Now if we identify g = go + ig o with go + J go , it follows that g is a complex Lie algebra of linear transformations on Vo which acts irreducibly. Since W has rank 1 with respect to the complex structure on Vo , it follows from Corollary 2.6 (since W 2 :;c 0) that g must be the Lie algebra of all complex linear transformations in Vo . Hence go must be the Lie algebra of all skew-Hermitian operators on Vo. This completes the proof.

3.1 Application. Let M be an irreducible symmetric Riemannian manifold. Let 0 EM be a point of M and let Vo be the tangent space at o. Furthermore let gil and Riikl be as usual, the metric and curvature tensors at o. We recall that if u = Xi and v = yi are two linearly independent vectors in Vo and [w] is the plane spanned by u and v, then the sectional curvature K[w] is defined by K[w]

=

see for example [1; 20]. More generally, let F2(VO) designate the space of 2nd order contravariant skew-symmetric tensors at o. Let P be the real projective space of line elements in F2(VO) and 11": F2(VO) - (0) ~ P the natural map of the non-zero elements of F2(VO) onto P. If a E F 2(V O), a :;c 0, we let [a] = 1I"a. Then the function K defined above can be regarded as being defined on the submanifold N of P corresponding to the set of 2-planes in Vo. Moreover, if we let w = xiyi yixi E F2(VO) ' then the notation above is consistent with this identification. Now K naturally extends to P where if a = (a ii ) E F 2 (VO), a :;c 0, then

103

A CHARACTERIZATION OF THE CLASSICAL GROUPS

119

We shall let KN be the restriction of K to N and we shall speak of it as the restricted sectional curvature. It is well known, of course, that KN completely determines K. Before proceeding it will be convenient to express the above in invariant notation. Let (u, v) designate the inner product in Vo as given by the metric tensor. Let a designate the Lie algebra of all skew-symmetric operators on V othat is, all operators A on Vo such that (Au, v) + (u, Av) = 0 for all u, v E Vo . In a we define the positive definite inner product Bo given by (A, B)o = - tr AB. Now let T: a ~ a be the operator on a defined by T(A) = B where

and where B = (b\) and A = (a';). Finally, let (T: F2(V o) ~ a be the isomorphism defined by lowering an index. That is, if a = a i ; E F 2 (VO), we let (T(a) = A where A = a\ = aikgk;' It is clear that if a ;e 0 and (T(a) = A, then (3.1.1)

K[a] =

.! (T A,

A)o. 2 (A, A)o

3.2. Now from the well-known identities ([1, (1.56)]) it is clear that T is a symmetric operator on a with respect to the given inner products. Hence T may be "diagonalized." That is, if ao is the kernel of T and X, , i = 1,2, ... , k are the non-zero distinct eigenvalues of T (necessarily real) and a, are the corresponding eigenspaces, we may write

where ri : a ~ a, , i = 0, 1, '" , n is the orthogonal projection of a on ai • It is clear, of course, that a = L:7~o a; is an orthogonal direct sum. Let go be the range of T. Of course go = L:7=1 a; . Now it is a simple consequence of the relation (3.1.1) that K as a function (analytic) on P has [a] as a critical point-that is, the differential, dK, vanishes at raJ-if and only if A is an eigenvector for T. We wish to show that for an irreducible symmetric space M, dim M > 1, K can never have a critical point in N with non-zero critical value unless K is constant (in which case, of course, every [a] is a critical point). Since it is well known that for an irreducible symmetric space, one must either have (a) K 2:: 0 or (b) K ::; 0 it follows that the maximal value of K in case (a) or the minimal value of K in case (b) can never be obtained in N unless K is constant. Interpreted in terms of the notation of §3.1 this means for such a space M an operator WE a of rank 2 cannot be an eigenvector of T with non-zero eigenvalue (that is, W f. a; , i = 1,2, ... ,n) unless T is a scalar multiple of the identity. Thus it suffices to show that if WE a; , i > 0, rank of W = 2, then T is a scalar multiple of the identity. 3.3. N ow if M is a Riemannian symmetric space then, as shown by E. Cartan (see e.g. [3; 265]) the curvature tensor satisfies the following algebraic identity:

104

120

BERTRAM KOSTANT

(3.3.1) By raising the index r we see easily that this is equivalent to the condition (3.3.2) for all A a'; (: go , the range of T. On the other hand, if ad A: II ~ II is the operator ad A (B) = [A, B], then it is straightforward to verify that (3.3.1) is equivalent to the condition that ad A commutes with T. Hence, in terms of the notation of §3.1, the algebraic identity (3.3.1) is equivalent to the condition ad A commutes with T for all A (: go . On the other hand, it is clear that an operator S on II commutes with T if and only if it leaves the eigenspace lli , i = 0, 1, ... , n, invariant. Hence if A (: go and B (: ll,. , then [A, B] (: lli. But then if A (: lli , B (: lli where i, j > 0, then [A, B] (: lli ( \ lli and hence [A, B] = 0 if i ~ j. We see then go is a subalgebra and that the lli C go are disjoint ideals in go . Conversely, of course, if go is a subalgebra of II and lli C go , i = 1, 2, ... , n, are ideals in go , then ad A commutes with T for all A (: go. It has to be observed, of course, that ad go leaves llc, (the orthogonal complement) invariant. THEOREM 3.3. The curvature tensor R.ikl satisfies the identity (3.3.1) if and only it the range go of T is a subalgebra of ll, and the eigenspaces Il. for i > 0 are ideals in go .

3.4. The fact that the range go of T is a subalgebra of II is, of course, well known for a symmetric space. In fact it was shown by Cartan that go is just the Lie algebra of the holonomy group, see [3; 223]. We obtain here the additional information that the eigenspaces of T, for the non-zero eigenvalues, are ideals in go . N ow since M is assumed to be irreducible symmetric, the holonomy algebra go acts irreducibly on Vo. We recall that we wish to show that if any ideal lli(i> 0) contains an operator of rank 2, then T is a scalar multiple of the identity. We now observe that it suffices to show that go = ll. Indeed if n ~ 4, then II is a simple Lie algebra and hence by Theorem 3.3 T can have at most one eigenvalue. If n = 4, then II = Ql + Q2 where Ql and Q2 are simple ideals both isomorphic to Lie algebra of SO(3). But neither Ql nor Q2 contains an element of rank 2. (In fact every non-zero element of Ql and Q2 has rank 4. This is true since exp Qi = Hi is isomorphic to the unit quaternions, and its action on Vo is equivalent to left multiplication on the quaternions in case i = 1 and right multiplication in case i = 2.) Thus if n = 4 and we show go = II under the assumption that T has an eigenvector in its range having rank 2, then Ql and Q2 are eigenspaces for the same eigenvalue. Hence even in this case T acts like a scalar. But now, assuming only that go contains an operator W of rank 2, it follows already from Corollary 2.7 that either (1) go = II or (2) Vo is even dimensional and admits a complex structure and a Hermitian metric with respect to which go

105

A CHARACTERIZATION OF THE CLASSICAL GROUPS

121

is the Lie algebra of all skew-Hermitian operators. We will now show that under the additional assumption that W E Q. for some i > 0, the second case (2) cannot occur unless dim Vo = 2 (in which case both (1) and (2) coincide). We begin with: LEMMA 3.3. Let c C go be the center of go. If c ~ 0, then for some i, c =

(t ••

Proof. Let h = go + Vo. We make h into a Lie algebra by (1) retaining the bracket relation in go , (2) defining [A, v] = - [v, A] = Av for A E go , V E Vo , and (3) for x = Xi and y = yi in Vo we have [x, y] = 2R i ikl X'yi Ego. From the standard identities on the curvature tensor and Theorem 3.3 it follows that h is a Lie algebra. In fact, as shown by Cartan, h is isomorphic to the Lie algebra of all infinitesimal motions on M; see e.g. [10; 80-83]. Consequently h is semisimple, see [5; 119]. Now if BI is the Killing form on h, that is, (X, Y)I = tr ad X ad Y for X, Y E h, it is immediate that go and Vo are orthogonal supplementary subspaces. Hence since BI is non-singular on h, it follows that its restrictions to go and Vo are, respectively, non-singular. But the restriction of BI to Vo is invariant under the action of go. Hence since go acts irreducibly on Vo , it follows that there exists a scalar b such that b(x, y)l = (x, y) for all x, y E Vo. Let B2 = 2bB l . When X = A Ego, we will write ad,A and A for the respective restrictions of ad A to go and Vo . We now show that for any A, B E go (3.3.3)

(A, TB)2

= (A, B)o .

For any x, y E Vo let W(x, y) E go be defined by W(x, y) = X'Yi - yiXi. Let l' = L~-I 1', so that 1': (t ~ go is the projection of (t on go. To show (3.3.3) it suffices by linearity to show (3.3.3) for all B of the form 1'(W(x, y)). But now T1' = T so that if B = 1'(W(x, y)), then TB = TW(x, y) = - [x, y] so that - (A, TB)2 = (A, [x, y])2 = (Ax, Y)2 = 2(Ax, y). But now 2(Ax, y) = 2a';x i y; = tr A· W(x, y) = - (A, W(x, y))o = - (A, 1'W(x, y))o = - (A, B)o . Thus - (A, TB)2 = - (A, B)o which proves (3.3.3). Now assume BE (ti ; then TB = XiB and hence by (3.3.3) Xi(A, B)2 = (A, B)o for all A Ego. Thus for all A Ego Xi ·2b tr ad A adB

= -tr

AB.

= -tr

AB

Conversely, if there exists X such that X·2b tr ad A ad B

for all A E go , then clearly B is an eigenvector of T with eigenvalue X. On the other hand, if B E c, then ad A ad B vanishes on go and hence tr ad A ad B = tr AB. Thus if X = -1/2b, we see that every element B E C is an eigenvector of T with eigenvalue -1/2b. Conversely, if BEg is an eigenvector of T with eigenvalue - 1/2b, then tr ad A ad B = tr AB for all A E go. Hence tr adgA adgB = 0 for all A E go. But this implies B E C since, admitting a faithful irre-

106

122

BERTRAM KOSTANT

ducible representation (its action on V o), go must be reductive. Thus, if c 'F- 0, there exists i such that - I/2b = Xi and Qi = c. This concludes the proof. Now it is a well-known result of E. Cartan that in case there exists a skewsymmetric J, such that J2 = - 1, and J commutes with go , then J I: go ; see [4; 250 and 257; IIIl. Hence if the case is that of (2) of Corollary 2.7, it follows that the J of (2), Corollary 2.7 belongs to go. In fact J I: C C g where c is the center of g. Now assume that (2) of Corollary 2.7 occurs. Since go acts irreducibly on Vo , it follows then that c is one-dimensional. Now by Lemma 3.3, c = Qi for some i > O. Let W I: Q; , j > 0, be of rank 2. Then if j = i, it follows that W = aJ for some number a. But then dim Vo = 2, in which case it is obvious that go = Q. Now assume that dim Vo > 2. Then j rE i and hence (W, J)o = O. We assert this implies W J = O. Indeed let U C Vo be the range of W. Then dim U = 2 and since J commutes with W, it follows that J leaves U invariant. But since J I: Q and since all skew-symmetric operators on a 2-dimensional space are proportional, it follows that there exists a scalar b such that bJ - W vanishes on U. Furthermore, since W vanishes on the orthogonal complement of U, this implies W(bJ - W) = o. But then tr W J = 0 implies - tr W 2= (W, W)o = 0 and hence W = O. But this is a contradiction. Hence only when dim Vo = 2, can (2) of Corollary 2.7 occur when we make the assumption that there exists W I: Q; for j > 0 of rank 2. But in this case we already have go = Q. We have proved ThEOREM 3.4. Let M be a Riemannian symmetric space. Let 0 I: M and let Vo be the tangent space at o. Let the ssctional curvature K be defined (at 0) as in §3.I on the projective space P associated with all bivectors at o. Then K takes a non-zero critical value on the submanifold N C P associated with the planar elements at 0 if and only if M has constant curvature.

REFERENCES 1. S. BOCHNER AND K. YANO, Curvature and Betti numbers, Annals of Mathematics Studies, no. 32, Princeton, 1953. 2. A. BOREL AND 1. DE SIEBENTHAL, Les S0U8-groupes ferm~s de rang maximum des groupes de Lie clos, Commentarii Mathematici Helvetici, vol. 23(1949), pp. 200-221. 3. E. CARTAN, Le~ons sur la Geometrie des Espaces de Riemann, Paris, 1946. 4. E. CARTAN, Sur une classe remarquable d'espaces de Riemann. I, Bulletin de la Societe MatMmatique de France, vol. 54(1926), pp. 214-264. 5. E. CARTAN, Sur une classe remarquable d'espaces de Riemann, I, II, Bulletin de la Societe MatMmatique de France, vol. 55(1927), pp. 114-134. 6. E. CARTAN, La. goometrie des groupes de transformations, Journal de MatMmatiques Pures et Appliquees, vol. 6(1927), pp. 1-119. 7. E. B. DYNKIN, The structure of semi-simple algebras, Uspehi Matematicheskih Nauk (N.S.)2, no. 4(20), (1947), pp. 59-127. American Mathematical Society Translation no. 17. 8. H. FREUDENTHAL, Zur Berechnung der Charakrere der halbeinfachen Lie8chen Gruppen. I,ll, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings of the Section of Sciences, no. 4, vol. 57(1954), pp. 369-376.

107

A CHARACTERIZATION OF THE CLASSICAL GROUPS

123

9. F. GANTMACHER, Canonical representation of automorphism& of a complex semi-simple Lie group, Matematiceski Sbornik, vol. 47(1939), pp. 101-146. 10. B. KOSTANT, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Transactions of the American Mathematical Society, vol. 80(1955), pp. 528-542. 11. J-L. KOSZUL, Homologie et cohomologie des algebres de Lie, Bulletin de la Societe Mathematique de France, vol. 78(1950), pp. 65-127. 12. A. I. MALCEv, On semi-simple subgroups of Lie groups, Bulletin of the Academy of Sciences URSS, Series on Mathematics, vol. 8(1944), pp. 143-174. American Mathematical Society Translation no. 33. 13. Seminaire Sophus Lie, Ie annee 1954/1955, Theorie des algebre de Lie, Topologie des groupes de Lie, Ecole Normale Superieure, Paris, 1955. UNIVERSITY OF CALIFORNIA

108

Reprinted from the TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 93, No. 1, October, 1959 pp. 53-73

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT(1) BY

BERTRAM KOSTANT

1. Introduction. 1. Let g be a complex semi-simple Lie algebra and f) a Cartan subalgebra of g. Let 7f).. be an irreducible representation of g, with highest weight A, on a finite dimensional vector space V". A well known theorem of E. Cartan asserts that the highest weight, A, of 7r" occurs with multiplicity one. It has been a question of long standing to determine, more generally, the multiplicity of an arbitrary weight of 7r". Weyl's formula (1.12) for the character of 7r" is an expression for the function X,,(x) =tr exp 7r,,(x) , xEf), on f) in terms of A and quantities independent of the representation. In the same spirit the author has always understood the multiplicity question to mean the following: Let I be the set of all integral linear forms on f). Let m" be the function in I which assigns to each integral linear form IIEI the multiplicity m,,(II) of its occurrence as a weight of 7r". Find a formula for the multiplicity function m" in terms of A and quantities independent of the representation. It is the purpose of this paper to give such a formula (1.1.5). Obviously a knowledge of the multiplicity function m" determines XA(x), xEf). That is, (1.1.1)

L

XA(x) =

mAC,,) exp ('" x).

vEl

On the other hand Weyl's formula asserts that

L (1.1. 2)

XA(X)

=

sg(cr) exp(cr(g

+ X), x)

crEW

--=-------

L

sg(cr) exp(cr(g), x)

crEW

where g is one half the sum of the positive roots and W is the Weyl group. Finding a formula for the multiplicity function m A in a sense then "accomplishes" the division indicated by the formula of Weyl. We hasten to addthis does not in any way detract from Weyl's formula since it still retains its overriding and quite remarkable feature of expressing what is in general a very complicated trigonometric polynomial on f) as a quotient of two relatively simple trigonometric polynomials. A direct interest in the multiplicity function arises from sources other than those mentioned above. Included are the following: Received by the editors April 26, 1958. (1) This research was supported by the United States Air Force through the Air Force

Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49(638)-79.

S3 B. Kostant, Collected Papers, DOI 10.1007/b94535_10, © Bertram Kostant 2009

109

54

BERTRAM KOSTANT

[October

(1) Let U be a compact connected Lie group and T';;. U a maximal toroidal subgroup. By virtue of the Frobenius reciprocity theorem the induced representation of U by a character of T is determined as soon as one knows mA(v) for all dominant X and a fixed suitable vEl. On the other hand such an induced representation is of special interest in algebraic geometry since as one knows it is equivalent to the representation of U defined by the natural action of U on the cross sections 9f the complex line bundle corresponding to v over the algebraic manifold (flag manifold) U/T. The knowledge of mA(v) for X~V supplements the Borel-Weil theorem with the information that 7rA occurs only on nonholomorphic cross-sections of the line bundle and does so with multiplicity mA(v). (See [1] for details.) (2) Concerning infinite dimensional representations a theorem of GelfandNeumark asserts that the restriction, to a maximal compact subgroup U of a complex semi-simple group G, of an irreducible unitary representation of G belonging to a nondegenerate series is given, as in (1), as soon as one knows mA(v) for all dominant X and a fixed suitable v. (See [5].) A means of computing mA(v) has been given by Freudenthal in [4]. The computation is based upon a recursive relation satisfied by the values mA(v), vEl, for a fixed X. This relation is an immediate consequence of what Freudenthal calls the Hauptformel. It is given as (see [3, 2.1 and 3.1])

(1.1.3)

L: mA(v +

kcp)·(v + kcp, cp) = mA(v) «A + g, A + g) - (v + g, v + g»

where the summation is over all positive integers k and all positive roots cp. For the purposes of finding a formula for mA(v), use of the relation (1.1.3) carries the repeated disadvantage of having always to divide by terms of the form «X+g, X+g) - (v+g, v+g» even in the case when v is not even a weight of 7r A• We could find no way in which (1.13) leads to a closed expression for mI.. Let P(p,) , p,EI, be the integer valued function on I defined by

pep,) = no. of ways p, may be partitioned into a sum of positive roots. I t follows from elementary considerations in representation theory that the inequality (1. 1. 4)

holds for all dominant X and all p,EI. Now one can show (and we exploit this fact) that, fixing p" for X sufficiently "far out" in the fundamental chamber and sufficiently far from the "walls" of the chamber the equality sign in (1.1.4) will always hold. It seems clear then that a formula for m A must necessarily involve the function P. It is the main result of this paper to establish the formula (1. 1. 5)

m,..(v) =

L: sg(u)P(u(g + A) aEW

110

- (g

+ v».

1959]

55

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

Putting A= 0 yields the following recursive relation for the partition function

P. (1.1. 6)

L

P(!J.) = -

sg(u)P(!J. - (g - ug»

aEW;D'~e

for ,u~O, ,uEI. (The recursive nature of (1.1.6) is further clarified when it is recalled that P vanishes outside the cone generated by the positive roots and g-ug lies in that cone. Also P(O) = 1.) 1.2. An auxiliary result is Theorem 5.1. Theorem 5.1 bears the same relation to the relation "totally subordinate" among representations, introduced in §4.4, as does a theorem of Dynkin (Theorem 4.3) to the relation "subordinate." Theorem 5.1 may be regarded as a weak generalization of the Clebsch-Gordan theorem. 2. Preliminaries. 1. Let g be a complex semi-simple Lie algebra of dimension n. Let 1 be the rank of g and let g be a Cartan sub algebra of g(dim g= l). Let B be the Cartan-Killing bilinear form on g. The value B assigns to vectors x, yEg will be denoted by (x, y). One knows that the restriction of B to g is nonsingular and hence one may identify g with its dual space. In particular ~, the set of roots of g with respect to g, is then a subset of g. Let go be the real subspace of g generated by~. Then one knows that go has real dimension l, the restriction Bo of B to go is positive definite, and

g = go + ig o is a real direct sum. Let W be the Weyl group of g regarded as operating on g. Elements x, yEg are said to conjugate under W if ux = y for some uE W. One knows that go is invariant under W. In fact it is only the action of Won go which is of interest to us. For each root cf>E~Cgo let g",~go designate the hyperplane orthogonal to cf> and therefore given by

g",

= {x

E go I (x, cjJ)

=

Also let R",E W designate the reflection of algebraically by R",x

for

go

O}. through

g",.

This

1S

given

2(cjJ, x)

= x - ---cjJ (cjJ, cjJ)

xEgo. The open set ill C 60 defined by ill

= go - U g", "'Ell.

is called the set of regular elements in go. The connected components D~ of ill are called open Weyl chambers. One knows that there are w of them where

111

56

BERTRAM KOSTANT

[October

w is the order of Wand that in fact they may be indexed by TV in such a way that if

is the decomposition of CR into its connected components D~ =u(DO) for any uEW and DO=D~, where e is the identity element of W. The closure D" of D~ will be called a closed \\leyl chamber or simply a Weyl chamber. Obviously one has

and D,,=u(D). Having fixed D--now called the fundamental chamber-among all the equally suitable Weyl chambers, we will say that an element xEgo is dominant if xED. We will say x is strongly dominant if xEDo. That is, x is strongly dominant if it is both dominant and regular. Each chamber D", uE W, decomposes A into a union of two disjoint subsets At and A;;- where A;;- = ( -1)A,t and c/JEAt if and only if (c/J, x) ~ 0 for all xED". Conversely, one knows that xED" if and only if (c/J, x) ~O for all c/JEA,t. Of course the inequality ~ becomes a strict inequality > when xED~. Write A+ and A- for At and A;-. The elements of A+ are called positive roots and they are in fact just the positive elements of A with respect to a suitable lexicographical ordering in go. We shall assume from now on that such an ordering is given in go. 2.2. Consider the lattice I (also, a discrete subgroup of go) of integral elements in go. By definition }LEI if and only if 2(}L, c/J)/(c/J, c/J) is an integer for all c/JEA. The set I is the set of all weights of all representations of g. Let n CA+, n = {aI, a2, . . . , al} , be the set of simple positive roots. The elements ai, i = 1, 2, ... , 1 form a basis of go. Let fi, j = 1, 2, ... , 1 be the dual basis to the basal elements 2a;/(ai' ai), i= 1, 2, ... , t. That is (2.2.1)

2(fj, ai)

- - - = fJ;j.

(a;, ai)

ThenjjEI,j=I,2, ·,1 and in fact these elements form a basis of I. That is, if }LEgo then upon writing I

j.J.

=

L

n;J;

}L E I if and only if the ni are integers. On the other hand 1; E D for j = 1, 2, . ,land in fact if xEgo then upon writing I

(2.2.2)

X

=

L i-I

112

c;fi

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

1959]

57

xED if and only if Ci~O, i= 1, 2, ... , land xEDo if and only if c.>O, = 1, 2, ... , t. Important in representation theory is the intersection ID=lrlD, the set of dominant integral elements in 1)0 and IDo=lrlDo, the set of strongly dominant integral elements in 1)0. 2.3. Now let 7r be a representation of 9 on the finite dimensional complex vector space V ... We shall always assume that the representation is complex linear. In such a case one knows that there is a unique decomposition of V.. as a direct sum of weight spaces V .. (JL), JLEI. That is

i

V .. =

L

V .. (/L)

pEl

where V .. (JL) is defined by V .. (IL)

= {v E V .. 1 7r(x)v

(IL, x)v for all x

=

E 1)}.

Of course V .. CJL),=O for only a finite number of JL. An element JLEI such that V.. (JL)'=O is called a weight of 7r. We will let A(7r) CI designate the set of weights of 7r. Now for any JLEI let m .. (JL) =dim V .. (JL). One always has m.. (/L) = m.. (u/L)

for any JLEI, uE W. Now in case 7r is irreducible the convex set in 1)0 generated by all the weights of 7r has as its extremal points a unique dominant weight X and all its conjugates {uX}, uE W. Anyone of these extremal points will be called an extremal weight. The weight X is the highest weight of 7r relative to any lexicographical ordering in 1)0 making A+ the set of positive elements in A. It is an already classical theorem, due to E. Cartan, that m .. (X) = 1 and that 7r is characterized by its highest weight. Furthermore, since any element vElD is the highest weight of some irreducible representation of g, we may use I D as the index set for the set of equivalence classes of all irreducible representations of g. In fact, for simplicity, for each XEID we choose a fixed irreducible representation of 9 with highest weight X and designate it by 7r).. The vector space for this representation will be designated by VA and, for simplicity, we will write V).(JL) for V ..,,(JL), m).(JL) for m..,,(JL) and A(X) for A(7r A). 3. The partition function P. 1. Now g admits the direct sum decomposition 9 = 1)

+L

(ei' i= 1,2, ... ,r, are the positive roots indexed so that cf>ii+1 and ~i' rh 'Y/k, 1 ~i~r, 1 ~j~l, 1 ~ k ~ r are non-negative integers, designating, of course, the powers of the corresponding basal elements. We shall need a simplified notation for this basal element. Towards this end let 0 designate the n-tuple (~i' rh 'Y/k) and write eO for the basal element (3.1.2). Let A designate the index set of all n-tuples 0 with non-negative integer coefficients. Thus the most general element PES(g) may be uniquely written p = 2: aoe o oeA

where a8 are complex numbers, only a finite number of which are distinct from zero. 3.2. Now since b..CI we can define a mapping of A into I as follows: the image of OEA is denoted by (0) and (0) is defined by (3.2.1)

(8) =

t

;=1

(~i -

71i)cf>i

where 0 is the integral n-tuple (~i' rh 'Y/k). The significance of (0) will be ap parent from the following: consider the infinite dimensional (purely algebraic) representation p of g on S(g) defined by p(x)p= [x, p] where xEg, PES(g). Then we observe that S(g) admits the direct sum decomposition (3.2.2)

S(g)

=

2: S~(g) ~eI

where

S~(g),

the "weight space for the weight

fJ,"

is defined by

S~(g) = {PE8(O)! [x,p] = (JL,x)P,xElJ}.

Indeed it is clear from (3.1.1) and (3.2.1) that e 8 ES(8)(g) and that in fact the set of all eO such that (0) = fJ, forms a basis of 8~(g). Since the set {eo}, OEb.., forms a basis of S(g) we have (3.2.2). Now any representation 7r of g on a vector space V". admits a unique extension to S(g) as a homomorphism of S(g) into the algebra of operators on V".. We shall always regard 7r as so extended. The significance of the decomposition as far as representation theory is concerned is that 7r(p) maps V .. (v)~ V .. (v+fJ,) for every PESI'(g). That is (3.2.3)

7f: S~(g) X V".(v) ~ V".(JL

+ v)

where 7r(p, v) =7r(p)v for PESI'(g), vE V,..(v). This is, of course, clear from the definition of SI'(g) and V".(v).

114

1959]

59

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

3.3. The space el'(g) is clearly infinite-dimensional. For many purposes it suffices to consider a particular finite dimensional subspace of el'(g) , the subspace generated by eq, where cf> is positive. Let n+, a maximal nilpotent subalgebra of g, be the Lie subalgebra spanned linearly by the root vectors eq, where cf>Et:.+. Then as usual one may regard e(n+) , the enveloping algebra of n+, as a subalgebra of e(g). Let A+ designate the subset of A consisting of all n-tuples ~i' 'f/k such that r;='l/k=O,j=l, 2,···, l, k=l, 2,···, r. We will use the letter ~ to designate elements of A+. Now we observe (applying [7, Theoreme 1', p. 1-07] again) that the elements ee, ~EA+ form a basis of e(n+). Let

r;.

e,,(n+) = e,,(g) II e(n+).

Then it is clear that if A+(M), MEl, is defined by A+(.u) = {~E A+ I (~> =

.u}

the elements eE, ~EA+(M) form a basis of e,,(n+). Furthermore it is also clear that e,,(n+) is finite dimensional. The function on I which assigns to each MEl the dimension of e,,(n+) plays a central role in this paper. Thus for any MEl let P(.u) = dim el'(n+) = number of elements in A+(.u).

We wish to make the following observations about P(M). First of all each may be regarded as a "way" of writing M as a sum of positive roots. Since repetitions of roots are permitted and the order in which the roots occur does not enter, ~ may be regarded as a "partition" of M into a sum of positive roots. Thus P(M) is, in effect, a partition function counting the number of partitions of M as a sum of positive roots. Upon writing ~EA+(M)

I

(3.3.1)

.u =

L:

niCl:i

i-1

it is obvious that P(M) =0, if for some i, ni is not a non-negative integer. Also note that P(O) = 1 (2). These facts are used in the recurrence formula given in §6 for P. We note more generally that if ni is a non-negative integer for all i then P(M) ~ 1. Indeed the formula (3.3.1) provides a way of writing M as a sum of simple positive roots. Since the simple positive roots are linearly independent there is only one such way of writing M. Now what was defined above for n+ we define similarly for n-, the Lie sub algebra generated linearly by all the root vectors e_q" cf>Et:.+. Let A- be (2) Recall that e(n+) contains the scalars and that ~i =li ='l/k =0 defines an element ~EA + such that eE= 1.

115

60

BERTRAM KOSTANT

[October

all n-tuples ~i' rio 'l/k such that ~i=rj=O, i= 1,2, ... ,r,j= 1,2,· ., t. We will use the letter '1/ to designate an element in A-. Analogously 81'(n-) has as basis the elements e~ where 'l/EA-(p,). Obviously dim 81'(n+) = dim 8_I'(n-) for any p,EI. 3.4. Returning to representation theory, for any }..EID let vAE VA be a weight vector belonging to the highest weight}... That is, (vA) = V A(}..). It is a well known and simple fact that every vector vE VA may be put in the form v=7r A(p)v A where PE8(n-). Indeed to prove this it suffices to know, (1), that the root vectors Cai and Lap where ai and aj run through the simple positive roots, generate 9 and, (2), that [e a " e"j]=5ijai. Furthermore it follows from (3.2.3) that for any p,EI every vector vE VA(}..-p,) may be written v=7r A(p)v A where PE8_I'(n-). That is, 7r).(8_I'(n-)v). = V).(X - JJ.).

It follows immediately then that (3.4.1) for any }..Eln . We will show that given any p,EI, }.. can be chosen so that the equality holds in (3.4.1). This and more will be needed in §6.2. 4. Theorems of Dynkin and Brauer. 1. For any}..Eln let}..*Eln be the highest weight of the contragredient representation to 7r A. 'vVe may always choose VA> so that VA> is the dual space to V). and 7r).o(x) is the negative transpose of 7r A(x) for any xEg. Now for any }..Eln we recall that A(}") = -A(}..*) and in fact mA(p,) =mA>( -p,). Note that this implies -}..* is the extremal weight of 7rx which lies in the chamber - D. It follows then that the one dimensional space V). ( -}..*) may be characterized by V).(-X*)

= {v E V A I7r).(x)v = 0 for all x E

n-l.

Now let 7r be any representation of 9 on a vector space V ... Define the subspace Z .. c V .. as follows:

Z". = {v E V .. I 7r(x)v = 0 for all x E n-} .

It follows immediately that (4.1.0)

dim Z .. = C(7r)

where C(7r) is the number of irreducible representations appearing in the decomposition of V .. into irreducible components. Now let }..l, }..2Eln and consider the case when 7r=7r).2®7rA~' the tensor product of the representations 7rA2 and 7rA~. It is well known that we may iden-

116

A FORMULA FOR THE MULTiPLiCITY OF A WEIGHT

1959]

61

tify the vector space V .. = V x• ® Vx~ with the space L( VXl' VA,) of all linear transformations A mapping V XI into V x• and that with respect to this identification 7r(x)(A)

= 7rx,(x)A - A7rA,(X),

But then we find that Z(7rx. ® 7rx~)

=

{A E L(Vx]! VA.) I 7rx.(x)A = A7rA,(X) for all x E n-}.

That is, Z(1I' X2®1I'X~) is the set of all intertwining operators for the pair of restriction representations 1I'Ali n- and 1I'x.1 n-(3). But then we note that if A EZ(1I'A2®1I'X~) 7rx.(p)A

(4.1.1)

=

A7rxI(p)

holds for all PE8(n-). On the other hand since every vector vE VAl may be put in the form V=1I'XI(P)VXl where PE8(n-) and (Vx) = VX(A), it follows that every A EZ(1I'x.®1I'x;) is uniquely determined by what it does to the single vector Vx,. Define the subspace WA.(A I ) C V x• by WX.(X 1) =

{v E

VA.

Iv =

Avx! for some A E Z(1I'X2 ® 7rx~)}.

Then, as we have just noted, the mapping (4.1.2) defined by (T(A)

= A1!Al

for A EZ(1I'A2®1I'A~) is an isomorphism onto. Recalling (4.1.0) we note in passing that we have proved LEMMA 4.1. Let AI, A2EID • For any finite dimensional representation 11' of g let C(1I') denote the number of irreducible representations occurring in the complete reduction of 11' into irreducible components. Then

(4.1.3)

C(7rA2 ® 7rA~) = dim W".(X,)

(4.1. 4)

~

dim V A2 .

We will be interested in the case when equality holds in (4.1.4). That is, when W".(AI) = V).. 2' Towards this end we wish to characterize the space WA.P'-l).

4.2. For any AEID and any vE VA let the left ideal S(v, A) in S(n-) be defined by 8(v, X)

= {p E S(n-) 17rx(P)V = o}.

(3) If rCg is a subalgebra and". is a representation of g denote by ". to f.

117

,..1 f

the restriction of

62

BERTRAM KOSTANT

[October

The following lemma is then an elementary fact in general ring theory: LEMMA 4.2. Let A!, A2EID ; then when vE VX2' VEWX2(Al) if and only if 8(vxl! AI) C8(v, A2)' That is, if and only if

7f)'t(P)VXl = 0 implies 1I"x.(p)v = 0

for any PE8(n-). Proof. If vE Wx.(Al) it is obvious from (4.1.1) that the condition of Lemma 4.2 is satisfied. Conversely if the condition is satisfied then setting A (1I).1(P)VXl)

= 1I).2(P)V

for all PE8(n-) defines (in a well defined way) an element A EZ(7rX2®7rX~) such that AVXl =V. 4.3. In [3], Dynkin introduces (Definition 3.2, p. 283) the notion of one representation being subordinate to another. In the case of irreducible representations, say 7rXl and 7r~2' Dynkin's definition is as follows: 7rx. is said to be subordinate to 7r~1 (or simply A2 is subordinate to AI) in case v~.E W~l ().2) (4). Recall (vx.) = V~2().2). Thus according to Lemma 4.2, A2 is subordinate to Al if and only if 8(v~J! Al)C8(v~2' A2)' That is, if and only if for all PE8(n-) (4.3.1) Dynkin then goes on to prove the following theorem (Theorem 4.3 below, Theorem 3.15 in [3, p. 285]), which asserts in effect that A2 is subordinate to Al if and only if (4.3.1) holds for a much smaller class of elements p. First however, we recall the following well known facts in representation theory. Let 7r be a representation of g on V ... Let cf>Etl, JLEtl(7r), v be any vector in V .. (JL) which is also an eigenvector of 7r(Lt/>et/». (It is known that one may find a basis in V .. (JL) which has this property.) Then (4.3.2)

2(cp, p.) (cp, cp)

=

p- q

where p is the smallest value of j such that 7r( (e_t/» i+ 1 )v = 0 and q is the smallest value of j such that 7r(~+l)V = O. For use later on we note the following easy consequence of (4.3.2). Let Mt/>(A) be defined by 2(cp, p.) Mt/>(X) = max - - ; I'E.1(X) (cp, cp)

then Mt/>(A) is the smallest value of j such that (4.3.3)

;+1 1I"x(et/> ) = O.

(') Note that in [3] extremal vector means weight vector for the highest weight, not any extremal weight.

118

63

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

1959]

The equivalence of the two statements in the following theorem arises from the fact that if AEID and A= L:=l C;ji then by (2.2.1) and (4.3.2) Ci is the smallest value of j such that

i=1,2,···,l. THEOREM (DYNKIN) 4.3. Let AI, 'A 2 EI D. Then and only if'A I -A2EID. That is, writing

71""2

is subordinate to

71",,\

if

k = 1,2 7I"A2

is subordinate to

7I"AI

if and only if 1 2 C, ~ Ci

for i = 1, 2, ... , l. Equivalently andi=l, 2,···, 1

where

(VAk) =

7I"A 2 is

subordinate to

71",,\

if and only if for all j,

VAk(Ak), k=l, 2.

4.4. For any fJ-EI, 'AEID, vE VA let e_~(v,

X) =

e_~(n-)

n

e(v, X).

Then the inequality (3.4.1) in fact becomes (4.4.1)

Dynkin's theorem asserts that as the coefficients of A go up the right side of (4.4.1) goes down. We shall need the fact (proved later) that it can be made zero. Let 'AI, A2EID; we will now say that 'A 2 is totally subordinate to Al in case V A2 = WA 2 ('A l ). Obviously totally subordinate implies subordinate. From the point of view of general ring theory the notions of subordinate and totally subordinate are very easy to describe. Let k = 1, 2, then V"k may be regarded as a cyclic module over the ring e(n-), with cyclic vector VAk. Now associated with every cyclic module over a ring are two prominent subrings (1) the left ideal (here e(VAk' 'Ak)) of all ring elements which annihilate the cyclic vector and (2) the two sided ideal-now written as Jk-of all ring elements which annihilate the entire module. Obviously JkCe(VAk' 'A k). Now we observe that A2 is subordinate to 'AI when e(VAl' 'AI) s;e(VA2' A2) and totally subordinate to Al when e(VAI' 'AI) CJ2. Continuing from §4.1, obviously Lemma 4.1 implies

119

64

BERTRAM KOSTANT

[October

LEMMA 4.4. Let Xl, X2E I D; then X2 is totally subordinate to Xl if and only if

C(1I'X 2 ® 1I'X;) = dim Vx!. 4.5. If X2 is totally subordinate to Xl it follows immediately from Lemma 4.2 and (4.3.2) that 2(a" Al) 2(a;, fJ.) --->---(ai, ai) = (a;, ai)

for all }LEA(X2). Thus by §2.1, (4.3.2) and (4.3.3) we have LEMMA 4.5. Let XI, X2 EID. If X2 is totally subordinate to Xl then Xl - }LEID for all }LEA(X2) or equivalently 1I'A 1 (e!_aJ Vx,

= 0 implies 1I'x2(eL

aj )

= 0

for all j and all i = 1, 2, . . . , 1. 4.6. We will prove an analogue of Dynkin's theorem (Theorem 5.1) for the notion of totally subordinate (instead of subordinate). Theorem 5.1 asserts in effect that the condition of Lemma 4.5 is also a sufficient condition for totally subordinate. For this we need a theorem of Brauer. First, however, we wish to observe, LEMMA 4.6. Let XI, X2 EID then X2 is totally subordinate to Xl if and only if

X: is totally subordinate to Xi.

Proof. It is obvious that 1I'x: ®1I'XI is the contragredient representation to Hence C(1I'x; ®1I'x 1) = C(1I'X2®1I'A~). Since, of course, dim VA' = dim VA, the result follows from Lemma 4.4. 4.7. Let a(I) designate the group algebra over I. We admit into a(I) only functions h on I with finite support. It is also convenient to regard elements of a(I) as finite formal combinations of the elements of I. However, since the group operation in I is written additively we will designate the function (Dirac measure at v) which is 1 at v and zero at}L for all}L ~v by 5•. Thus when h is regarded as a formal combination of elements of I, h is written uniquely as

1I'A2®1I'X~.

h

=

:E a.B• • F.l

where only a finite number of the a. are distinct from zero. When regarded as a function, h(v) = a •. The function mA, defined in §2.3, which assigns to each vEl its multiplicity in 11'1\ is an element of a(I). Now for each vEl let F.E a (I) be defined by F. =

:E sg(uB).,•.

.,ew

Note that F.= sg(u)F.,. and as a function F.(}L) = sg(u)F.(u}L). Next we observe that F.~O if and only if vE.;.

But by Lemma 4.6 this implies A2 is totally subordinate to Ai. We have thus proved the following theorem. The tensor product aspects of Theorem 5.1 may be regarded as a weak generalization of the Clebsch-Gordan theorem. A generalization, since if g is the Lie algebra of all 2 X 2 complex matrices of trace zero then for any pair At, A2EID, either Ai is totally subordinate to A2 or vice versa. Weak, since for general g not every pair Ai, A2EID are so related. Applying Lemmas 4.4 and 4.5 we have

122

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

1959]

67

5.1. The following statements are all equivalent. Let AI, A2EID (1) A2 is totaUy subordinate to AI, (2) AI-MElD for all ME.!l(A2), (3) IfA1= L~=l c;f; then ci~Ma;(A2)' i=l, 2, ... ,l (see §4.3), (4) 'lrXt (tI-aJVXt = 0 implies 'lr X2(tI- a ,) = 0 for i = 1, 2, ... , l; j = 1, 2, (5) C('lrX2®'lrX~) =dim V X2 where C('lrX2®'lrX~) is the number of irreducible representations appearing in the decomposition of the tensor product 'lrA2 ®'lrA~ into irreducible representations. (6) For any 'hElD the representation 'irA appears m A2 (M) times, where M= Al - A, in the decomposition of the tensor product 'lrX~ ® 'lrAt into irreducible representations. THEOREM

5.2. Let g be as in §4.7. Consider now the "one-parameter" family of representations 'lrkg, k=O, 1, 2,···. But now for any AEID where A= L~=l cif. it follows immediately from (4.7.1) and Dynkin's theorem, Theorem 4.3, that 'irA is subordinate to 'lrkg if and only if k ~ maXi Ci. In particular we note that 'lrkg is subordinate to 'Ir(k+1)g for all k. Thus for all MEl (5.2.1)

Since as we recall (5.2.2)

it follows that the left side of (5.2.2) is monotone decreasing with increasing k. Next as an immediate consequence of (1) and (3) in Theorem 5.1 we have LEMMA

5.2. Let AEID. Then

k

'irA

~

is totally subordinate to 'lrkg if and only if

max Ma;(X). i

5.3. But now we can prove easily that the right side of (5.2.2) vanishes for k sufficiently high. From (5.2.1) it obviously suffices to show that for any PEBI'(n-) there exists k such that 'lrkg(P)VkuO=O. [This insures that we can always strictly drop the dimension of BI'(Vkg, kg) by choosing k large enough.] But now by Theorem 1 in [6] for any PEB(g) there exists AEID such that 'lrx(p) 0=0. In particular if PEBI'(n-) there exists vE VA such that 'lrx(p)vO=O. On the other hand by Lemma 5.2 for any k ~ maXi Ma;(A), 'irA is totally subordinate to 'lrkg. Thus from the definition of totally subordinate 'lrkg(P)VkuO=O. Thus we have proved LEMMA

5.3. Let MEl be arbitrary. Then there exists a positive integer N such

that for all

k~N.

Now for every O'E W let the subset T(O') C.!l+ be defined by

123

68

BERTRAM KOST ANT

[October

I

T(u) = {cp E Ll+ u- 1(cp) E Ll-l

and let s(u) EI be defined by

L

s(u) =

cp.

ET(v)

It is clear that if we write I

(5.3.1)

s(u)

L

=

bi(u)a;

i=l

then all the integers bi(u) are non-negative. If u¢e then b;(u) ~ 1 for at least one ~. Now observe that

L

u(g) = -1 ( 2

L

1/1 -

!l-E(A+-T(v»

cp) .

ET(v)

Hence we see that (5.3.2)

s(u)

= g - u(g)

for all uEW. In particular then it follows immediately from §2.1 and §4.7 that the elements s(u), uEW are all distinct. These elements of I will play a fundamental role in the remaining portions of this paper. In the definition of the function P on I we were concerned with the ways of expressing an element J.LEI in terms of the positive roots. Now we shall be concerned with expressing J.L in terms of the elements s(u), uEW. However, although in general there are far more elements s(u) than there are roots the matter is nevertheless simplified since now we will be concerned with the number of ordered ways of doing this-in fact ordered ways with a signature. Hence a recursion formula can immediately be given in this case. In the case of P(J.L), until we establish the equality given in Lemma 6.2 no recursive formula is apparent to the author. 5.4. For convenience write the elements s(u), uE W, as s;, i = 1, 2, ... ,w where w=order W. We may choose the ordering so that Sl=e and if s;=s(u) then -sg(u) = (_1)i. Now let r be the set of all finite sequences 'Y,

"Y

= (it,

i 2,

••• ,

iq)

such that i j is a positive integer and 2 ~ i j ~ w. For any J.LEI let the subset rl'cr be defined by

r I'

=

{"Y E r

IE

si;

=

It} .

Observe that since 'Y;~2 for all 'YEr it follows that rl' is a finite set. Now for any 'YEr define

124

1959)

A FORMULA FOR THE MUL TIPLlCITY OF A WEIGHT

69

Observe that if 'Y is the empty sequence, that is, q=O, then sg('Y) = 1. We can now define the function Q on I. For any p.EI let

Q(P) =

2: sg('Y)' 1'er"

If we write

we observe the following properties: (1) Q(J.L) =0 in case b; is not a nonnegative integer for some i, (2) Q(O) = 1, by a remark made above. Finally, (3), we note that if J.L¢O, Q satisfies the following recursive relation. (Here we revert to the original notation, s(u).) (5.4.1)

2:

Q(P) = -

sg(u)Q(P - s(u».

"'EW,cr~e

From (1) and (2) above, and from (5.3.1) we note that (5.4.1) provides an effective computation for Q. I t is especially interesting from the geometric point of view to write (5.4.1) in the form

2: sg(u)Q(J.L -

s(u» = 0

.. ew

for all J.LEI, p.¢0. 6. The multiplicity formula. 1. Let 'AEID • We now recall the formula of \Veyl for the character of the representation 1r~. For any xE1) Weyl's theorem (see [8] or [7, p. 19-07]) (6.1.1)

tr exp 'lrlo.(x) =

2: Fg+lo.(w) exp(w, x) 2: Fg(w) exp(w, x)

weI -",=------weT

where the (known) functions F.ECi.(I), pEl, are given in §4.7. On the other hand in terms of the (unknown) function mlo., obviously (6.1.2)

tr exp 'lrlo.(x)

=

2: m~(w)

exp(w, x).

weI

Setting (6.1.1) equal to (6.1.2) and clearing the denominator in (6.1.1) it follows that

Fa+"A

= Fg * mlo.

where * designates multiplication (convolution) in the group algebra G,(I). Thus recalling the definition of Fg and the usual expression for convolution

125

70

BERTRAM KOSTANT

(6.1.3)

L: sg(u)mA(w -

Fg+A(w) =

[October

u(g».

vEW

This immediate but none the less important consequence of Weyl's formula was first pointed out to us by Raoul Bott. Now let v=w-g and define G(v) = FII+A(w) = Fg+A(v

+ g).

Then GEa(J) has the following properties G(v) =

(6.1.4)

o for {sg(u)

v ;;tf. u(A

+ g)

for v = (T(A

- g,

+ g)

- g.

Substituting v+g for w in the right side of (6.1.3), solving for the case when u=e, and recalling that g-ug=s(u), we get (6.1. 5)

L:

mA(v) = G(v) -

sg«(T)mA(v

+ s(u».

G'EWiG'~e

Like (1.1.3) this sets up a recursive formula for mA(v). On the other hand (6.1.5) has one significant advantage over (1.1.3)-it does not necessitate the division of a term which might vanish. Now as in §5.4 write s(u) = Si where -sg«(T)=(-l)i and u;;tf.e implies i~2. Then applying (6.1.5) a second time we get w

mA(v) = G(v)

+ L:

w

(-l)ilG(v

+ Si) + L:

(_l)it+i2mA(V

+ Sil + Si2)'

In fact repeating the substitution k times we derive the relation k-l

111A(V) =

L:

w

L: w

+

L:

But now by (5.3.1) there exists an integer lvI such that for all k ~ lvI and any sequence 'Y= (iI, i 2 , • • • , i k ), W~ii~2, mA(v+Sil+Si2+ ... +Sik) =0. The same is true for G. Thus we can let k-t 00 and obtain from the definition of Q the relation (6.1.6)

mA(v) =

L: Q(w weI

Now recalling (6.1.4) we obtain

126

v)G(w).

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

1959]

71

6.1. Let XEI D , vEl then

LEMMA

m,,(v) =

:E sg(u)Q(u(g + X)

- (g

.. ew

+ v)) .

6.2. We will now proceed to prove the second major point of this paper, namely P=Q. LEMMA

6.2. The functions P and Q on I are identical.

Proof. Let p.EI be arbitrary. By Lemma 5.3 there exists an integer N such that (6.2.1) for all k~N. Now write p.

=

:E

biOli

i=1

and let N 1 =max i bi • Then by (5.3.1) for all integers

k~Nl

and all uEW,

UF-e

(6.2.2)

Q«k

+

1)(u(g) - g)

+ p.)

=

o.

This is clear since (k+1)(u(g)-g)+p.=p.-(k+1)s(u) and if we expand (k + 1)s(u) in terms of the (Xi, at least one of the coefficients must be negative. Let N 2 =max (Nit N). Apply Lemma 6.1 where X=kg, v=kg-p. and k~N2. Then

p. -

(6.2.3)

mko(kg - p.)

=

:E sg(u)Q(u«k + l)g)

- «k

+

1)g - p.)).

t7eW

But u«k

+ 1)g)

- «k

+ 1)g -

p.) = (k

+ l)(u(g)

- g)

+ p..

Hence by (6.2.2) all terms but one (u=e) drop out of (6.2.3). That is, mku(kg - p.) = Q(p.).

But then by (6.2.1) Q(p.) =P(p.). Q.E.D. The following is our main theorem. Summarizing from above we have proved THEOREM 6.2. Let g be a semi-simple Lie algebra with Cartan subalgebra g. Let leg be the discrete group of integral linear forms on g (see §2.2). Let P be the function on I,-the partition function- which assigns to every p.EI the number of ways p. can be partitioned into a sum of positive roots. (By a partition is meant multiplicities are permitted and the order is discounted. See §3.3.)

127

72

BERTRAM KOSTANT

[October

Let ~+ be the set of positive roots and let

g

1

L:

=-

~E,.(v) =0 in case v is not a weight of 7r>,. and in case v is a weight of 7r>,. let m>,.(v) be the multiplicity of the weight v. Then the number m>,.(v) is given by the formula

(6.2.5)

m>,.(v) =

L: sg(q)P(q(g + > E 6,+ let a (cf» be the sum of the coefficients of cf> relative to the basis of simple positive roots. Let ble be the number of roots cf> such that a (cf» = le. Then ble - bk+1 is the number of times le occurs as an exponent of g. For example, if we apply this to the case where

+

+

* Received December 19, 1958. Research supported by Air Force Contract AF 49 (638) -79. • See R. Bott, "An application of the Morse theory to the topology of Lie groups," Bull. Boc. Math. France, t. 84 (1956), pp. 251-281. 1

973 B. Kostant, Collected Papers, DOI 10.1007/b94535_11, © Bertram Kostant 2009

130

974

BERTRAM KOSTANT.

g is Lie algebra of the spedal linear group SL (n, 0) then the fact that the number of matrix units e'j such that j - i = 7c, where 1 < 7c < n -1, is greater by one then the number of matrix units eij such that j - i = 7c 1 accounts for the fact that the exponents of g are 1, 2,' . " n-1. After Shapiro informed us of this counting device for the exponents we observed that it could be reformulated as follows: The principal three-dimensional subalgebra a o of g is a uniquely defined (up to conjugacy) three dimensional simple subalgebra (TDS) of g which can be readily distinguished from other TDS by its properties. [It was discovered almost simultaneously by Dynkin and de Siebenthal (see [6] and [13]) and later used extensively by these authors (see e. g. [7])]. Using standard facts in the representation theory of a TDS it is not difficult to show that the observation of Shapiro is equivalent to the observation that if we decompose the adjoint representation of a o on g into a direct sum of irreducible representations then the number of irreducible components is l and the dimensions of these components are ri., i= 1, 2,' . " l, where the d, are given by (1.1.1). However, this reformulation of the procedure still does not supply a proof. A second empirical procedure for finding the exponents was discovered by H. M. Coxeter. He recognized that the exponents can be obtained from a particular transformation y in the Weyl group, which he had been studying, and which we take the liberty of calling a Coxeter-Killing transformation, in the following manner (see [5]) : Let h be the order of y. Coxeter observed that (1) h satisfies hl = 2r, where r is the number of positive roots, (2) m, < h for all i and (3) the eigenvalues of yare wm" i= 1,2,' . " l, where w = e21fi /l&. A proof of (2) and (3) would provide, among other things, a proof of duality in the exponents mi observed by Chevalley (see [3], p. 24) since nonreal eigenvalues of y necessarily occur in conjugate pairs. Requiring (1) hl = 2r as the only empirically observed fact such a proof was recently obtained by A. J. Coleman (see [4]). A proof that hl=2r will be given in this paper. A second question posed in [4] of showing that h = 1 0 (tf!), where tf! is the highest root, will also be settled here. It will be the main result of this paper to establish a direct relationship between the principal TDS, its adjoint representation of g, and the transformation of Coxeter-Killing. The proof that the procedure of Shapiro yields the exponents is then a direct consequence of this relationship. A major role here is played by a particular conjugate class in G, the elements of which we call principal elements of G. It is shown that an element of G which induces a transformation of Coxeter-Killing on a Carlan subalgebra is necessarily a principal element. On the other hand a principal element of G

+

+

131

COMPLEX SIMPLE LIE GROUP.

975

belongs to the subgroup corresponding to a principal TDS and is sufficiently specialized in that subgroup so that the adjoint representation of a principal TDS on g is determined by its eigenvalues. That the conjugate class of principal elements is truly a distinguished one in G may be judged from the following characterization (one of several) of principal elements. Let A be an arbitrary regular element of G. Let k be the order of A (possibly O()). Then k ~ hand k = h if and only A is principal. An important role is also played here by two distinguished classes of g, the elments of which we have, respectively, called principal nilpotent and cyclic (the latter name is derived from the transformation properties of such elements in the case when g is the Lie algebra of SL (n, 0) ). As is the case with the principal elements of G these elements can be given simple characterizations (Corollary 5. 3, Theorem 9. 2 and Corollary 9. 3). In this paper §§ 1-4 are devoted to the theory of the general TDS. The main theorems here are Theorems 3.6 and 4.2. Both concern conjugacy questions. The first is an extension of a well known theorem of JacobsonMorosov. A corollary of it puts the conjugate classes of TDS in g in a canonical one-one correspondence with the conjugate classes of nilpotent elements in g. The second is implicit in the proof of a weaker theorem of Malcev. In § 5 the theory of the principal TDS is taken up. For the most part the theorems here are devoted to characterizing principal TDS among all TDS and principal nilpotent elements among all nilpotent elements. The result here which is used most often in the remainder of the paper is Corollary 5.3. The main results of the paper are given in §§ 6-9.

2.

Preliminaries and the complex three dimensional simple Lie algebra.

1. Let g be a complex semi-simple Lie algebra. Let nand l be respectively the dimension and rank of g. As usual the linear transformation y~

[x,y]

is designated by ad x and x ~ ad x is the adjoint representation of g on itself. If u is a Lie subalgebra of g the mapping x ~ ad x for x E u will be called the adjoint representation of u on g. We distinguish two types of elements in g. An element x Egis called nilpotent if ad x is nilpotent and is called semi-simple if adx is completely reducible, that is, (since we are dealing with

12

132

976

BERTRAM KOSTANT.

complex numbers) if ad x is diagonizable. 3 Only the zero element is both nilpotent and semi-simple. We recall that any element x contained in a Cartan sub algebra 1) C g is necessarily semi-simple and conversely every semi-simple element may be embedded in a Carlan subalgebra. (See e.g. [8], p. 119). For any element x E g let g'" designate the kernel of ad x. We recall that x is called regular when x is semi-simple and g'" is a Cartan subalgebra. 2.2. Let G designate the adjoint group of g. Since this is the only group associated with g that we shall consider the operation of exponentiation (Exp) will be always understood to go from g to G. Elements x and y in g are called conjugate if there exists A E G such that Ax=y. 2.3. The simplest complex semi-simple Lie algebra (up to isomorphism) is 0 1 , the Lie algebra of all complex 2 X 2 matrices of trace zero. In this case n = 3 and l = 1. One knows that any three dimensional complex semisimple Lie algebra is isomorphic to 01' By conjugating any element of 01 into Jordan canonical form the following is apparent: (a)

every non-zero element in

01

is either semi-simple or nilpotent,

(b) the set of all non-zero nilpotent elements in jugate class,

01

form a single con-

(c) if x, Y E 01 are semi-simple, x ~ 0, then y is conjugate to a unique, up to sign, scalar multiple of x. (Note also that the set of non-zero semisimple elements coincides with the set of regular elements in od. 2.4. Let 0 be a 3-dimensional complex simple Lie algebra. We recall further facts in the structure theory of o. Let x E 0 be a regular element. Then the eigenvalues of ad x are a, 0 - a for some non-zero complex number a. We may modify x by scalar multiplication so that a and - a take the values 1 and -1. This defines x uniquely up to conjugacy. We thus isolate a particular conjugate class in o. We thus isolate a particular conjugate class in o. The elements in this class will be called mono-semisimple. Let x Eo be a mono-semisimple element. Let

B+

Eo be a non-zero eigen-

• It is not difficult to show that this very same definition [of semi-simple and nilpotent elements] may be achieved using any faithful representation of g in place of the adjoint representation. This follows, e. g., from !.emma 5. 4 and well known facts in the representation theory of g.

133

977

COMPLEX SIMPLE LIE GROUP.

vector of ad x for the eigenvalue 1; e+ is unique up to a non-zero scalar. Then e_, an eigenvector of ad x for the value -1, is uniquely determined by condition [e+, e_J = x. One thus has the commutation relations

(1)

[x,e+J=e+

(2)

[x, e_ J = -

(3)

[e+, e_J = x

(2.4.1)

e_

for the basis x, e+, e_ of o. 2. 5. matrices

When

0 =

01

the elements x, e+, and e_ may be realized by the

__~(O0 1)0 '

e+- 2

e=2

_5(01 00) .

For each positive integer d there exists up to equivalence one and only one linear irreducible representation of 0 having that dimension. (For a complete treatment of the representation theory of 0 1 see [12, Expose no. 10].) To describe an irreducible representation 71'rl of g on a d-dimensional vector 1. It is of course space V first define the number k by the relation d = 2k to be noted that k is an integer if and only if d is odd. We may then find a basis vi> j ~ k, k -1,· .. , - k of V, where the vectors are each unique up to a scalar factor, satisfying the following condition:

+

The behavior of 71'rl(e+) and 71'rl(e_) on the one dimensional spaces (Vi) is given by

71'a(e+) (Vj)

=

(Vj+1)

71'a(e_) (Vi)

=

(vi-d

where Vk+1 = V-k-1 = o. Thus Vj is an eigenvector for 71'a(e+)7I'a(e_) and 71'a(e_)7I'a(e+), where in fact

and

It follows that with respect to the ordering Vk, Vk-1,· . ., V-k of the basal elements, when the latter are suitably modified by scalar multiplication, one obtains for 71'a(x), 71'a(e+) and 71'a(e_) the matrices

134

978

BERTRAM KOSTANT.

k

0

k-1 'lra(X)

=

o o

-7c

o

(1(27c))i o (2(2k-1»)i

o

o

o

(2k(1»i

o o

o (1(2k))1 0 (2(2k-1»)1

0

°

o

(27c(1))1

Several facts are to be noted.

0

Among those we shall require are

(a) The dimension d of V is odd or even according as the eigenvalues of 'Ira (x) are all integers or all half-integers. 4 (b) The eigenvalues of 'lra(x) all occur with multiplicity 1 and the real number j is an eigenvalue of 'Ira (x) if and only if d - (2 I j I 1) is a non-negative even integer.

+

+

( c) The number k, where 2k 1 = d, may be characterized as the highest eigenvalue of 'Ira (x). Furthermore, the one-dimensional eigenspace for this eigenvalue k may be characterized as the kernel of 'lra(e+). Now assume 'Ir is an arbitrary, not necessarily irreducible, representation of a on the finite-dimensional vector space V. One knows 'Ir may be decomposed into a direct sum of irreducible representations. It follows then that one knows 'Ir up to equivalence as soon as the dimensions of the irreducible components of 'Ir are given. That is, if nk denotes the number of such components having dimension 2k 1, then 'Ir is given when the sequence nk, k = 0, i, 1,' . . is known.

+

• We use the word half-integer to designate all numbers of the form m m is an integer.

135

+ t. where

979

COMPLEX SIMPLE LIE GROUP.

( d) The problem of finding the sequence nk can be reduced to an investigation of the kernel W C V of ?T (e+). In fact, it follows from (c) that dim W = no

+ ni + n + .. '. 1

Furthermore W is stable under ?T(x) and if wE W is any eigenvector of ?T(x), w may be embedded in an irreducible component of?T. Hence if leI, le 2 , ' • " lep are the eigenvalues of ?T(x) on W, the dimensions of the irreducible com1, 2le 2 1,' . " 2lep 1. (N ote that the ponents of ?T are respectively 2le l lei are non-negative.)

+

(e)

+

+

The space V admits a canonical direct sum decomposition

V=VE+ VO, where VE is spanned by eigenvectors of 7r(x) belonging to half-integral eigenvalues and VO is spanned by eigenvectors of ?T(x) belonging to integral eigenvalues. It follows immediately from (a) that VE and VO are both stable subspaces for the representation ?T and that in the complete reduction of 'IT IVE only irreducible representations of even dimension appear and the complete reduction of ?TI VO yields only irreducible representations of odd dimension. 5 (f) Now let V; be the eigenspace of ?T(x) for the eigenvalue j. Clearly dim V; = dim V _i' Furthermore, if j is non-negative, it follows from (b) that dim Vj=n;+ nj+1 n;+2

+

+ .. '.

The statement (f) has the following 2 consequences: (g) The dimension of V o, that is the dimension of the kernel of 7r(x) (nullity of ?T(x», equals no n 1 n 2

+ + +.

(h)

If j is non-negative,

dim V; - dim Vj+l = nj. Finally, we shall require (i)

If le is the maximal eigenvalue of 'IT(x),

2k

is a direct sum and

~ p=1

"I U

V p / 2 lies in the range of ?T(e+).

• If 7r is a representation on a vector space V and U C V is stable under'll", then denotes the representation on U obtained by restricting'll" to U.

136

980

BERTRAM KOSTANT.

3.

Nilpotent elements and TDS.

1. Let 9 be a complex semi-simple Lie algebra. Consider the question of determining all three dimensional simple subalgebras (TDS) in g. If a egis a TDS, then by considering the adjoint representation of a on 9 it follows from the representation theory of a outlined in § 2. 5 that any nilpotent element of a is necessarily nilpotent in 9 and any semi-simple element of a is a semi-simple element of g (Also see footnote 3). Since a has only semi-simple and nilpotent elements (see § 2. 3 (a) ), the question arises (1) which nilpotent and semi-simple elements of 9 can be embedded in a TDS and (2) how does one find all such sub algebras. We shall first consider the case of nilpotent elements. A theorem of Morosov asserts that every nilpotent element of 9 can be embedded in a TDS. (See [11].) However, his proof was incomplete. Later in [9] Jacobson gave a correct proof of this result. Since the proof leads into Theorem 3.6, we shall give it here. The proof requires Lemma 3.3. With the exception of the proof of Lemma 3. 3 the proof is the same as the one given by Jacobson.

3. 2. A famous result of Jacobson asserts that if A and B are linear transformations on a finite dimensional space V with the condition that [A, B] commutes with A, then [A, B] is nilpotent. If in addition A is assumed nilpotent, the following lemma (which is no doubt known) asserts that AB is nilpotent. 3.2. Let A and B be linear transformations on a finite dimensional space V. Assume A is nilpotent and LEMMA

[A, [A,B]] =0. Then AB is nilpotent. Proof. Let Vk be the kernel of Ak. We wish to show AB leaves Vk invariant. The result is obvious if k = o. Assume the result is known to be true for k = r. Let x E Vr+l ABx = [A, B]x

Apply Ar+l to both sides.

+ BAx.

Then

Ar+l(AB)x = [A, B]Ar+lx =

0

+ Ar+1BAx

+ k"(AB)Ax.

But Ax E Vr and by our assumption (AB)Ax E Vr. Hence ABx E Vr+1.

137

Thus Ar(AB)Ax =

o.

COMPLEX SIMPLE LIE GROUP.

Assume AB is not nilpotent. vector x E V, x:;l= 0 such that

981

Then there exists a scalar A, A:;I= 0 and a ABx=AX.

(3.2.1)

Let k be the smallest integer such that x E Vk+l Now AkABx=Ak[A,B]x AkBAx.

+

But AkB=BAk+k[A,B]Ak-l.

Thus

AkBAx = BAk+lX

=

0

+ k[A,B]AkX

+ k[A,B]AkX.

Hence AkABx= (k

+ 1) [A,B]AkX.

But by (3.21)

+

Hence [A, B]AkX = (Alk 1)AkX. Since A kX :;1= 0, this contradicts Jacobson's lemma asserting the nilpotence Q.E.D. of [A,B].

3.3.

Now for any x, y E g let (x, y)

=

tr ad x ad y

be the Cartan-Killing bilinear form B on g. Using the non-singularity of B on g we can now prove the following lemma. Lemma 3.3 is crucial in the proof that any nilpotent element e E g can be embedded in a TDS of g. It is clear that if e is contained in a TDS then e must lie in the range of (ad e) 2 since according to § 2. 3 (b), e can play the role of e+ in the commutation relations (2.4. 1) . In particular, it is interesting enough to observe then that for any non-zero nilpotent element e, (ad e) 2:;1= O. (The degree of nilpotency of ad e is greater than 2.) LEMMA

3. 3.

Let e E g be a nilpotent element.

Then e is in the range

of (ad e )2. Proof. The invariance of B under the adjoint representation implies that, for any z E g, ad z is skew-symmetric with respect to B and hence (ad z) 2 is symmetric. In particular, this is true for z = e. But now if A is a symmetric operator (with respect to B) on g and if Hi and KA are, respectively~ the range and kernel of A, then

138

982

BERTRAM KOSTANT.

By the non-singularity of B, then, to show an element z lies in RA , it suffices to show (z, y) = 0 for all y E K A' Letting.A = (ad e) 2, to prove the lemma it suffices to show that

[e, [e, y]]

(3.3.1)

0

=

implies (e,y) =0. But under the adjoint representation (3.3.1) becomes [ade, [ade,ady]] =0. Since ad e is nilpotent, we apply Lemma 3.2 to assert that ad e ad y is nilpotent. But then by definition of B it is clear that (e, y) = O. Q. E. D. 3.4.

By Lemma 3.3, e may be written as

(3.4.1)

[[f,e],e]=e

for some f E g. Let x = [f, e] so that [x, e] = e. LEMMA 3.4. Let gO be the kernel of ad e. Then gO is invariant under ad x. Furthermore if m is the smallest integer such that (ad e) m+l = 0, then m

II

(adx-p/2)

p=O

vanishes on ge. Proof. The space (ad e )Pg is the range of (ad e )p. We define a sequence of subspaces bp of gO, p = 0, 1,' . " m 1, where

+

(3.4.2) by letting First observe that ge is invariant under ad x. Indeed, if y E ge, [e, [x,y]]

=

[[e,x],y]

=-

[e,y]

=0. We now show that (3.4.3) Let y E bp.

Then y = (ad e)pz for some z E g.

Now since [x, 6] =

[ad x, (ade)p] =p(ade)p.

139

6

clearly

983

COMPLEX SIMPLE LIE GROUP. Thus [ad x, (ad e)p]z = py, or

[x, y] - p. y = (ad e)p[x, z] (3.4.4)

=

(ad e)p[[f, e]z]

=

(ade)P+1[z,f]

+ (ade)Padf[e,z].

But [(ad e)p, ad f]

1'-1

=

-l: (ad e) i ad x(ad e)l'-l-(

'=0 = ip(p -1) (ad e)l'-l_ pad x(ad e )1'-1. Applying this to [e, z] we obtain (ad e)p ad f[e, z]

=

adf(ad e)p[ e, z] + ip (p -1) (ad e )pz - p[x, (ad e)pz]

[f, [e,y]] +tp(p-1)y-p[x,y] = ip (p -1) Y - P [x, y].

=

But then (3.3.4) becomes

(p

+ 1) [x,y] -lp(p + l)y= (ad e)P+1[z, f]

or

[x, y] -tpy E Dp +1 • This proves (3.4. 3) . It then follows immediately from (3.4. 2) that m

II (adx-p/2) p=0

vanishes on ge. Q. E. D. It is an immediate consequence of Lemma 3.4 that ad x is completely reducible on ge and that its eigenvalues are restricted to non-negative integers and half-integers. In particular, what is essential for us at this point is COROLLARY 3.4. on ge.

The linear transformation ad x

+1

is non-singular

We can now prove THEOREM 3.4 (Jacobson-Morosov). Every nilpotent element of a complex semi-simple Lie algebra can be embedded in a TDS.

Proof. Let e be nilpotent, e oF 0, and let f and x be defined as in § 3. 4. 6 • The theorem holds when e = 0 once we know the existence of a single TDS. The existence of a TDS follows from the proof since g contains non·zero nilpotent elements.

140

984

BERTRAM KOSTANT.

In case [X, f] === - f we would be done, that is, x, e, f would satisfy the desired commutation relations. The problem is to modify f so that this relation is satisfied without destroying the relation (3.4. 1) . Even if [x, f] f =1= 0 we still have

+

[[x,f] +f,e] =0

+

as one easily checks. Thus [x, f] f E gO. But now by Corollary 3.4, since ad x 1 is non-singular on gO, there exists a unique g E go such that

+

[x,f] +f= [x,g] +g.

Then writing e+ for e and letting e_ = f - g it follows that [x, e+]

(3.4.5)

= e+

[x, e_] =-e_ [e+, e_]

=

x.

This proves Theorem 3. 4 as soon as one notes that x, e+ and e_ must be linearly independent. 3. 5. Any set of non-zero elements x, e+ and e_ in g satisfying the commutation relations (3.4.5) will henceforth be called an S-triple (to be written {x, e+, e_} ). The element x will be called the neutral element of the S-triple and e+ (resp. e_) will be called the nil-positive (resp. nil-negative) element of the S-triple. It is obvious that the elements of an S-triple form a basis of a TDS. Two S-triples are called conjugate if there exists A E G which carries one set onto the other. Given a non-zero nilpotent element e E g we wish now to find all S-triples which contain e as the nil-positive element. (By § 2. 3 (b) this yields all TDS which contain e). First we note the following corollary of the proof of Theorem 3.4. COROLLARY 3. 5. Let e E g be nilpotent, e =1= 0, then x and e are respectively the neutral and nil-positive elements of an S-triple if and only if (1) x is in the range of ad e and (2) [x, e] = e. Furthermore if x and e satisfy these conditions such an S-triple system is unique (and hence x and e are contained in just one TDS).

Proof. The first part of Corollary 3. 5 follows from the proof of Theorem 3.4 and the definition of x used in the proof (see (3.4.1». To prove the uniqueness of the nil-negative element assume that {x, e, fd and {x, e, f2} are two S-triples. It is obvious then that [e,fl-f2] =0

141

985

COMPLEX SIMPLE LIE GROUP.

so that

h-

f2 is an eigenvector of ad x

+ 1 on

ga.

By this implies f1 = f2.

Q.E.D. 3. 6. As a consequence of Corollary 3. 5 the problem of finding all S· triple systems containing e as the nil-positive element reduces to finding all elements x E g which satisfy the conditions of Corollary 3.5. Towards this end define, for e E g, the subspace ga=ade(g)

n ga,

the intersection of the range and kernel of ad e. We now observe that g. is a Lie subalgebra of g. Indeed, ge is a Lie sub algebra of g. Therefore it sufficies only to show that [u,v] E ade(g) if u,vE g•. Writing V= [e,w] we have, since u E go, [ u, v]

=

[u, [e, w]]

=

[e, [u, w]].

This proves [u, v] Ega. N ow let G. be the subgroup of G corresponding to the subalgebra g.. Note that among other things the elements of Ga leave e fixed. We can now state THEOREM 3.6. Let e E g, e oF 0, be nilpotent. Let g. and G. be as above. Then the elements of g. are all nilpotent (and hence the elements in G. are unipotent). Let x E g be such that x and e are, respectively, the neutral and nil-positive elements of an S-triple. Then the linear coset x g. of g. is the set of all neutral elements taken from all S-triples containing e as nil-positive element. Furthermore, any two elements in x g. are conjugate. Moreover the conjugation can be performed by an element in G. so that e is fixed under the conjugation. In fact for any A E G., Ax Ex g. and the map

+

+

+

defined by making A, A E G., correspond to Ax is one-one and onto. In other words (recalling Corollary 3.5) if {x,e,f} is the S-triple containing x and e, the map A~ {Ax,e,Af} sets up a one-one correspondence of the group G. onto the set of all S-triples containing e as nil-positive element. Furthermore Ax ranges over x g •.

+

142

986

BERTRAM: KOSTANT.

Proof. Assume {X, e, f} and {y, e, g} are S-triples and e nil-positive in both cases while x and yare neutral. Since [x,e] = [y,e] =e,

it follows that y - x E gO. But clearly [e, g - f] = y - x. That is, y - x E ad e (g) • Hence y - x E g. or y E x go. Oonversely, if y E x g., then clearly [y, e] = e and y E ad e (g). Thus applying Oorollary 3. 5 Y is a neutral element of an S-triple containing e. N ow according to Lemma 3.4 gO admits the direct decomposition

+

+

where m is the smallest integer such that (ad e) m+l = 0 and Op/2 is the eigenspace of ad x in gO belonging to the eigenvalue p/2. It is of course clear that (3.6.1) Let a be the TDS spanned by x, e and f. Now if7rd is an irreducible representation of a on a vector space of dimension d it is clear from § 2. 5 (i) that the eigenspace belonging to the highest eigenvalue (i(d-1», of 7ra(x) lies in the range of trd (e) if and only if d > 2. Now decompose g into irreducible subspaces under the adjoint representation of a on g and apply this fact to the irreducible components. Recalling 2. 5 (d) it becomes clear then that the subspace go of gO can be written m

(3.6.2)

gO=~OP/2 p=1

and hence, in particular, ad x is non-singular on go. One immediate observation from (3.6.1) and (3.6.2) is that go is a nilpotent Lie sub algebra of g. Furthermore, since the eigenvalues of ad x on go are strictly positive it is clear from a relation similar to (3. 6. 1) that ad w is nilpotent for every wE go.7 That is, the elements of ad go may be simultaneously triangulized with zeros appearing along the diagonal for every element. It follows then from well known facts concerning linear nilpotent Lie algebras that in such a case G. is closed, simply connected-that is, Ge is homeomorphic to Euclidean space-and the exponential map Exp: 7

g.~

Go

That is, if gj is the eigenspace of ad (J) on g for the eigenvalue j then clearly

[gj, lid C g,+j.

143

987

COMPLEX SIMPLE LIE GROUP.

is one-one and onto. That is, every A E G. may be uniquely written A for a unique w E g.. But then

(3.6.3)

Ax=x+ [w,x] +-Hw[w,x]]

=

Exp w

+ ....

Since g. is stable under ad x and since g. is a Lie subalgebra, all terms starting from the second on the right side of (3.6.3) lie in g. so that AxE x+ g•. Now let v E ge. Assume that there exists a unique element wj E g. such that (1) and (2)

m

Exp Wj(x) -

(x

+ v) Ep=j+1 ~ Op/2

Now let Zj+1 be the component of Exp wj (x) if

Wj+1 = Wj

(x

+ v)

in 0(j+1) /2. Then

+ (2jj + 1)zJ+1

it is clear that [Wj+1,x] = [Wj,x]-zJ+l. On the other hand, for i>l it follows from (3.6.1) that the components of (ad wj+d iX and (ad Wj) iX in OS/2 are the same for all s < j 1. Thus

+

i+1

Wj+1 E ~ 08 / 2 8=1

and

m

Exp Wj+1(X) -

(x

+ v) E. '=i+2 ~ 0

8 /2

and furthermore that in satisfying these conditions Wj+1 IS unique. If we define W1 = - 2v., where v. is the component of v in 0., then W1 uniquely satisfies (1) and (2) when j = 1. Thus we have proved inductively that there exists a unique W E g. such that

Expw(x) =x+v.

Q.E.D.

Note. It is useful to observe that the proof, above, of the statement that Ax ranges over x ge when A ranges over Ge depends essentially on just two facts, (1), the nilpotence of ge and (2), the non-singularity of ad x on ge. We can now supplement Theorem 3.4 with

+

COROLLARY 3. 6. Let e E g be nilpotent, e =1= 0, and assume e E 0 1 n 02, where 0 1 and 02 are two TDS. Then 01 and 02 are conjugate to each other. Furthermore, the conjugation can be chosen so as to leave e fixed.

144

BERTRAM KOSTANT.

988

P1·00f. According to § 2. 3 (b) we may find two S-triples each containing e as nil-positive element and which, respectively, are bases for a i and a2· Corollary 3. 6 then follows immediately from Theorem 3. 6. Q. E. D.

3. 7. Corollary 3. 6 is actually just a special case of the following corollary. The set of all TDS in g breaks up into conjugate classes under the action of G. Concerning these classes we have COROLLARY 3.7. The conjugate classes of TDS in g are in a natural one-one correspondence with the conjugate classes of non-zero nilpotent elements in g. The correspondence is established by associating to the conjugate class of a, a TDS in g, the conjugate class of any non-zero nilpotent element in a. That is, two TDS ai and a2 are conjugate if and only if ei and e2 are conjugates, where ei E ai, e2 E a2 and ei , e2 are non-zero nilpotent elements. Proof.

Follows immediately from § 2. 3 (b), Theorem 3.4 and Corollary

3.6.

Q.E.D.

4.

Semi-simple elements and TDS.

1. We now consider the semi-simple elements of a TDS and take up questions of conjugacy. Let {x, e+, e_} be an S-triple with x and e+, respectively, as the neutral and nil-positive elements. We wish first to determine all S-triples which contain x (necessarily as neutral element). Let a be the TDS spanned by x, e+ and L By considering the adjoint representation of a on g it follows from § 2. 5 that the eigenvalues of ad x on g are integers and half-integers. In fact, recalling § 2. 5 (f) and § 2. 5 (i), if gp/2 is the eigenspace of ad x for the eigenvalue p/2 and if k is the maximal value of ad x, then dim gp/2 = dim g-1l/2 and 2k

g= ~

gp/2.

p=-2k

In this case, however, we have the addtiional relation

(4. 1. 1) Since we are concerned with S-triples containing x, interest focuses on gi since any nil-positive element in an S-triple containing x obviously belongs to gi. In particular, e+ E gi. The question arises which other elements of Qi

145

COMPLEX SIMPLE LIE GROUP.

989

belong to S-triples containing :1: and whether such S-triples are conjugate to {:1:, e+, e_}. To settle this question we first consider go = g"'. It follows immediately from (4. 1. 1) that g'" is a Lie subalgebra. Let G'" be the subgroup of the adjoint group corresponding to the subalgebra gIll. It also follows from (4. 1. 1) that each of the subspaces gp/2 is stable under the adjoint representation of g'" on g and hence these spaces must be stable under Ga;. We are particularly interested in the action of G'" on gl'

4. 2.

N ow for each element e E gl it follows from (4. 1. 1) that ade: go~ gl'

Let T. be the restriction of ad e to go. Our interest now centers on what will be shown to be an important subset of gl' Define gl= {eE gl

I T. maps go

onto gd.

That is, e E l'it if and only if the rank of T. equals the dimensions of gl' We now observe LEMMA 4. 2A. A necessary condition that an element e E gl be the nil-positive element of an S-triple containing :1: is that e E gl' In particular,

e+E

(it.

Proof. Indeed, assume e and :1: belong to an S-triple. Let a' be the TDS which contains e and:1:. If we apply § 2. 5(i) to the adjoint representation of a on g it follows that gl is the range of ad e. On the other hand, it is clear from (4.1.1) that ad e (g) n gl = ad e (go) n gl' Hence T. must map go onto gl' Q. E. D. The following topological properties of III are needed for the proof of Theorem 4.2 (Here one is inspired by the use of regular elements in the usual proof of the conjugacy of any two Cartan subalgebras.). LEMMA

4. 2B.

The set gl is an open, dense and connected subset of gl'

Proof· It follows from Lemma 4. 2A that gl is not empty (e+ E gl)' Choose a basis of go and a basis of gl' For any e E gl set T.o equal to the dim go X dim gl matrix determined by T. and the given pair of bases. It is clear then that gl is the set of all e E gl such that at least one dim gl X dim gl minor of T.o is not zero. But it is an easily verified general fact that if Fh j = 1, 2,' . " m, are m non-zero polynomials on a complex vector space V then the complement -V to the set of common zeros in V of all the F J is open, dense and connected. Indeed, if u E V and v E V, then there are only a finite

146

BERTRAM KOSTANT.

990

number of complex scalars>.. such that F/(>..) =FJ(>..u+ (1->")v) vanishes for at least one j. Q. E. D. Now observe that gl is invariant under ()m. The major point in the proof of Theorem 3.4 is the observation contained in the following lemma. LEMMA 4.20. Let e E ih; then the orbit GllJe of e under the action of GIJJ is an open subset of gl (and hence of ?h).

Proof. The mapping A ~ Ae of GIJJ into gl is analytic. Thus it suffices to show that the differential of this mapping carries the tangent space to GIlJ at 1 onto the tangent space of gl at e. But the image of the former, under the differential, when translated to the origin of gl is just the subspace ad glJJ (e) of gl. But glJJ = go and since e E {h, this space coincides with gl, by definition of gl. Thus GlJJe is open in gt. Q. E. D. We have shown ()me is open in gl for any e E gl. But for el, e2 E gl, GlDel and GIJJ e2 are the same sets or else they are disjoint. But this fact taken together with Lemma 4.2B (the latter asserting the connectivity of fit) implies that there can be at most one orbit. That is, gl is itself a single orbit of GIJJ. But then recalling Lemma 4. 2A and observing that x is fixed under the action of GflI. we see that the following theorem has been proved. THEOREM 4.2. Let x E g be the neutral element of an S-triple {x, e+, e_}. (See § 3. 5.) As in § 2.1 let glJJ be the centralizer of x in g and let GIJJ be the subgroup of G corresponding to gllJ.

Define gl = {e E g I [x, e]

=

e}.

Then (4.2.1) for any e E gt. Let e E g. Then e and x are, respectively, the nil-positive and neutral elements of an S-triple if and only if (1) e E gl and (2) the map (4.2.1) is onto. Moreover, any two S-triples which contain x are conjugate to each other and the conjugation can be performed by an element in GID. In other words, if gl = {e E gl I (4. 2. 1) is an onto map}

then ?h is the conjugate class of e+ under the action of GID. That is

147

COMPLEX SIMPLE LIE GROUP.

991

Furthermore, ?h coincides with the set of all nil-positive elements taken from all S-triples having x as neutral element.

The statement of Theorem 4.2 gives a complete and simple description of the set of all nil-positive elements which "go" with a given neutral element. However, it is implicitly contained in Malcev's proof of the following corollary. Furthermore our proof of Theorem 4.2 although found without knowledge of Malcev's proof of Corollary 4. 2 (yet with the knowledge that it had been proved) amounts to only a technical simplification of Malcev's proof. Corollary 4. 2 provides the basis by which all the TDS in any complex simple Lie algebra have been classified (see [7], § 8). COROLLARY 4.2 (Malcev). Two TDS in 9 are conjugate if and only if any mono-semisimple element of one is conjugate io any mono-semisimple element of the other (see [10]). 4. 3. Now, continuing with the notation of § 4. 1 and § 4. 2, the subset ii1 in gl is of course only a part of the conjugate class of e+ in g. It does not seem likely that one can give a simple description of the entire class. Nevertheless, it is easy to describe a part of this class which is yet larger than gl (see Theorem 4.3, Theorem 4.3 is required for the proof of Theorem 5. 3) and we shall do this now. For any t = 0, 1,· .. , 2k let 2k

nt/2 =

~ gp/2·

p=t

It is clear from (4.1. 1) that nt/2 is a Lie sub algebra of g. Let Nt/2 be the subgroup of G corresponding to nt/2. We now have, in terms if the preceding notation,

THEOREM 4.3. The orbits Nox and Noe+ of e+ and x under the action of the group No are as follows: Nox=x+n. and

NOe+=gl+~.

In fact, G" is a subgroup of No and Nt is a normal subgroup of No. Moreover, the elements of N. are unipotent linear transformations of g. Furthermore, No can be written as a semi-direct product

13

148

992

BERTRAM KOSTANT.

(every element C E No is uniquely written C = AA', where A E Nt, A' E Gil) and the correspondence A~Ax

sets up a one-one mapping of Nt onto x A defines a mapping of No onto e+

+ nt while

the correspondence

~Ae+

+ tq.

+

Proof. Since no = g'" nt is a semi-direct sum (that is, [g"', nt] en.) it follows that every element C E No can be written, C = AA', where A E Nt and A' E Gill. On the other hand, if A E N. and y E gt/2, then Ay = y w,

+

21
E A be representative root vectors so that we have usual direct sum decomposition g = f)

+~

(e.p).

E A, then cf> may be identified with an element in f) by the relation

[x, e.p] for all x E f).

=

(x, cf> ) e.p.

One knows then if e.p and e_.p are normalized only in so far as

(5.1.0) as we shall assume, then (5.1.1) N ow let f)# be the real linear space in f) spanned by the roots. knows that

One

is a real direct sum. We recall that B is positive definite on f)# (see e.g. [12], p. 10-04). In particular then, (x, cf» is real for all x E f)# at all cf> E A. This means f)# can be characterized as the set of all elements x E f) such that ad x has real eigenvalues. Hereafter, if Y is any subset of f), we will let Y# = Y n f)#. Now for any cf> E A let

f)[cp] = {xE f) I (cf>,x) =O}. Then, clearly, if R is the set of all regular elements in f),

R# = f)- U f)[cf>]. 0 for

all cp E A+}.

994

BERTRAM KOSTANT.

The closure D of DO is called a Weyl chamber. It can be defined in the same way as DO except that one replaces the strict inequality > by the inequality > . The set D is a fundamental domain for the action of G on the set of all semi-simple elements y in g such that ad y has real eigenvalues. That is, if y is such an element there exists one and only one ([12], p. 16-08 and [7], Lemma 8. 2) element x in D which is conjugate to y. Now let IT = {1X1' 1X2, ' • " IX!} be the set of simple positive roots. The simple positive roots form a basis of f) and for every root q" upon writing

one knows that the coefficients 14 are integers which are all non-negative or all non-positive according as q,E A.+ or q,E A.-. We define the order o(q,) of q, by letting (5.1.2) Obviously, if q,l, q,2 and q,l

+ q,2 E A., then

(5.1. 3) Let

£i,

i

=

1, 2, .

. ,l, be the dual basis in h to the

IXj.

Since

(5.1.4) it is clear that

£i

E D.

More generally, if

(5.1. 5) then xED if and only if itt >0, i

=

1, 2,' . " l.

Indeed,

(5.1.6) Now assume xED is the neutral element of an S-triple {x, e+, e_}. Since the eigenvalues of ad x are integral multiples of i, it follows from that in the expansion (5. 1. 5) ai = imi for some non-negative integer mi' On the other hand, it is not possible for ai to be greater than 1. Indeed, since adx(e_) =-e_, it is clear that

where we emphasize the summation is over negative roots. Hence it follows from § 2. 5 (d) (with signs reversed. Also see Lemma 3.4) that if ai> 1, ad e_ (ea,) is a non-zero eigenvector of ad x with the positive eigenvalue itt -1.

151

995

COMPLEX SIMPLE LIE GROUP.

But then since xED, this implies [e_, eaJ is a sum of root vectors for positive roots. This, however, clearly contradicts the simplicity of ~i' We have proved then that ai = 0, i, or 1 for i = 1, 2,' . " 1. More than this we have (see [7], Theorem 8.3). LEMMA 5.1. Let the Weyl chamber D and the basis Ei, i = 1, 2,' . " 1 of q be given as above. Assume that xED is a mono-semisimple element of a TDS. Then

where for each i = 1, 2,' . " l, we have ai = 0, i, or 1. Furthermore, if X2,' . " Xb E D is the set (ordered) of all elements in D which happen to be mono-semisimple elements in at least one TDS (so that b < 3 1 ), then there are exactly b conjugate classes of TDS in g. In fact, if Xi is a monosemisimple element of the TDS a.; for i = 1, 2,' . " b, then the subalgebras a., i = 1, 2,' . ., b, are representatives of the conjugate classes.

Xl)

For each simple Lie algebra g Dynkin lists [7] which among the 3 1 - 1 choices are indeed mono-semisimple elements of a TDS. 5.2. We retain the notation of Lemma 5.1. Among the elements X;, j = 1, 2,' .. , b there is a distinguished one. This is the case when ai = 1, i = 1, 2,' .. , l. That is, for one of the X; all the a. (in (5. 15» take the maximal possible value. 5. 2. Let D and Xo E D be given by LEMMA

Ei,

i

=

1,2,' . ., l, be as defined in 5. 1.

Let

1

(5.2.1)

XO=~£i' i=l

Then Xo is a mono-semisimple element of a TDS Proof·

Since the elements of

~i

form a basis of

q#,

Xo may be uniquely

written I

Xo= ~ri~i' 1=1

N ow let Define (5.2.2)

C'h

i

=

1, 2,· . ·,1, be any 1 non-zero complex numbers. I

eo= ~ciea, .=1

and (5.2.3)

1

fo=~ (r./cj)e-a,. i=l

152

BERTRAM KOSTANT.

996

Clearly (ai, x o) = 1 for all i.

Thus

[xo, eoJ

=

eo

and On the other hand, since a. - aj is never a root, it follows from (5. 1. 1) that

[eo, foJ

=

Xo

Q.E.D. so that {xo, eo, fo} form as-triple. Let X o, eo and fo be as defined in the proof of Lemma 5.2. Let ao be the TDS spanned by the vectors. The TDS a o or any conjugate TDS is called (originally in [6]) a principal TDS of g. It and some of its properties were discovered by Dynkin and de Siebenthal (See [6J and [13]). We shall call the S-triple (or any of its conjugates) {xo, eo,fo} a principal S-triple. The matrices exhibited in § 2. 5 are an example of a principal S-triple in the Lie algebra of SL(d, C). One of the first properties we observe about the neutral element of a principal S-triple is that it is regular. In fact, Xo E f)# and for any cp E a clearly (5.2.4)

(see (5.1. 2) ). We shall call Xo or any element of g conjugate to Xo a principal regular element of g. Any element of the conjugate class of eo will be called a principal nilpotent element of g. One of the most significant ways in which a principal TDS is distinguished among all TDS is in regard to its adjoint representation on g. For any TDS a of g let n(a), nE(a) and nO(a) designate, respectively, the number of irreducible components occurring in the complete reduction of the adjoint representation of a on g, the number having even dimension and the number having odd dimension. THEOREM 5.2. Let a be any TDS of g.

a is principal if and only if n(a)

=

Then n(a) > l.

Furthermore

l.

Proof. Let {x, e, f} be an S-triple whose linear span is a. In fact, employing the notation of § 5. 1 we may assume xED C f). Since f) is contained in g'" (see §2.1), obviously dimg"'>l. But now according to 2.5(g) we have nO(a) = dim g"'. Thus

(5.2.5)

153

COMPLEX SIMPLE LIE GROUP.

997

which proves the first part of Theorem 5.2. N ow if a is principal x is regular and hence dim g'" = l. On the other hand, in this case all the eigenvalues of adx are integers (see (5.1.2)) so that nE(a) =0 (see §2.5(e)). Thus if a is principal, n(a) = nO(a) = l. Now, conversely, assume n(a) = l. But I

then by (5.2. 5) dim g'" = 1 which implies x is regular. Writing x =

~

ait:i as

(=1

in §5.1, it follows then that ai=i or 1 for all i. But since nO(a) =n(a), that is, n E (a) = 0, this means (see § 2. 5 (e)) the eigenvalues of ad x are all integral. Thus ai =1= t by (5.1. 6) and hence x

I

=

~ t:i

so that a is principal

.=1

by Corollary 4. 2. As a corollary of the proof we have

Q.E.D.

COROLLARY 5.2. Let a be any TDS of g; then nO(a) > l. Furthermore, n (a) ° = 1 if and only if the mono-semisimple elements of a are regular in g. Incase a is principal n E (a) = O. 5.3. The following corollary of Theorem 5.2 distinguishes the principal nilpotent elements in the set of all nilpotent elements in g. The proof follows immediately from § 2.5 (d) and Theorem 5.2. COROLLARY 5.3. Let e be a nilpotent element tn g; then dim ge > 1 and dim ge = 1 if and only if e is principal nilpotent. Now let {xo, eo,fo} be the principal a-triple defined as III § 5. 2. We will now apply the theory of § 4. 1 to this a-triple. First of all, since the eigenvalues of ad Xo are integral, we observe that gp/2 = 0 whenever p/2 is not an integer and for j a non-zero integer gj= ~ o(¢)=;

(5.3.1)

(e.p)

and We will write n for nt

=

n1 and

{l

for no.

n=

~

(e.p)

Then

¢Ell.+

(as is well known) is a maximal Lie subalgebra of nilpotent elements and f) n is a maximal solvable Lie sub algebra of g. Let H, N and be the subgroups of G corresponding to f), nand {l. Recalling that

{l =

+

a

eo

=

z ~ Cie""

0=1

154

998

BERTRAM KOSTANT.

where c. =F 0, i = 1, 2, . this case

. ,l it follows from Theorems 4. 2 and 4. 3 that in I

H eo = gl = {e E gl I e = }: aa,ea" where aa, =F 0, i = 1, 2,· .. , l}

.=1

and (since 11i = n2) (5.3.2)

Se O=gl+n2= {eE n I e=}: a4>e~, where a~=FO when ~€d+

cp = ai, i=1,2,· . ·,1}. We emphasize that (5.3.2) above implies that" almost all" the elements in the maximal subalgebra of nilpotent elements n are principal nilpotent (and hence lie in a single conjugate class). We wish to prove now that Se o contains every principal nilpotent in n, that is, we have the following simple characterization of those elements in n which are principal nilpotent. THEOREM 5. 3. elements given by

Let n C g be the maximal Lie stlbalgebra of nilpotent

Let e En,

be arbitrary. Then e is principal nilpotent if and only if i = 1, 2,· . ·,1.

a~

=F

°for cp

=

ai,

Proof. By (5. 3. 2) if e satisfies the condition stated in Theorem 5.3, then e is principal nilpotent. Conversely, let e E n be principal nilpotent and .assume aaJ = 0, for some j. Assume first that g is simple. Let 1/1 E .6. be the highest root. We recall I

two basic facts about the highest root.

One, upon writing 1/1 = }: qia, the

.=1

integers qi satisfy (5.3.3) and {5.3.4)

0(1/1)

> o(cp)

for any cp E .6., cp =F 1/1. Indeed, both of these facts are immediate consequences I

of the following single fact: If cp =

}: ti(Ji i=1

(5.3.5)

155

is any root, then

999

COMPLEX SIMPLE LIE GROUP.

for i = 1, 2, ... ,l. Finally, (5.3. 5) is a consequence of well known facts in representation theory; in fact, the adjoint representation of g is irreducible and by definition of 1/1, e", is the weight vector belonging to the highest weight, 1/1. One knows then that e.p, for any cp E ~, is obtained by applying polynomials (non-commutative) in the operators ad e-a" i = 1, 2,' . " l, to e",. This proves (5. 3. 5). Let I

q= ~ q.= 0(1/1). (=1

Now let Qo be as in § 5. 2 and consider the adjoint representation of Qo on g. It follows immediately from § 2. 5 (d) and (5. 3. 4) that (1) e", lies in an irreducible component 0 C g, (2) dim 0 = 2q + 1, (3) e_", E 0 and (4) any other irreducible has dimension less than dim 0. A direct consequence of these facts is that and where a =1= O. N ow since e is conjugate to eo it follows that (ad e) 2q =1= o. hand, since e E n, it follows that for any cp E ~ we can write

On the other

(5.3.6) and, moreover,

b~ =1= 0

implies

oa) > o(cp)

+ 2q.

But this together with (5.3.4) implies that (5.3.6) vanishes for all Thus (ad e) 2q ( e-tJt) =1= 0 and in fact, using (5. 3. 4) once more,

cp =1= - 1/1.

( ad e) 2qe-tJt = a' e", for some non-zero scalar a'. Now write e = e1 + e2, where e1 E gl and e2 E n2' Expanding (ade)2q= (ade 1 +ade 2)2q it is clear again from (5.3.4) that But for any i, writing ( ad e1) • ( e-tJt)

=

~ b'~e~,

C€a

it follows that since a,,} = 0, one can have b'~ =1= 0 only when upon writing

156

1000

BERTRAM KOSTANT.

we have tj = - qj. In particular, setting i = 2q this means qj = - qj or qj = O. This contradicts (5.3.3) and hence Theorem 5.3 is proved when g is simple. The general case follows immediately upon writing g as a direct sum of its simple ideals. The component of e in each ideal is necessarily principal nilpotent in that ideal by Corollary 5.3. Q. E. D. 5.4. A theorem proved in [1] (See [1], Remark p. 66.) asserts that every solvable sub algebra {l1 of g is conjugate to a subalgebra {l'l of {l. If in addition it is assumed that the elements of {ll are all nilpotent (hence {ll is a nilpotent Lie algebra by Engel's theorem), it is clear then that {l'l C n. In particuar, it follows then that every nilpotent element e Egis conjugate to some element e' E n. Actually, we don't need the result of [1] referred to above to prove this. For completeness, we observe that this follows from Lemma 5.1. LEMMA

5.4.

Any nilpotent element e Egis conjugate to an element

e' En. Proof. Applying Theorem 3.4 it suffices only to show that the nilpositive element of any S-triple containing Xj (using the notation of Lemma 5.1) as neutral element lies in n. But this is clear since Xj ED. That is, for cf>E fl., (xj,cf» =1 implies cf>E fl.+. As a corollary to Theorem 5. 3, its proof, and Lemma 5. 4 we obtain another characterization of principal nilpotent elements in case g is simple. COROLLARY 5.4. Assume g is simple. Let 1{1 be the highest 1'00t and let q = 0 (1{1 ) • Let e E g be any nilpotent element. Then e is principal nilpotent if and only if (ad e) 2q =1= O. However, if e is principal nilpotent (ad e)q+l = O.

5.5. Corollary 3.7 sets up a natural one-one relation between the conjugate classes of nilpotent elements and the conjugate classes of TDS. It is clear that in this correspondence the class of principal nilpotent elements corresponds to the class of principal TDS. Regarding the latter as distinctive among all conjugate classes of TDS will be given further justification then when it is shown that the former is distinctive among conjugate classes of nilpotent elements. The following corollary shows this very clearly. COROLLARY 5. 5. The set of principal nilpotent elements (a conjugate class of the adjoint group G) in g forms an open, dense and connected subset of the set of all nilpotent elements in g.

Proof.

Openness follows easily from Corollary 5. a by choosing a basis

157

COMPLEX SIMPLE LIE GROUP.

1001

of g and considering the (n - l ) X (n - l ) minors of the matrix defined by ad e, e nilpotent, with respect to the basis. Denseness follows from Theorem 5.3 and Lemma 5.4. Connectivity is immediate also since the set of principal nilpotent elements is an orbit of the group G. 5.6. Principal nilpotent elements behave like the "regular" elements in the set of all nilpotent elements. That is, one can make a strong case for the following analogy: The set of principal nilpotent elements is to the set of all nilpotent elements as the set of all regular elements is to the set of all semi-simple elements. We cite for example Corollary 5.3 and Corollary 5.5. In this analogy, between the semi-simple elements and the nilpotent elements, the Cartan subalgebra clearly corresponds to the maximum Lie subalgebra of nilpotent elements (see Lemma 5. 4 § 2. 1. Also all maximum Lie subalgebras of nilpotent elements are conjugate-see § 5.4). One knows that a regular element can be characterized by the property that it lies in one and only one Cartan subalgebra. Corollary 5. 6 asserts that also in this regard the analogy still holds. COROLLARY 5. 6. Let e E g be nilpotent. (One knows that e can be embedded in at least one maximal Lie subalgebra of nilpotent elements.) Then e is principal nilpotent if and only if e lies in one and only one maximal Lie subalgebra of nilpotent elements of g.

Proof. By Lemma 5. 4 we may assume e E n (using the notation § 5.4) . Assume that e is not principal. We will prove e is contained in a second (different) maximal Lie sub algebra of nilpotent elements n/. N ow we may write

By Theorem 5. 3 since e is not principal, aa, now it is known (and easy to verify) that

=

0 for some value of i.

But

is a maximal Lie subalgebra of nilpotent elements of g. (In fact, n is carried onto n' by any element of G which (1) leaves ~ invariant and (2) whose restriction to ~ is the reflection Ra,-see § 7.1). Obviously e E n/. N ow assume e is principal. In fact, we may take e = eo, where we use the notation of § 5. 2. Assume eo E n/, where n' = An, A E G. We shall use

158

1002

BERTRAM KOSTANT.

the prime (') on previous notation to indicate the effect of conjugation by A. Applying Theorem 5.3 to n' it follows that

eo =

~

a.pe.p',

= rt.;, i = 1,2,· .. , I. A1 E N' such that

N ow by Theorem 4.3 there exists

I

A1eo = ~

aai ea /.

i=l

But then since xo' (= Ax ) and A 1 eO are clearly the neutral and nil-postive elements of an S-triple (see proof of Lemma 5.2), it follows from Theorem 3.6 that However, since x o' is regular upon applying § 2.5 (d) and (5.1. 2) in the case of this S-triple, it follows that gA l 6 0 en'. But then by Theorem 4.3 there exists A2 EN' such that A2AIXO = x'o. But now since Xo is regular, we must have A 2 A 1 l) = l)'. But this of course implies l) C s' because A 1 -IA 2-1 E S'. But this means s' and hence n' = [s', s'] are stable under ad l). But then n' must contain and in fact must be spanned by root vectors associated with l) (since obviously n'

n l) =

I

0) . But eo =

~.

c;e a , E n' and

Ci

¥= O. Thus eat En',

i=l

i = 1, 2,· .. , I. Hence since the eal generate n, it follows that n en'. But then of course n = n'. Q. E. D. 5. 7. We continue with previous notation. Now consider goo, the kernel of ad eo. By Corollary 5. 3 dim g60= I.

+

+

In the special case when g is the set of all (l 1) X (I 1) complex matrices of trace zero one sees easily that g60 is a commutative Lie algebra (of nilpotent matrices). This and other evidence suggested to us that perhaps g60 is commutative in the general case. However, since among other things we were unable to construct a "useable" basis of g60 we could not settle the question using purely algebraic methods. Nevertheless, it is true that g60 is commutative (Corollary 5.8) in the general case. The proof which we have found is very simple but uses limit arguments. It was Corollary 5.8 which orignally suggested the validity of Theorem 6.7. Theorem 5.7 or Corollary 5.7 may be regarded as a generalization of the fact that a Cartan sub algebra of a semi-simple Lie algebra is commutative.

159

COMPLEX SIMPLE LIE GROUP.

1003

THEOREM 5. 7. Let 9 be a complex semi-simple Lie algebra of rank l. Let y E 9 be arbitrary. Then g1l contains an 1 dimensional commutative Lie subalgebra.

Proof. It is well known that the set of regular elements in 9 is dense III g. Thus we may find a sequence y'n, n = 1, 2,' .. of regular elements converging to y. Now consider the Grassmann manifold of alll planes in g. This, of course, is compact and hence we may find a subsequence Yn of the sequence y'" with the property Yn ~ y and the Cartan subalgebras g1l.. converge to an l-plane u in the Grassmann manifold. Now if Wi, i = 1, 2,' . " l, is any basis of u we may find elements Wi" E g, n = 1, 2,' . " i = 1, 2,' . " l, such that Win E g1lft and Win ~ Wi as n ~ 00 for i = 1, 2,' . ',1. Since [y", Wi"] = 0, it follows immediately by taking the limit that u C g1l. But [w.;n, wt] = O. Again taking the limit this obviously implies u is commutative. Q. E. D. COROLLARY 5. 7. tative. 5. 8.

Let y E g.

Assume dim g1l = 1.

Then g1l is commu-

Corollary 5. 7 and Corollary 5.3 imply

COROLLARY 5. 8.

Let e be a principal nilpotent element in g.

Then ge

is commutative. 6.

The principal element of G and the duality theorem.

1. We shall assume from now on that 9 is simple. The theorems to be proved are either true in the general semi-simple case or can be obviously modified to be true in that case. The extension from the simple case to the semi-simple case in any event is immediate. We consider only the former mainly for notational simplicity. Recall what is meant by a compact form f of g. This may be defined as a real Lie sub algebra of 9 with the property

g=f+if

(1) is a real direct sum. (2)

(That is, f is just a "real form" of 9 and

The restriction of B to f is negative definite. One immediate consequence of (2) is

(a) every element of f is semi-simple. Next we recall some facts in the Cartan subalgebra theory of f (which is somewhat different from that of g).

160

1004

BERTRAM KOSTAN'f.

+

(b) If t is a Cartan sub algebra of f, then f.lt = t it is a Cartan subalgebra of g and f.lt # = it. (c) A Lie sub algebra t of f is a Cartan subalgebra if and only if it is maximal commutative. An immediate consequence of (c) which we shall soon use is (d) Any commutative Lie subalgebra u of f can be imbedded in a Cartan sub algebra of f. Corresponding to the adjoint operation (conjugate transpose) in the space of matrices we introduce a *-operation in g by defining for any z E g

z*=x-iy, where z is written for x, y E if. Generalizing the notion of a normal matrix or normal operator we will say z Egis normal with respect to f whenever

[z,z*] =0

+

Writing z = x iy, x, y E if it is clear that z is normal if and only if [x,y] =0. It is well known of course that a normal matrix is diagonizable (semisimple). The following generalization of this fact (and also of (a)) is a useful criterion for semi-simplicity in g.

Let z, an element of g, be normal with respect to a compact form f of g. Then z is a semi-simple element of g. LEMMA

6.1.

+

Proof. We may write z = x iy, where ix, iy E f. But now since z is normal [ix, iy] = o. Thus by (d) there exists a Cartan subalgebra t of f which contains ix and iy. But by (b) 91 = t it is a Cartan sub algebra of g. But then z E f.lt and hence z is semi-simple. Q. E. D. When the root vectors, relative to a Cartan sub algebra, are suitably normalized (Weyl's normal form) Weyl has given a basis of a compact form of g in terms of these root vectors and the Cartan subalgebra. In previous sections we required no other normalization of the eq>, cp E~, other than (5. 1. 0). We will assume from here on, unless specified otherwise (in Theorem 8.4), that the root vectors are normalized into Weyl's normal form, with the exception that (eq>, e_ ) = -1. (In such a normal form one may choose the root vectors ea"

+

161

COMPLEX SIMPLE LIE GROUP.

1005

i = 1, 2,' . " l, arbitrarily and hence we need not regard

eo as having been altered). Then one knows that the linear span, with real coefficients, of iU# and the vectors et/> - e_t/>, iet/> ie_t/> for all cp Ea+ is a compact form f of g. (See remark, p. 11-11 in [12J). One sees easily then that (et/» * = e_t/> or

+

(6.1.1) for any set of complex numbers at/>, cp E a. 6. 2. We recall now (see § 6. 1) that g is simple. Let if! E a be the 1 roots obtained by adjoining highest root.s Let n Q C a be the subset of l - tf; to the simple positive roots. That is, n Q = n u (- if!). The notion of simple root and highest root are notions which of course are relative to the choice of a lexicographical ordering in U# or rather to the choice of a Weyl chamber (see § 5.1). We will say then that the roots in n Q are Q-simple relative to the chamber D. (We shall not require the fact but it can be shown that the number of Weyl chambers which give rise to the same set of Q-simple roots is equal to order of the fundamental group of G; that is, to the order of the center of the simply connected covering group of G.) Now an element z E g will be called cyclic if there exists a Cartan subalgebra Ul and a set n 1 Q Cal of Q-simple roots relative to some Weyl chamber Dl in Ul # such that z can be written

+

(6.2.1) where the at, ~ E n 1 Q, are non-zero complex numbers and the e~ are root vectors for fh corresponding to the roots ~ E a 1 • 9 Cyclic elements playa major role in the remainder of this paper. If z is given by (6. 2. 1), observe that in effect we have formed the cyclic element z by adding a-'/Jle-'/Jl to a principal nilpotent element, where tf;l is the highest root in a 1 relative to D 1 • It shall be shown that not only does this destroy nil potency but the cyclic elements are in fact regular. First we shall need LEMMA

6.2.

Let the cyclic elements z, z' E g be given by

L

Z=

afJefJ

fJ(ITO

and z' =

~ a'fJefJ,

fJ (ITO

8 The properties of t/I which will be required are all consequences of (5.3.5). • The set III is the set of roots associated with the Cartan subalgebra lit.

162

1006

BERTRAM KOSTANT.

where the coefficients ap, a' p are all arbitrary non-zero complex numbers. Then if H is the subgroup of G corresponding to ~ there exists an element A E H and a non-zero scalar A such that AZ=Az'. Proof. Let

£.;,

i

=

1, 2,· . ., l, be as in § 5. 1. Let y E ~ be given by

,

y = ~ Log (a'a.!aa,)£.;. i=1

Clearly Exp y E Hand ,

z

Expy( ~aa,ea,) =~a'a,ea,. ":::::1

-£=1

Now let c be any complex number and let Xo be given by (5. 2. 1). Then clearly ,

Exp(y

(6.2.1) Now let q =

+ cXo) ( ~ aa,ea,) = i=l

0 ("')

z

eC (

~ a' a,ea.). -£=1

as in § 5. 3. But then Exp cXo ( e_f/1)

=

e-cqe-,/!

and if b is defined by Exp y ( e-'/!) = be-,/!, then b =1= 0 and (6.2.2) N ow choose c so that Then (6.2.3) Hence if A = eO, A = Exp(y and (6. 2. 3) imply

+ cXo), one has A E Hand (6.2.1), (6.2.2), Q.E.D.

An immediate consequence of Lemma 6. 2 is the following theorem which asserts that up to scalar multiplication the set of cyclic elements forms a single conjugate class in g. 6. 2. Let z and z' be any two cyclic elements in g. Then there exists a scalar A, A =1= 0, such that z and Az' are conjugate to each other. THEOREM

Proof. If D l is a Weyl chamber in ~l #, where ~l is a Cartan subalgebra, there exists A E G such that AIh = Ih and ADl = D. This fact together with Lemma 6.2 proves Theorem 6.3.

Q. E. D.

163

1007

COMPLEX SIMPLE LIE GROUP.

6.3.

Lemma 6.1 is needed solely to prove the following lemma.

LEMMA

6.3. Cyclic elements are semi-simple.

Proof. N ow if we write z

if; = ~ qi~i,

(6.3.1)

;'=1

recall that the coefficients qi are positive integers. is defined by

(See (5.3. 3) ). Thus if e1

z

~

e1=

(q.)l;ea,

.=1

and Zl is defined by then e1 and Zl are, respectively, principal nilpotent and cyclic elements. To prove Lemma 6.3 it suffices by Theorem 6.2 to prove only that Zl is semi-simple. But to prove that Zl is semi-simple, by Lemma 6.1 it suffices only to prove that Zl is normal with respect to f. N ow by (6. 1. 1) and z

~

e1* =

(q;,)l;e--«,.

;'=1

But, obviously then, since if; is the highest root

[e 1,e.p]

=

[e1*,e_.p] =0.

Thus But z

[e1, e1*]

=

~

z

q.[eap e_a,]

=

;'=1

by (5. 1. 1) and (6. 3. 1).

;'=1

On the other hand, of course,

[e_.p, elf] Thus [Zl' Zl*]

=

~ qi~i= if;

=

-if;.

0 and hence Zl is normal with respect to f.

Q. E. D.

6. 4. To gain information on cyclic elements in general it suffices by Theorem 6. 2 to focus attention on a single fixed cyclic element. We choose this element to be Zo, where Zo is given by (6.4.1) Here, of course, eo is given by (5.2. 2) . 14

164

1008

BERTRAM KOSTANT.

As in § 5. 2 let ao be the TDS spanned by X o, eo and fo. N ow let nk be the multiplicity of the irreducible 2k 1-dimensional representation 'lI"2k+l in the complete reduction of the adjoint representation of a o on go. By Corollary 5.2, nk = 0 if k is a half-integer and

+

(6.4.2) Now it follows from § 2. 5 ( d) and (5. 2. 4) that the kernel geo of ad eo is contained in~. On the other hand, it is obvious that geo n f) = o. Therefore, since Xo E f) is regular, it follows that the eigenvalues of ad Xo on geo are positive. This implies two things, (1) geo C n (Actually, we have already noted this fact-see proof of Corollary 5. 6) and (2) by § 2. 5 (d), no = o. That is, only the zero element is annihilated by ad a o• Indeed, this also follows from § 2. 5 (h) which asserts no = dim go- dim gl

=l-l =0.

For the present we direct our attention to the first fact, ge o C n. Let n* be the maximal Lie subalgebra of nilpotent elements given by

Then, of course, (6.4.3)

g=n* + f)+n

is a direct sum decomposition. We shall let p (resp. p*) be the projection p:

(resp. p*:

g~n

g~n*)

of g onto n (resp. n*) defined by the decomposition (6.4. 3) . Now consider the kernel gZo of ad Zoo given by

6.4A. Let po be the restriction of the projection to the subThen po (gZo) C geo and in fact

LEMMA

space gZo.

A relation between geo and gzo is

is a linear isomorphism of gZo onto geo.

165

COMPLEX SIMPLE LIE GROUP.

1009

Proof. It follows immediately from the decomposition (6.4.3) of 9 and (5.1.3) that [e-t/J, n] C n* q [e_1/I, q] C (e_1/I) [e_1/I, n*] = O.

+

Now let y E gZo be arbitrary.

Write

y=v +x+u, where vE n*, xE q, and uE n. Now by (6.4.1) and § 5.1

0= [zo,y] = [eo, v] [eo, x] [eo, u] (x, 1/1 ) e-ljt [e-ljt, u] .

(6.4.4)

+

+

+

+

Applying the projection p to both sides of (6.4.4) we obtain

[eo, x]

+ [eo,u] =0.

But then recalling the relation [gi, gj] C g»j, it is clear that [eo, x] = 0 and [eo, u] = O. Thus since u = p (y) it follows that p (gzo) C geo• It also follows that since geoC n, we must have

x=O.

(6.4.5) Thus y =v +u and

o=

[eo, v]

+ [eo, u] + [e-ljt, u] .

To show first that po is one-one, assume u = O. But this and the last expression imply that v E geo. But geoC n. Hence v = O. Thus Y = 0 which proves that po is an isomorphism into. But now, by Corollary 5. 3, dim geo= l. On the other hand, dim gZo > 1 (see, for example Theorem 5. 7). Since po is an isomorphism, this means dim gZo=l and also that po is onto. A major consequence of Lemma 6.4A is COROLLARY

6.4.

Q.E.D.

Cyclic elements are regular.

Proof. As above let Zo be defined by (6. 4. 1). It suffices to show Zo is regular. But by Lemma 6.3 Zo is semi-simple. Hence gZo contains a Cartan subalgebra. But, by Lemma 6.4, dim gZo = l. Hence Zo is regular. Corollary 6. 4 implies

166

BERTRAM KOSTANT.

1010 LEMMA 6.4B. subalgebra.

Let Zo be defined by (6.4.1).

Then gZo

~s

a Carlan

We will let ~' designate the Cartan subalgebra gZo. In the proof of Lemma 6.4 we considered the decomposition of any element y E ~' and showed that its projection x in ~ vanishes. (See (6.4.5». We state this as LEMMA 6.4C. Let Zo be defined by (6.4.1). Let p and p* be the projections of g on nand n* also defined as in § 6.4. Then the projection 1 - (p p*) of g on ~ vanishes on ~'.

+

Perhaps a simpler way of expressing Lemma 6.4C is to state that the Cartan subalgebras ~' and ~ are orthogonal to each other (with respect to B). 6.5. Now geo is invariant under ad Xo (see § 2. 5 (d) ). Let Ui, i = 1, 2,' . " l, be a basis of geo and also eigenvectors of ad Xo. Let ki' i = 1, 2,' . " l, be the corresponding eigenvalues. That is, Ui E gk" i = 1, 2, . . " l. We may regard the basis so ordered that ki < ki +1' At this stage we already know two of the ki' namely, the extreme ones kl and k!. We also know an inequality, kH k!.


O. On the other hand, since eo E geo, it follows that kl = 1. But also e1fi E geo since tf! is the highest root. Thus [x o,e1fi] =qe1fi implies k =q. Since o(¢) < q for every ¢#tf! (see (5.3.4)), it also follows that kH < k l • Q. E. D. Taking the proof of Lemma 6. 5A into account we will choose U 1 = eo and u!=e1fi. N ow let di , i = 1, 2,' . " l, be the dimensions, in non-decreasing order, of the irreducible components occurring in the complete reduction of adjoint representation of a principal TDS (e.g. 0 0 ) on g. Appplying §2.5(d) the eigenvalues ki yield the dimensions di by the relation (6.5.1)

+

+

Lemma 6. 5A implies d1 = 3, dl = 2q 1 and d1r-l < 2q 1. N ow by Lemma 6. 4A there exists a unique basis Yl, Y2,' . " Yl of the Cartan subalgebra ~' with the property that p (Yi) = Ui. Define Vi by (6.5.2)

Yi=Ui+ Vi'

167

1011

COMPLEX SIMPLE LIE GROUP.

By Lemma 6.4C it follows that Vi E n* for i = 1, 2,· .. , l. Since U 1 = eo and since p(zo) = eo, it follows from (6.4.1) and Lemma 6.4A that V1 = e_tfJ and Y1 = zoo Now u! = etfJ. We shall have to know what V! is. LEMMA 6.5B. Let Ci and qi, i = 1, 2,· .. , l, be defined respectively by (5. 2. 2) and (6. 3. 1) . Then V! is the element given by I

L (qil Ci) e_a,.

V! =

(6.5.3)

.=1

Proof.

As in the proof of Lemma 6.3 I

[zo, etfJ

+ L (qJCi)e-aJ .=1

I

=

[e_tfJ, etfJ]

+ ~ qi [ea" e_a.] ;'=1

I

=-t/I

+ ~ qiai= O. i=l

I

Thus etfJ

+ ~ (qJ Ci) e_a, E ~'.

It follows immediately then from Lemma

;'=1

I

6.4A that

v!=~

(qJci)e-a,.

Q.E.D.

;'=1

6. 6. N ow it is clear (for example from the matrix representation of a TDS given in § 2. 5) that there exists an element A in the subgroup of G corresponding to 0 0 such that Axo = - Xo. But since Xo is regular, it is clear then that

In fact these relations are already contained in the more general fact

A:

gj~

g_j

for any - q < j < q. It follows then that we may interchange the roles of 6.+ and -6.+, nand n* and also gl and g-l in the results of §§ 6. 2-5. But then one sees that V! is principal nilpotent and that y! = u! V! is cyclic. But then by Corollary 6.4 it follows that gYI = ~'. Finally, applying Lemma 6. 4A we obtain

+

6. 6. We retain previous notation. Let p* 0 be the restriction of Then (1) V! is a principal nilpotent element, (2), P*o(~') C gVI and

LEMMA

p* to ~'.

in fact

168

1012

BERTRAM KOSTANT.

is isomorphism onto. In other words, the elements Vi, i = 1, 2,' . " l, form a basis of gVI. Also, the elements Vi commute with each other. (See Oorollary 5.8). It is obvious from (5. 2. 3) that VI is a nil-negative element of a principal S-triple containing Xo as neutral element or an nil-positive element of a principal S-triple containing - Xo as neutral element. Thus gVI is stable under ad Xo and since all principal S-triples are conjugate the eigenvalues of ad Xo on gVI are -lei, -lel-l,' . " -lel in non-decreasing order. The elements Vi are a basis of gVI by Lemma 6.6 but it is not at all clear yet that they are eigenvectors of ad Xo. 6.7. We now isolate a particular conjugate class in G, elements of which, will playa major role in the remainder of this paper. An element PEG will be called a principal element of G if there exists a principal regular element x E g (see § 5. 2) such that P can be written

P

(6.7.1)

=

Exp(27ri/s)x,

+

where s = q 1 and q, as usual, is the order of the highest root.p. Note that since x lies in a principal TDS, the principal element P lies in a subgroup of G corresponding to a principal TDS. (We shall make no use of the fact here but it can be shown that the number of principal regular elements x which satisfy (6.7.1) for a fixed principal element PEG is equal to order of the fundamental group of G). Various characterizations of principal elements will be given in §§ 8 and 9. Throughout the remainder of the paper we will let w be the primitive s root of unity defined by w = e27ri / B• Let Po be principle element in G defined by letting Po = exp (27ri/s) Xo. It is clear then that

for u E gj, - q < j < q. eigenvalues of Po and if

It follows then that wi, j = 0, 1,' . " q, are the Ui designates the corresponding eigenspaces, then

(6.7.2) where it is understood that g-8 denotes the zero subspace. It is an immediate consequence of (6. 7. 2) that Zo is an eigenvector of Po. In fact, (6.7.3)

Pozo =wZo.

But this clearly means that the Cartan sub algebra ~' = gZo is stable under Po.

169

COMPLEX SIMPLE LIE GROUP.

+

On the other hand, the elements Yi = Ui Let us apply Po to these basal elements.

1013

Vi, where Ui Egk" form a basis of 1)'. Then by § 6. 5

But the elements P OYi belong to 1)' and furthermore since, obviously, P oV. E n*, it follows that

But now if we apply Lemma 6.4A we must have

Thus % Ui and hence also Vi are contained in Uk,. But since Vi E n*, it follows from (6. 7. 2) that Vi E gk8' Thus not only did we prove that the Vi are eigenvectors of ad xo but more we obtain a duality relation among the integers lei' LEMMA Vi

E gk,-., i

Let

6. 7.

Vi,

i

=

1, 2,' . " l, be defined by (6. 5.2).

Then

= 1, 2,' . " l. Also the integers lei satisfy the following duality

law, s=lei

+ le =le Z

2

+le H =· . ·=le z + lei'

Proof. Only the second part of Lemma 6. 7 is not yet proved. By Lemma 6.6 the elements Vi are a basis of gV' and the first part of Lemma 6.7 asserts that Vi is an eigenvector of adxo with eigenvalue lei-so On the other hand, as we have seen, the eigenvalue;; of ad xo on gV' are, in non-decreasing order, -le z< -kH < .. ·1 ~

CijOlj.

Since the coefficient of Ol, in CPt is one, it follows that cpt is positive. Furthermore, since the coefficient of Olj in cpt for j < i is zero, it is clear that CPt are linearly independent and hence form a basis of f). But clearly and hence

Since the coefficient of Ol, in y (cpt) is minus one, CPi is a positive root which changes sign under y. Thus N (y) > l. N ow it is well known that N (R",,) = 1. In :fact, Ol. is the only positive root which changes sign under R"". This is clear since only the coefficient of Ol, in cP, :for any cP E ~, is affected by R"". N ow assume cP is a positive root which changes sign under y. Since y(cp) E ~-, there clearly exists a maximal value i such that

That is, R""+l· .. R""cp E ~+.

It :follows then that

This, however, means that cp = CPl. Thus N (y) = l. Q. E. D. Now it is well known that one is not an eigenvalue o:f a Coxeter-Killing trans:formation. A proo:f of this resting on the work of Coxeter is given, :for

176

BERTRAM KOSTANT.

1020

example, in [4]. (See [4], p. 352). However, since a direct proof of this fact, making use of Theorem 8.1, can be given we shall include it. LEMMA 8.1. Let y be the Coxeter-Killing transformation given as in § 8. 1. Then one is not an eigenvalue of y. Let £i, i = 1, 2,· .. , l, be the basis of Clearly

Proof.

q given

as in § 5. 1.

Let

l. = 2£i/ (ex., ex.).

yl. = R a, · .. Rail. l.

=

+ Ra, · .. Ral+l (- exl)

by (7. 1. 1). On the other hand, it is clear from the definition of CPo in the proof of Theorem 8.1 that R Ra,•, (- ex.) = y (cp.). Thus we have

a, · ..

yl. = l.

+ y(cp.).

z Assume now that x E q is fixed under y. Write x = ~ ail..

Then

.=1

x=yX z

=

~ a.;(l.

0=1

+ 1'( CPo))

Z

=x+ ~ai'Y(cp.). i=l

z

Thus

~ .=1

acy (cpt)

=

o.

However, by Theorem 8. 1 the vectors CP. and hence

y (CPi) are linearly independent. x=o.

8. 2.

Thus a. = 0, i

=

1, 2,. .. , l, and hence Q.E.D.

Lemma 8. 1 is needed solely to prove

LEMMA 8. 2. Let r. CA, i = 1,2,· .. , L, be the orbits in A under the action of y. Then for i = 1, 2,· .. , L

Proof. This follows immediately from Lemma 8. 1 since the sum of the roots in any orbit of y is obviously left fixed by y. Q. E. D. The following theorem is proved in [4] THEOREM 8.2 (Coleman). Let h be the order of the Coxeter-Killing transformation y defined as in § 8.1. Let v = e27ri / h • Then there exists a regular eigenvector Zl of I' whose corresponding eigenvalue is v. Coleman also observed and used in [4] the following consequence of Theorem 8. 2. We repeat his proof.

177

1021

COMPLEX SIMPLE LIE GROUP.

8. 2. As in Lemma 8. 2 let r, C a, i = 1, 2,· . ., L, denote the distinct orbits of "Y with respect to its action on the set of roots a. Then each orbit r. contains exactly h roots so that in particular hL = 2r, where 2r is the total number of roots. COROLLARY

Fro of. Let cp E a. Assume ymcp = cpo It suffices to prove that h divides m. Now, where Zl is given as in Theorem 8. 2, (Zl' cp)

(Zl' ymcp) (y-mZl' cp ) y-m (Zl' cp) •

= = =

But since Zl is regular, (Zl' cp) # 0 and hence divides m.

y-m =

1.

This implies h Q. E. D.

8.3. Now one knows that the Poincare polynomial Fa(t) can be put in the form

Fa(t)

=

(1

+t

2mt+1 )

(1

+

t2m2+1) • • •

(1

+t

2ml +1 ) ,

where the mi, i = 1, 2,· .. , I, are positive integers in non-decreasing order. The integers mi (sometimes m. 1) are called the exponents of g (or W as in [4J). When the values of the exponents for the simple exceptional Lie algebras were announced by Chevalley at the International Congress at Cambridge in 1950, Coxeter recognized a rather remarkable coincidence. He observed (1) that in all cases hI = 2r, so that in our notation h = s, and hence y=ro (See Corollary 6.8), (2) mi. Now since A./, reduces to the identity on g, we can write A./, = Exp x for some x E g. It follows that A~ = e(IIJ.¢) for any cf> E A. But then /&-1

(A~)"=n A-y'~ ~=O

=

e(IIJ.¢+'Y¢+ .•• +'Y"-1¢)

=1 by Lemma 8.2. Thus A~ is an h root of unity. This shows that A./" = 1. We can now prove that h = s and hence that L = l. More than this we have THEOREM

subalgebra

g.

8.4. Let'Y be a Coxeter-Killing transformation on the Cartan

+ ~ q., where the integers J

Let h be the order of 'Y' Then h = 1

i=1

q. are the coefficients of the highest root relative to a basis of simple positive roots. That is, h = s. Moreover, hl = 21', where l' is the number of positive roots so that there are l distinct orbits r. C A, i = 1, 2,' . " l, in A under the action of 'Y and each orbit contains h roots. Furthermore, if we take 'Y = Ra,Ra.· .. R a, and let cf>i' i = 1, 2,' . " l, be the positive roots which change sign under 'Y (see Theorem 8. 1), then we can choose an ordering of the orbits so that cf>i E r i , i = 1, 2,' . " l. Now let A'Y E G be any extension of 'Y' Then A./, = 1. That is, A'Yhe~ = e~ for any cf> E A so that we can renormalize the root vectors e.p in such a way that A'Y'e~ = e'Y'~

for any cf> E A and any integer i. Now let w. E g, i = 1, 2,' . " l, be defined by

w.=

~ e~

¢Er,

179

COMPLEX SIMI'],E UE

1023

(movl'.

and let ~ be the subspace spanned by the elements Wi. Then 1) is a Cartan subalgebra of 9 (so that the elements Wi are semisimple and commute with each other). Furthermore, ~ = gA~. Proof. Let y, A-y and A~ for rp E ~ be as previously defined. It has been = 1. In particular A-y is semisimple. But if A EGis semishown that simple it is well known that gA contains a Cartan subalgebra of 9 (See e. g. [8J). For A = A'Y the proof is somewhat more direct since A'Y' having finite order, lies in a maximal compact subgroup of G. But then A'Y = Exp x for some x in a compact form of g. We could then apply § 6. 1 (d) ). Let 1) be a Cartan sub algebra of 9 contained in gA~. N ow let Ii;, i = 1, 2,· . ., L, be defined by (8. 4. 1) . Consider the decomposition

kr

L

g=~+LIii. ;=1

Obviously gA~ = ~

L

n gA~ + ~ Iii n gA~ i=l

(8.4.3)

by Lemma 8. 1. N ow let rp E rio As we have already noted A~ = A'Y'~ for any integer i. Thus A-yh reduces to the scalar A~ on the space Iii. But then if A~ =1= 1, A'Yh has no non-zero fixed vectors in Iii and hence certainly Iii n gA~ = O. On the other hand, if A~ = 1, then clearly Iii n gA~ is the one dimensional subspace 1&-1

spanned by

~ A'Yie~.

Thus if L1 is the number of integers i, 1 < i < L, such

i=O

that A~ = 1 for all rp E r i , it follows from (8. 4. 3) that Ll = dim gA~. In fact, since ~ C gA~, we then have

L > L1 = dim gA'l' > 1 and hence in particular L > l. Now we assert that for any i, r i n~+ and r i n fJ.- are both non-empty. Indeed assume, without loss, that r i n fJ.- is empty. Then L rp, in the ¢€r,

lexicographical order of ~#, must be strictly positive and hence cannot vanish. This contradicts Lemma 8. 1 and hence the assertion is proved. N ow since r i n fJ.+ and ri n fJ.- are non-empty, it is clear that there exists a root rp E r i n ~+ such that yrp E r i n fJ.-. That is, each orbit r i contains at least one positive root which changes sign under y. But then if we apply Theorem 8. 1 it follows obviously that L < l. Thus L = Ll = dim gA~ = l. That is, A~ = 1 for all

O. Hence we conclude bi > 1

182

1026

BERTRAM KOSTANT.

and !

S

> ~,qibi. i::=l

!

But

s-l=q=~q.

(see (6.3.1)).

Hence it follows that bi =l for all i.

i=1

That is, or

But xo is a principal regular element of g and A'Y = Exp (2'1J"ijs)xo. Thus A'Y is a principal element of G. In fact A'Y=APA-l. Furthermore, since the restriction y of A'Y to ~ has w for an eigenvalue, it follows by definition (See § 7. 3) that ~ is in apposition to ~ with respect to A'Y. We have proved 8. 6. Let ~ be a Cartan subalgebra of g. Let W be the Weyl group operating on ~ and let yEW be a Coxeter-Killing transformation. That is, y = RO/lRO/.· .. , RO/I> where (Xi, i = 1, 2,· . ., l, are simple positive roots 1'elative to some lexicographical order in ~# and RO/, E Ware the reflections they define. Let A'Y be any element of the adjoint group G of g which extends y. Then A'Y is a principal element of G (so that in particu~ar its order is sand all such extensions are conjugate to each other). Let ~ be the set of fixed elements of A'Y. Then ~ is a Cartan subalgebra (since principal elements are regular) and ~ is in apposition to ~ with respect to A'Y (see § 7. 3) . THEOREM

We derive a number of corollaries. The first is an immediate consequence of Theorem 7.3 and Theorem 8.6. 8.6. Let ~l and ~/l be Cartan subalgebras. Assume that ~/l is in apposition to ~l with respect to the principal element PEG. Then the restriction of P to ~/l defines a Coxeter-Killing transformation of ~/l. In particular, the restriction of Po to ~' is a Coxeter-Killing transformation (See Theorem 6.7). COROLLARY

8.7. In § 9. 2 we obtain a more general result (Theorem 9.2) than Corollary 8. 6. Taking § 2. 5 (h) into account the next corollary asserts the validity of the empirical method found by A. Shapiro for the determination of the exponents m •. COROLLARY

8. 7.

For i

=

1, 2,· . ., l let ki be as given as in § 6. 5 and

183

1027

COMPLEX SIMPLE LIE GROUP.

let m. be the exponents of 9 in non-decreasing order.

Then k i = m.,

i= 1,2,· .. , l. In other words, if di , i = 1, 2,· .. , l, are the dimensions of the irreducible components of the adjoint representation of a principal TDS on g, then

is the Poincare polynomial of G. Proof. This is an immediate consequence of the theorem of Coleman Q.E.D. (see §8.3), Theorem 8.4, Theorem 6.7 and Corollary 8.6. The proof of Theorem 8. 6 yields a characterization of principal elements of G. We know that a principal element of G is regular and that its order IS s. Among all regular elements in G we now show that its order is minimal. COROLLARY 8. 6. Let A E G be regular and let k be its order (possibly 00). Then k > s, where s = 1 q and q is the order of the highest root. Furthermore, k = s if and only if A is a principal element of G.

+

Proof. It suffices to assume the order k of A is finite. Now gA is a Cartan subalgebra. Without loss we may assume this to be g. Since A has finite order the eigenvalues of A have modulus 1. Thus we may write A = Exp 2'7rix, where x E g#. More than this, as argued in the proof of Theorem 8.6, by conjugating A, if necessary, we may asume x is contained in the fundamental simplex of the chamber D. That is, (x, ai) > 0, i = 1, 2, . . ., l, (x, t/t) < 1. On the other hand, since A is regular, (x, cf» is not an integer for any cf> E ~ so that strict inequalities hold in the inequalities z

just given.

Thus we may write x z

integers and

~

.=1

tiq.

< k.

=

~

.=1

(ti/k) E;, where the t. are positive

But then I

s -1 =

~

I

q.


(b, -v).

It follows then from (3.1.6) and (3.1.8) that if (3.2.2)

is the linear mapping defined by a(X) = «Bx)p, Xp) then ap is an isomorphism of gR into gpo 3.3.

Let W be any open set in M and let g~. be the Lie algebra of all

infinitesimal B-affine transformations defined on W.

ap as defined on

g~

For PEW we may regard

by replacing M by W in the previous definition.

In [7J,

p. 62 Nomizu has shown that if W is a sufficiently small neighborhood of p EM then ap defines an isomorphism of gJ~' onto gB and furthermore M is a locally B·affine homogeneous space with g:. as the Lie algebra of the local group of B·affine transformations mapping pinto W.

It follows then that M has an

underlying analytic structure and that the affine connection is analytic with respect to it.

But then a theorem of Nijenhuis asserts that the Lie algebra of

the restricted holonomy group at p (the holonomy algebra at P) is obtained by contracting successive covariant derivatives of

RR

at p.

Since

RB

is con·

stant this implies (See [7J, p. 50). LEMMA

4.

Let B be an affine connection which is invariant under paral-

lelism on a manifold M. Let p E M and let the Lie algebra !lp oj endomorphisms of the tangent sjJace Vp at p, be given by (3.2.1).

Then!lp is the holonomy

algebra at p. Let X E

gB

and let p E M. Then if (Bx)p E sp it follows from the equation

(3.1.5) that (Bx)q E!lq for any point q E M. Let g={XEgRI (Bx)oE!lo at some (and hence any) point OEM} and for any p E M and let gp E sf>

gp

(3.3.1)

be the subspace spanned by !lp and Vp • Since

is an ideal in Op it follows from the equation (3.1. 5) that 9 is an ideal in

198

44 gR

BERTRAM KOST ANT

and

{lp

is an ideal in

gZ

and that ap :

5.

LEMMA

g

-->

(3.3.2)

gpo

Let the notation be as above. Assume M is simply connected.

Then for any P E M the map

defines an isomorphism of gB onto g% and an

ap

isomorphism of g onto gpo Proof. Given b E

Op,

V E

such that (Bx)p = b, Xp = v. curve joining p and o.

Vp it suffices to show that there exists X Let

0 E lit[

gB

and let c be a piecewise differentiable

By solving the linear differential equations(3.1.3) and

(3.1.5) along the curve c with initial value (b, v) we obtain at The fact that bo in

(b', v') E go.

E

an element

0

Sq ~ Oq

follows from (3. 1. 5) and the fact that

00

for all points q. Assert that the element (b', v') is independent of the curve c. To prove this it suffices to assume c is a closed curve, that is p show that b = b' and v

=

= 0,

and to

v'. But now since M is simply connected by a standard

deformation argument it suffices to assume that c lies in an arbitrarily small neighborhood W of p (recall that p is an arbitrary point of M and (b, arbitrary element in gp). ap(g~·)

v)

is an

But now, as noted above, W can be chosen so that

= gZ. It follows then that

b'

= b and v' = v.

Now let XE 3'1(M) be the vector field defined by letting Xo

W' is a neighborhood of a such that ao(gR,,)

= g~

= b'

v'. Now if

then it is clear from the equa-

tions (3.1. 3) and (3.1. 5) that the restriction of X to W' lies in g~,. that X is of class Coo, (Bx)o

=

and that XE gB.

This proves

It also follows that apeX)

= (b, v).

Q.E.D.

3,4.

An element X E gB will be called an infinitesimal transvection at p if

(Bx)p =

o.

It is clear that if X is an infinitesimal transvection at p then X

E

g.

The following lemma is an immediate consequence of Lemma 5. LEMMA

9.

Assume M is simply connected. Let PEM

spar;e of all infinitesimal transvections at p and let f)p such that Xp = o.

£;;;

Let

nip £;;;

g be the

g be the set of all X

E

g

Then mp = apl( Vp ) and f)p = aj;l(sp) so that g = hp + mp is a

direct sum with f)p a subalgebra 0/ g and [f)p, mp]

£;;;

mp.

Furthermore mp gener-

ates g. In fact 9 = [mp, mp] + mp (non-direct in general). The reason for restricting attention to 9 instead of gB from this point on is given in

199

45

A CHARACTERIZATION OF INVARIANT AFFINE CONNECTIONS

LEMMA

7.

Let the ideal 9 in the set of all infinitesimal B·affine transfor-

mations be defined by (3.3.1). Let S be a tensor field on M

If S is covariant

constant with respect to B then S is invariant under g (that is, LxS = 0 for all XE9). Furthermore

~r

M is simply-connected then S is covariant constant with

respect to B if and only if S invariant under g. Proof. Assume

r:s = 0

for all YE e-:yI(M). But then for any p E M, Sp

is invariant under the restricted holonomy group at p.

bEsp.

Thus b( Sp) = 0 for all ButV~S=Oandsince

NowletXEg. Then (LxS)p=(r:S)p+(Bx)pSp.

(Bx)p E sp it follows that (Bx)pSp = O. Hence S is invariant under It Now assume M is simply connected and S is invariant under g. Let YE 3'"l(M) and

p E M. Then by Lemma 5 there exists XE mp such that Xp = Yp. Thus (P'~S)p = (P'fS)p

= (LxS)p.

But LxS = O. Thus (P'~S)p

But (Sx)p = O.

=0

and hence S is

covariant constant.

Q.E.D.

Assuming only that M is simply connected we now prove a part of Theorem 2 in the infinitesimal sense. LEMMA

8. Let A be an affine connection on a manifold M

Assume that

there exists an affine connection B on M such that (1) A is rigid with respect to Band

(2)

B is invariant under parallelism. Let g be the ideal in the Lie

algebra of infinitesimal B·a.ffine transformation defined by (3.3.1). Then 9 is a Lie algebra of infinitesimal A-a.ffine transformations. Proof. Let S = B - A. By definition S is covariant constant with respect to B. But then by Lemma 7 S is invariant under g.

Writing

r: = P'~ -

Sy for

any ¥ES,.l(M) one immediately verifies (3.1.4) for any XEg when A is substituted for B in (3.1.4). 3.5.

Q.E.D.

Now one knows that the group of all B·affine transformations of M

is a Lie group. See [2J.

Let

e

B

be the simply connected covering group of

the identity component of this group.

We may regard

by passage to the group of B-affine transformations. B

may identify the Lie algebra of G with a subalgebra § ll.

to Q n

Now Q/.

Q

n QI

is an ideal in g/.

e

B

as operating on M

It is clear then that we Q'

of gB.

(See Note 1 in

Let G be the subgroup of G B corresponding

e

Since G R is simply connected and

R

also a simply connected closed subgroup of G

200

,

is normal one knows that G is

46

BERTRAM KOST ANT

Now assume M is simply connected. Then one knows that every element X

E

gB generates a one parameter subgroup of B-affine transformations, that is

g' = gB, if and only if M is complete with respect to B and that in such a case G B is transitive on M.

(See [4J, p. 35 and [8J, p. 77).

We wish to observe

now that in this statement g may be substituted for gB and G for G B •

Indeed

if M is complete then it is general theorem (see Prop. p. 67 in [8J) that every Conversely if X Egis an

element in g (and in fact in gB) can be integrated.

infinitesimal transvection at p E M and X generates a one parameter group get) then get) • p is the geodesic through p with the tangent vector Xi> and it is defined for all values of the canonical parameter t.

This is immediate since

(Bx)p=O implies (PxX)p= (LxX)p=[X, XJp=O.

Furthermore Lx(Bx ) =0 so

that (Bx)q=O for all q=g(t).p with t arbitrary.

Finally to show that G is

transitive on M it is enough to show that G carries an arbitrary point into an open and closed set.

This, however, is immediate under the condition of com-

pleteness since the group generated by an element X E tnp for (See Lemma 6) any p

E

M lies in G.

LEMMA

Let B be an affine connection, which is invariant under paral-

9.

lelism, on a simPly-connected manifold M

Let the group G and the Lie algebra

of vector fields g be defined as above. Then M is complete with respect to B if and only if g is the Lie algebra of G. Furthermore if M is complete with respect to B then M is a reductive homogeneous space with respect to G. group at p then tnp

g=

In fact if p E M and Hp

~

G is the isotropy

f)p + tnp is direct sum where f)p is the Lie algebra of Hp and

is the space of infinitesimal transvections at p and

tnp

is invariant under

AdHp. Moreover the linear group defined on the tangent space at p by Hp (this is equivalent to the action of AdHp on at p.

tnp)

is the homogeneous holonomy group

One of the two statements of Theorem 2 is an immediate corollary of Lemmas 8 and 9. THEOREM

M.

4.

Let A be an affine connection on a simply connected manifold

Assume that there exists a second affine connection B on M such that (1)

B is invariant under parallelism

(2)

A is rigid with respect to Band (3) M is

complete with respect to B.

201

47

A CHARACTERIZATION OF INVARIANT AFFINE CONNECTIONS

Let 9 be the Lie algebra of infinitesimal B-affine transformations X on M such that (Bx)p

E ~p

for some (and hence every) point p EM where sp is the

(B) holonomy algebra at p.

Then M is a reductive homogeneous space with

respect to a connected simply connected Lie group G operating as A-alfine (and also B-alfi?ze) transformations on M. Furthermore G can be chosen so that 9 is the Lie algebra of vector fields on M defined by the action of G (See Note 1). 1. To prove

4. The definition of S on a reductive homogeneous space.

the second statement of Theorem 2 we shall assume in this section that M is a reductive homogeneous space with respect to a connected Lie group G1 • 91

Let

be the Lie algebra of C 1 • Now let

0 E

M and let HI

the Lie algebra of HI.

~

G1 be the isotropy group at

0

By assumption there exists a subspace ml

and let hi be ~

91 such that

(a) 91 = fh + ml is a direct sum and (b) ml is stable under AdHI . We will now show how the complement m] to f)! defines a tensor field S of type

(j i k)

on M as soon as an affine connection is given on M.

= r(9 1 ).

Let r be defined as in Note 1, § 1. 2. Let 9r

Since r is a homo-

morphism it is clear that we may regard AdG I as operating on 9r •

Now let

1) = r( 1)1) and m = r( ml). Since the kernel of r is contained in 1)1 note that (1) r

is faithful on ml, (2) 9r For any point p

E

=

1) + m is a direct sum and (3) m is stable under AdHI.

M let Vp designate the tangent space at p and let

"rp

be

the mapping

defined by putting "rpX= Xp where Xp is the value of X at p. Now for each p

E

M define the subspace tnp =

where g

E

G1 is such that g'

0 =

p.

I11p ~

9 by the relation

Adg(tn)

(4.1.0)

It is clear that

tnp

depends only on the

coset gHI and not solely on g. Now by (1) and (2) above it is clear that oro defines an isomorphism of tn onto Vo. But since Adg( HI) is the isotropy group at p where g'

0 =

is an isomorphism

P it is clear, more generally, that the restriction of tnp

onto Vp •

"rp

to

tnp

Let CPp : Vp

---)0

I11p

be the map which is inverse to the restriction of

202

"rp

to mp.

That is, CPP is such

48

BERTRAM KOST ANT

that ,prpp is the identity on Vp. Now assume that an affine connection A is given on M (A is not as yet assumed to be invariant under the action of G). E ~(M)

For any XES"l(M) let Ax

be defined by Ax= L x - J1~.

Now if

YE

yl(M) let Sy

E ~(M)

(4.1. 1)

be defined by (4.1.2)

where Yp

Vp is the value of Y at p.

E

It is obvious that the mapping Y

-->

Sy

(unlike X --> Ax) is a ,3"°(M)-linear from

210

334

BERTRAM KOSTANT

REMARK 2.2. If C is a vector space over C, the words orthogonal, orthonormal and orthocomplement will always be understood to be with respect to a positive definite hermitian structure {C} which has been defined on C and not with respect to a bilinear form (C) which may also have been defined on C.

2.3. It will be assumed from this point on that, unless statements are made to the contrary, every vector space considered in this paper is over C and that every homomorphism of one vector space into another is C-linear. More generally every homomorphism of one complex Lie group into another will be assumed to be holomorphic so that, in particular, representations of such groups are understood to be holomorphic. 2.3. Assume {C} is a positive definite hermitian structure on C. Let d be an operator on C such that d 2 = 0 and let d* be the adjoint of d with respect to {C}. Obviouslyd*2 = O. But we observe also that since {C} is positive definite d and d* are disjoint. Furthermore the laplacian L = dd* + d*d is self-adjoint and the decomposition (2.1.3) for 0 = d* is an orthogonal direct sum decomposition. REMARK

3. Cochain complexes defined by Lie algebras and hermitian structures

1. Let a be a complex Lie algebra. Then the exterior algebra A a over a together with the boundary operator 8 on Aa given by (3.1.1)

8(XI/\ ••• /\ x k )

=

Ei<j( _l)HJ+I[Xi'

Xj] /\

Xl'·· /\ Xi ••• /\

Xj ••• /\ X k

,

where Xi E a, is a chain complex which one denotes by C*(a). The derived space of homology is denoted by HAa). Covariantly the exterior algebra Aa' over the dual a' to a is canonically identified with the dual to An and Aa' together with the coboundary operator d, defined as the negative transpose of 8, is a cochain complex which one denotes by C(a). The derived space of cohomology (for the present we ignore its ring structure) is denoted by H(a). (See [9]). More generally let V be a vector space and let n: a -> End V

be a representation of a on V. Let d l be the operator on the tensor product Aa' ® V defined by putting d l = d ® 1, and also, regarding Aa' ® V as the space of all linear maps p from Aa to V, let d 2 be the operator on An' ® V given by (3.1.2)

d 2P(X I /\· •• /\ Xk ) = E:=l (-l)Hln(Xi)p(XI ••• /\ Xi ••• /\ x k ) .

Now define d" = d l

+d

2•

Then

d~ =

0 and the space Aa' ® V, together

211

GENERALIZED BOREL-WElL THEOREM

335

with the coboundary operator d", is a cochain complex which one designates by C(a, V). The derived space of cohomology is denoted by H(a, V). 3.2. Now let 9 be a complex semi-simple Lie algebra and let (g) be the Cartan-Killing form on g. The form (g) induces an isomorphism of 9 onto g' which extends to an algebra isomorphism of Ag onto its dual Ag'. The latter in turn induces a non-singular symmetric bilinear form (Ag) on Ag which, explicitly, is given by (Xl 1\ ••• 1\ Xp , Yl 1\ ••• 1\ Yq) = 0 if p *- q (3.2.1) if p = q. = det (Xi> Yj) 3.3. A real Lie subalgebra f of 9 is called a compact form of g if (1) 9 = f + if is a real direct sum, and (2) the bilinear form (g) is negative definite on f. A compact form of 9 denoted by f is henceforth assumed to be fixed once and for all. Let q = if. Then q is a real subspace of 9 on which (g) is positive definite. Where R denotes the real field let ARq be the subalgebra of Ag generated over R by q and R. Clearly (Ag) is positive definite on ARq and is a real direct sum. A *-operation is now introduced into Ag by defining (u

+ iv)* = u

- iv

for all u, v € ARq. It follows easily that this operation is a conjugate linear automorphism of Ag. Since (u*, v*)

for every u, v by putting (3.3.1)

€

= (u, v)

Ag we can define a

hermitian inner product {Ag} on AfJ.

{u, v} = (u, v*)

for all u, v € Ag. Since (Ag) is positive definite on ARq it follows immediately that {Ag} is positive definite on Ag. Let A € End Ag and let At, A * € End Ag be defined as in § 2.2. It follows immediately from (3.3.1) that At and A * are related by (3.3.2)

A*u

= (At(u*»)*

for every u € Ag. Substituting At for A in (3.3.2) and then A * for A and u* for u it follows at once that At* = A*t . (3.3.3)

212

336

BERTRAM KOSTANT

3.4. For any subspace 0 0*

Ag let = {u* e Ag Iu e o} ~

•

Obviously 0* is again a (complex) subspace of g. We now assume that the arbitrary Lie algebra a of § 2.3 is a Lie subalgebra of g. Since the *-operation is a conjugate linear automorphism of Ag it is obvious that (3.4.1) (Aa)* = Aa* . We now define a degree preserving linear mapping

1;: Aa* - Aa' by the relation (3.4.2)

=

End Ag

216

340

BERTRAM KOSTANT

denote the adjoint representation of g on Ag. Thus 8(y), for every y € g, is the unique derivation of degree 0 on Ag which on g satisfies 8(y)z = [y, z]. Since (g) is invariant under 8 it is clear that for every y € g (3.9.4)

(8(y»)t

=

-8(y) .

But now it is well known (when considered on Ag') that (3.9.5)

c(y)C

+ cc(y) = 8(y)

.

In fact since the left side of (3.9.5) is easily seen to be a derivation of degree 0 on Ag, it suffices to verify (3.9.5) on Ng. But then (3.9.5) is an immediate consequence of the definition of c. Now applying the operation A -> At, A € End Ag, to (3.9.5) and taking negatives, one also has by (3.6.2), (3.8.2) and (3.9.4) that (3.9.6)

c(Y)'y

+ 'Yc(Y) =

8(y) .

On the other hand by taking the adjoint of (3.9.5) one gets, by (3.9.2) and (3.9.3), the same expression as (3.9.6) on the left except that y* replaces y. It follows then that (3.9.7) for any y

(8(y»)* E

= 8(y*)

g.

3.10. Let p and t be two subspaces of g. We now observe that there exists a linear mapping such that for any y, z E g (3.10.1)

'Y:p,t(Y 1\ z) =

t([rpy, rtz] - [rpz, rty]) .

This is clear since the right side of (3.10.1) is alternating in y and z. But now by § 3.6 there then exists a unique derivation cp .t of degree 1 on Ag such that for any U E Rg and z E Nz (3.10.2)

(cP,tZ, u)

=

-(z, 'YP,tu) •

Although the right side of (3.10.1) is alternating in y and z, we now make the observation that it is symmetric in p and t. Consequently one has (3.10.3) Obviously if t1 and t2 are orthogonal subs paces of g one has (3.10.4) Let r be a subspace of g. We now observe that

217

341

GENERALIZED BOREL-WElL THEOREM

(3.10.5) for any z € g. Indeed since rr is a homomorphism (see § 3.7) it follows immediately that ryt,t = ryrr on /(g. But then (3.10.5) follows at once from the fact that -(ryrr)t = rr*c (see (3.7.1». If 1: = a where a is a Lie subalgebra of 9 it is clear from Lemma 3.8 and (3.10.5) that Ct,t will be significant in computing H(a, V). It would therefore be convenient if one could express Ct,t in terms of such computable operations as exterior multiplication and the adjoint representation. This seems to be unlikely in the case of a general Lie subalgebra a. On the other hand if 1: is any subspace of g, one can find, as will soon be shown, just such an expression for the derivation dr of Ag defined by putting (3.10.6) But the point is that under certain assumptions (which are satisfied for the cases which interest us) d r is a satisfactory replacement for CU' This is seen in comparing (3.10.5) and 3.10. Let 1: be a subspace of g. Assume 1:1. is a Lie subalgebra of g. Then for any u € A 1:* , LEMMA

dru

=

rr*Cu •

PROOF. Let z € 1:*. We first observe that cr1..r1.z = O. For this it suffices to show that (v, Cr1. .r 1.z) = 0 for any v € /(g. But, since 1:1. is a Lie subalgebra of g, it is obvious that ryr1..r1. maps /(g into 1:1.. Therefore since z* € r,

(v, cr1..r1.z)

= -(ryr1..r1.v, z) = - {ryr1..r1.v, z*} = O.

Thus for any z € 1:* one has drz = Ct,tZ. But then by (3.10.5) d r and rr*c agree on Nr*. On the other hand since the restriction of rr*c to A1:* is a derivation of A1:*. (See Remark 3.8; that is, because rr* is a homomorq.e.d. phism of Ag) it follows that d t and rt*c agree on A1:*. 3.11. Let 1: be a subspace of g. Let Zt, 1 ~ i ~ m, be an orthonormal basis of 1:. Let y € g. Writing (y, zi) for {y, Zt} it is obvious then that (3.11.1) Now let Ct = cr .g• To obtain the desired expression for d r we first observe that Ct may be given by the following simple expression. LEMMA 3.11. Let 1: be any subspace of g. Then if Zi, 1 orthonormal basis of 1:

218

~. i ~

m, is an

342

BERTRAM KOSTANT Cr

= ; E~=l e(zt)O(z,) .

PROOF. Since the right side of (3.11.2) is a derivation (see (3.6.1» it suffices to verify the equality for elements in /Xg. Thus if x, y, z € g, it is enough to prove the equality

y, z) = -t(x 1\ y, E~=le(ZnO(z,)z) , that is, to prove the equality (writing y 1\ x = -x 1\ y) (ryr.g X 1\

(3.11.2)

([rrx, y], z) - ([rrY, x], z)

=

E~=l(Y 1\

x, zi 1\ [z" z]) .

=q=2 [z" z]) = (y, zi}(x, [z" z]) - (x, zt)(y, [zc. z]) = (x, zt)([z" y], z) - (y, zt)([z" x], z) ,

But by (3.2.1) where p

(Y 1\ x, zt 1\

since O(z,)' = -O(z,). Summing over i the equality in (3.11.2) follows imq.e.d. mediately from (3.11.1). We now observe that (3.11.3) In fact by (3.10.4) c t = ct,t + ct.t.1 and Ct.1 = Ct.1.r.1 + Ct.1.t' But then (3.11.3) follows from (3.10.3) when p is replaced by t.1. Now recalling the definition of dr, (3.10.6), the proof of the following proposition follows from Lemma 3.11 and (3.11.3). PROPOSITION 3.11. Let t be any subspace of g. Let d r be the derivation of degree 1 of Ag defined by (3.10.6). Let z" 1 ~ i ~ n, be an orthonormal basis of 9 such that z, for i ~ m is a basis of t. Then dr = ~ (E~=le(Zt)O(z,) - E;=mHe(Zj)O(zJ») .

As an analogy with the definition d l in § 3.1, let dt.l be the operator on V defined by putting

Ag ®

(3.11.4)

dt.l

= dt ® 1 .

3.12. Let t be a subspace of 9 and let basis of t. We define an operator d t •2 on

z,. 1 ~ i Ag ®

~ m, be an orthonormal V by putting

(3.12.1) It is straightforward to verify that the definition is independent of the orthonormal basis chosen. Put (3.12.2) C2 = d g.2 • Obviously C2 = d t •2 + d rl.. 2 • Note then that we can write (and for "compatibility" with the expression for d r .l given by Proposition 3.11 and (3.11.4) it is convenient to do so)

219

343

GENERALIZED BOREL-WElL THEOREM

dt,2

(3.12.3)

= ~ (c 2 + d r.2 -

d r1-.2)

•

Now let t = a be a Lie subalgebra of g. Let z~ e a' be the basis of a' dual to the basis z, of a. It is then a simple matter to verify that the operator d 2 on Aa' ® V defined by (3.1.2) may be given by d 2 = E~=l e(zD

® l.I(z,)

,

where e(zD is left exterior multiplication on Aa' by z;. On the other hand since (z" zj) =

o'J ,

and since zj e a*, it follows from the definition of 1) that 1)(z;) = z:. But then for any p e Aa' ® V one obtains the relation (3.12.4) 3.13. Now let 8~:

9 --> End (Ag

®

V)

be the representation of 9 in Ag ® V formed by taking the tensor product of 8 and 1.1. Thus for any z e 9 8~(z)

Now let t

=

®1

8(z)

+ 1 ® l.I(z)

.

= a be a Lie subalgebra of g. Define da.~ =

(3.13.1)

da.1

+ da.

2

where d a.1 and d a.2 are given by (3.11.4) and (3.12.1). But then by Proposition 3.11, (3.12.1) and (3.12.3) we obtain, as a corollary of Proposition 3.11, the following expression for da.~'

Let a be a Lie subalgebra of g. Let Xii 1 ;;:;; i ;;:;; n, be an orthonormal basis of 9 such that for i ;;:;; n, z, is a basis of a. Let da.~ be the operator on Ag ® V given by (3.13.1). Then PROPOSITION 3.13.

da.~ = ~ (c 2

+ E~=l (e(zi) ® 1)8~(z,) - E;=m+1 (s(zj) ® 1)8~(zJ»)

,

C2 is given by (3.12.2) and 8~ is the tensor product of 8 and 1.1. The significance of the operator da.~ for a family of Lie subalgebras a of 9 which we call Lie summands (see § 4.1) is made clear by Lemma 4.1.

where

4. The laplacian in the case of a Lie summand 1. Let t

~

9 be a subspace of g. Define to

=

{z e 9 I (z, y)

=

0

for all

yet}.

Now let a be a Lie subalgebra of g. We will say that a is a Lie summand

220

344

BERTRAM KOSTA NT

of 9 if aO is also a Lie subalgebra of g. The name Lie summand is derived from the following immediate proposition. PROPOSITION 4.1. Let a be a Lie subalgebra of g. Then a is a Lie summand if and only if a-L is a Lie subalgebra of g. PROOF. The proof is an immediate consequence of the obvious fact that a -L = (aO)* and that by Remark 3.9, aO is a Lie subalgebra if and only if (aO)* is a Lie subalgebra. q.e.d. We recall that the representation n of § 3.1 is here the restriction of v to a. The following lemma states that in case a is a Lie summand the coboundary operator d" on Aa' ® V corresponds to the restriction of d a" to Aa* ® V under the mapping r; ® 1. LEMMA 4.1. Let a be a Lie summand of g. Then for any P E Aa' ® V, r;

®

l(d"p) = da,,(r;

®

l(p») .

PROOF. By (3.12.4) and by the definition of da" and d" it suffices only to show that r; t

®

1(d 1P)

= da,l(r; ® l(p»)

.

But this is an immediate consequence of Lemma 3.8, Lemma 3.10 with replaced by a, and Proposition 4.1. q.e.d.

Ag ® V, C, = C + C

4.2. Now, as an operator on

1

put

C1

= C ® 1 and let

2 •

In the case of a Lie summand the problem of finding a suitable expression for the operator on Aa* ® V which corresponds (under r; ® 1) to d" is settled by Proposition 3.13 and Lemma 4.1. The corresponding problem for d; is much easier. It is settled for all Lie subalgebras a by LEMMA 4.2. Let a be a Lie subalgebra of g. Let pEA a' ® V. Then r;

®

l(d;p) = c:(r; ® l(p») .

PROOF . We first prove

® l(dip) = ci(r; ® l(p») . To do this first observe that ci = d~,2 + d;-L,2 (see (3.12.2». Next we note

(4.2.1)

r;

that Aa* ® V is stable under d~,2 and d;-L,2' This is clear since both of these operators, by (3.5.1) and (3.9.3), are a sum of operators of the form t(y) ® )"i(z) where y, Z E 9 and by (3.6.3) Aa* ® V is stable under every operator of this form. But now since Aa* ® V is stable under d a ,2 and its adjoint dci,2 and since r; ® 1 is an isometry mapping Aa' ® V onto Aa* ® V it must follow from (3.12.4) that

221

345

GENERALIZED BOREL-WElL THEOREM

r; ® l(dip) = d~'2(r; ® l(p») .

To prove (4.2.1) therefore, it suffices only to show that d;.L,2 vanishes on Aa* 0 V. But d;.L,2 is a sum of operators of the form t(y) 0 ;rr(z) where y € a.L. Therefore for any x € a* one has (y, x) = {y, x*} = o. Consequently by (3.6.3) t(y) vanishes on Aa* and hence d;.L,2 vanishes on Aa* ® V. This proves (4.2.1). To conclude the proof one need only show that r;0 l(dip) = ci(r; 0 l(p» or more simply r;d*g

(4.2.2)

= c*r;g

for any g € Aa' since V is not involved. But now c* = 7 by (3.9.2) and since a* is a Lie subalgebra of 9 (by Remark 3.9) it follows that Aa* is stable under c*. Therefore one need only show that for any f € Aa' {r;d*g, r;f} = {c*r;g, r;f} .

But {r;d*g, r;f} = {d*g,j} = {g, df} = {r;g, r;df}· On the other hand {c*r;g, r;f} = {r;g, cr;f} = {r;g, ra*cr;f}. But r;df = ra.cr;(f) by Lemma 3.8. q.e.d. 4.3. It follows immediately from Lemma 3.10 that c. = dg ,•• Replacing a by 9 (obviously 9 is a Lie summand) in Lemma 4.1, it then follows that c. is equivalent under a linear mapping to the coboundary operator of the cochain complex C(g, V). Consequently c~ = o. Moreover one also knows that the relation (3.9.5) generalizes to (4.3.1)

(t(z) ® l)c.

+ c,(t(z) ® 1) =

e.(z) .

Indeed (4.3.1) is an easy consequence of (3.9.5) and the easily verified relation s(y)t(z)

+ t(z)s(y) =

(y, z)l

where y, z € 9 and 1 denotes the identity operator on But now (4.3.2)

(e.(z»)*

Ag.

= e,(z*) .

This is an obvious consequence of (3.5.1) and (3.9.7). Thus if we take the adjoint of (4.3.1) we obtain (4.3.3)

(s(z*)

0

Obviously c~ = 0 implies replacing z* by z that (4.3.4)

+ c~(s(z*) ® 1) = e.(z*) . (C~)2 = o. It follows then from

l)c~

c~e.(z) = e.(z)c~

for all z € g.

222

,

(4.3.3) after

346

BERTRAM KOSTANT

4.4. Let RY € End V denote the Casimir operator corresponding to (g) and the representation I.J of 9 on V. If Zh 1 ;;;:; i ;;;:; n, is an orthogonal basis of 9 we note that RY may be written (4.4.1) This is clear since

(Zj,

zj)

=

Ojj'

Ag ® V the following relation: c c: + c:c = 1 ® RY .

LEMMA 4.4. One has on

2

2

PROOF. Let Zj, 1 ;;;:; i ;;;:; n, be an orthogonal basis of g. Note that since 1 ® I.J(Zj) commutes with ci [1

(4.4.2)

® l.J(z,), c:l = =

[1

® l.J(z,), cil

E/(zj) ® l.J([z" Zj]) ,

since

ci

= EAzj) ® I.J(Zj)

by (3.12.1) and (3.12.2), after substituting the orthonormal basis zj for the basis Zj' However since Eyzj ® zj € 9 ® 9 is invariant under the adjoint representation of 9 on 9 ® 9 it follows from (4.4.2) that (4.4.3)

[1

® I.J(ZI), c:l = EA[zj, z,]) ® I.J(Zj) .

But

c2c: (4.4.4)

+ c:c = E, (s(zi) ® l)c: + c:(s(zi) ® 1»)1 ® I.J(Zi) + E, (s(zt) ® 1)[1 ® l.J(z,), cn = E,Oy(zi)(1 ® I.J(Zi») + Ej(E,s(zi)c([zj, z,])) ® I.J(Zj) 2

,

by (4.3.3) and (4.4.3). On the other hand (4.4.5) since both sides of (4.4.5) are derivations (see (§ 3.6» of degree 0 of and both sides are easily seen to agree on I\g. Thus by (4.4.4)

c2c:

+ c:c = E, (Oy(zi) 2

Ag

- O(zi) ® 1)(1 ® l.J(z,»)

= 1 ® RY •

q.e.d.

We can now give an expression for the operator on Aa* ® V which corresponds under 1) ® 1 to the laplacian L" on Aa' ® V . THEOREM 4.4. Let a be a Lie summand of 9 (see § 4.1.). Let L" be the laplacian on the cochain complex C(a, V) defined as in § 3.5. Let 1) be the mapping defined as in § 3.4. Let z" 1 ;;;:; i ;;;:; n, be an orthonormal basis

223

GENERALIZED BOREL-WElL THEOREM

of 9 such that z, for i 1) Q9

l(L"p)

~

m is a basis of a. Then for any p

347 €

Aa' Q9

V

= ~(1 Q9 R' + E;:1 0,(Z;,,)O,(z,) - E7"'m+I O,(zf)Ov(Zj»)(1) Q91(p»)

where 0, is the tensor product of the adjoint representation 0 and 1.1 and R' is the Casimir operator corresponding to 1.1. PROOF. By Lemmas 4.1 and 4.2, 1) Q91(L"p)

= (da.vc; + c;da.v)(1) Q9 l(p») .

But now substituting the expression for da., given by Proposition 3.13 in da.,c; + c;da.v and recalling that c; commutes with 8,(zt) (see (4.3.4» the result follows from (4.3.3) and Lemma 4.4. q.e.d. 5. The spectral resolution of the laplacian and cohomology for a family of nilpotent Lie summands .

1. Let 1) be a Cartan subalgebra of 9 and let l (the rank of g) be its dimension. One knows that the restriction (1) of (g) to 1) is non-singular and hence one can define a map P--> x". of 1)' onto 1) by the relation

(x, x".) = <x,

p>

for all x € 1). On the other hand, the mapping defines a non-singular bilinear form (1)') on 1)' given by (p, A,) = <x"., A,>. Now let A ~ 1)' be the set of roots associated with 1) and let erp, cp € A, be a corresponding set of root vectors so that for any cp € A, x € 1)

[x, erp] = <x, cp>eip .

(5.1.1)

One knows that the erp can be chosen so that (erp, e",) = 0 (5.1.2) =1

if"'" =1= -cp if"'" = -cp •

In such a case it is immediate that (5.1.3) [erp, e_rp] = Xrp. If 1)# is the real subspace of 1)' spanned over R, by A, we recall that (1)') is positive definite on 1)#. If 1: ~ 9 is a subspace which is stable under O(x) for all x € 1), we will let A(1:) ~ A be the subset defined so that 1:

= 1: n 1) + ErpeMt) (erp) .

Thus if U ~ 9 is a Lie subalgebra then A(U) is defined if 1) lies in the normalizer of u. In particular then A(U) is defined if 1) ~ u. Denote by the operation of addition in A in case the sum again lies in A. It is clear that A(U) is closed under in case u € 9 is a Lie subalgebra.

+

+

224

348

BERTRAM KOSTANT

Now let G be a simply-connected group whose Lie algebra is 9. REMARK 5.1. If U~ G is the subgroup corresponding to a Lie subalgebra u ~ 9 and u contains a Cartan subalgebra of 9 then U is necessarily closed. To prove this it suffices to show that u equals its own normalizer in 9. But, using (5.1.1), this is immediate. 5.2. Let 0 ~ 9 be a maximal solvable Lie subalgebra which will be regarded as fixed once and for all. Let V be the collection of all Lie subalgebras u such that 0 ~ u. If U ~ G is the subgroup corresponding to u € CU, then it is due to Wang [10] that (see Remark 5.1) the space (left cosets, a U, a € G) (5.2.1)

X =

G/ U

is compact, has positive Euler characteristic and one obtains, up to a biholomorphic map, all such complex homogeneous spaces of G this way. Incidentally one knows also that X is algebraic (admits a holomorphic embedding into complex projective space) and that (over all 9) one obtains, up to a biholomorphic map, all simply connected algebraic homogeneous spaces this way. Obviously 0 € CU. Let Y denote the generalized flag manifold y= G/B

(5.2.2) where B

~

G is the subgroup corresponding to O.

5.3. Let u (5.3.1)

€

CU and put 91

= un u* .

It is clear that 91 is a Lie subalgebra of 9 and that 91 is closed under the *-operation so that (see Remark 3.9) 91 is reductive in 9. Now put nt = 0°. One knows that nt is a maximal nilpotent Lie subalgbra of 9 and that m is the set of all nilpotent elements in o. We note then that 0 is a Lie,summand. This, however, is a special case of

PROPOSITION 5.3. Let u € CU. Then u is a Lie summand of 9. In fact if n = UO then n is both the maximal nilpotent ideal in u and the set of all nilpotent elements in the radical of u. Furthermore if 91 is defined by (5.3.1) then (5.3.2)

9

=

n*

+ 91 + lt

is an orthogonal direct sum and

u = 91 + n. Moreover 91 lies in the normalizer of both nand n*.

(5.3.3)

225

GENERALIZED BOREL-WElL THEOREM PROOF.

349

It is obvious from (3.9.4) that n is stable under O(z) for all

z e u. But b ~ u. Hence one must have (5.3.4) Thus n is an ideal in u which proves, in particular, that u is a Lie summand. Now since u* = (nO)* = n.L it follows, by definition, that gl is the orthocomplement of n in u. This proves (5.3.3). Furthermore n* = (UO)* = u.L, and this proves (5.3.2). Now let c be the center of gl so that s = c + n is the radical of u. But now the center of u is zero since u contains b. Thus [z, n] = 0, for z e c, implies z = O. But since O(z) is diagonalizable for any z e c this implies that n is the maximal nilpotent ideal of u. Furthermore since n ~ m it is clear that the elements of n are nilpotent. On the other hand by simultaneously triangularizing O(x) for all xes = c + n it becomes obvious that n is the set of all nilpotent elements in s. Since n is an ideal of u it follows that gl lies in the normalizer of n. Applying the *-operation it also lies in the normalizer of n*. q.e.d. 5.4. Now let l be the rank of g and let r (5.4.1) and

dim g

= dim m so that

= l + 2r

dim b = l + r . It follows from (5.4.1) that dim b n b* ~ l. But since b n b* is a Lie subalgebra which is both reductive in g and solvable, it follows that it is commutative. Hence b n b* is a Cartan subalgebra of g. From this point we fix the subalgebra 1) of § 5.1 so that 1) = b n b* • It follows then that e; is a root vector for -qJ. Hence by (5.1.2) we may choose the root vectors e", so that in addition to (5.1.2) they form an orthonormal basis of 1).L. It is immediate that this is equivalent to (5.1.2) and the condition (5.4.2) for all qJ e ~. Now put ~+ = ~(m) and ~_ = -~(m). One knows then that (a) ~ = ~+ U ~_ is a disjoint union and (b) ~+ (and hence ~_) is closed under +. Let II ~ ~+ be the set of simple roots corresponding to ~+. For any

"1 2 _ I g 1:10 on Ag ® V. Thus since X U2 e g, we can apply (5.7.2) once more and obtain (5.7.3)

L"

= ~(I g + >"1 2- I g 12)1 -

(;3(x U2 )

+ !RIl)

where 1, here, denotes the identity operator on C(n, VA). But XU2 lies in the center of g, by Lemma 5.5. Hence L" reduces to a scalar on C(n, yAy. To determine the scalar it suffices to compute L" on a highest weight vector p e C(n, VA) End H(n, VA) of g, on the cohomology space H(n, VA). On the other hand it is obvious that 13 is equivalent to the sub-representation of ;3 defined by Ker L". But then since L" is positive semi-definite we obtain, immediately, the following corollary of Theorem 5.7. 5.7. Let t; e D,. Then if the multiplicity of vi in ;3 is positive one must have COROLLARY

Ig+>"I~lg+t;I·

Furthermore if I g + >"1 > I g + t; I then the multiplicity of vi in 13 = 0' and if I g + >..1 = I g + t; I then the multiplicity of vi in 13 = multiplicity' of vi in ;3.

5.7. (A) Another way of expressing the statement in Corollary 5.7 is as

REMARK

231

GENERALIZED BOREL-WElL THEOREM

355

"*

0, then p is a cocycle which is not cohomolofollows. If P E C(n, Vh) I g + ~ I. At a later point we will make important (for us) use of the following fact (contained implicitly in Corollary 5.7). (B) Every irreducible component of jJ is inequivalent to any irreducible -component of the sub-representation of f3 defined by 1m d n • 5.8. Let z+ ~ Z be the semi-group generated by ~+. Writing an element

Y' E Z as a linear combination of simple roots it is clear that Z+ can be characterized by (5.8.1)

Z+

= {'o/ E

Z I (p,

'0/)

~

0 for all P E D} .

Now let)., E D and let ~h denote the set of weights of the irreducible representation ).Ih of g. One knows that if P E Z then a necessary condition for P E a h is that (5.8.2) 'The following lemma is a consequence of this fact. LEMMA 5.8. Let).,H).,2 E D. Let PI

E ~hl,

P2 E

~h2.

Then

(5.8.3)

.and equality holds in (5.8.3) if and only if there exists a

W such that

+ P2 . W be such that T(PI + P2) E D. a(A.1

+ ).,2) =

E

PI

PROOF. Let T E For i = 1, 2, put "fr, = A., - Tp,. Since TP, E ~h£ it follows then from (5.8.2) that "fr, E Z+ and hence "fr E Z+ where "fr = "fr1 + '0/2. Now put P = TPI + TP2 so that P s D. But then ~ + ).,2 = P + "fr. Consequently, since Ipi = I PI + P21, one has

I ).,1 + ).,21 2 = I PI + P2 12 + I"fr 12 + 2(p, "fr) . But by (5.8.1) (p, '0/) ~ O. This proves the inequality (5.8.3). Furthermore if equality holds in (5.8.3) then obviously "fr=0. But since "fr="fr1 +"fr2 and '0/17 "fr2 E Z+, it follows that '0/1 = "fr2 = O. That is, )." = TP .. i = 1, 2. The lemma follows in one direction by putting a = T- 1 • The other direction is obvious. q.e.d. REMARK 5.8. Let the notation be as in Lemma 5.8. Let a E W. Then the proof of Lemma 5.8 also yields the statement (by putting T = a-I) that a(A.I + ).,2) = PI + P2 implies a).,l = PI and a).,2 = 11..

"*

5.9. We recall that an element P E Z is called regular if (p, rp) 0 for all rp E~. One knows that P E Z is regular if and only if ap = p, a E W,

232

356

BERTRAM KOSTANT

implies (] is the identity element of W. We recall that g € D and that g is regular. In fact both of these statements are consequences of the well known relation (5.9.1)

=

(g, a)

(a, a) 2

for any a € II. One obtains (5.9.1) from the easily verified fact that a is the only root in ~+ which "changes sign" under Ta,. That is, T",~_

Consequently REMARK

T",

n ~+ =

(a) .

g = g - a. But by (5.5.9) this is equivalent to (5.9.1).

5.9. Freudenthal has proved (see e.g., [6, 6.1])

(5.9.2)

*

for any A. € D and any P € ~ \ P A.. We observe that, since g is regular and g E D, (5.9.2) follows from Lemma 5.8 by putting A.I = PI = g. We now wish to consider the irreducible representation ),IU of g whosehighest weight is g. Weyl has given a formula for the dimension of a representation in terms of its highest weight. Weyl's formula asserts. that for any A. € D (5.9.3)

This formula generally proves to be quite awkward for computational purposes. However in the special case when A. = g we observe that (5.9.3} immediately yields (5.9.4) where r (= dim m) is the number of roots in ~+. We wish to determine the weights of ),Ig and their multiplicities. For any subset ~ ~+ let where CI> ~ ~+ and that the multiplicity of f is equal to 2[1/ 2] times the number of subsets CI> ~ .6.+ such that

f =

g - (4;>.

In particular we note that g is a weight of v 0 B and that its multiplicity is at least 2[1/2]. On the other hand we now observe that every weight vector corresponding to g is necessarily a highest weight vector. To prove this, it suffices to note that if rp e ~+ then g + cp is not a weight of voB. Indeed if it were we would have g + rp = g - (CI» or rp + (CI» = 0 for a subset

234

358

BERTRAM KOSTANT

~ A+. But this is impossible since Z+

the multiplicity of

J.)g

in u 0 0 is at least

n-

2[1/2].

= O.

This proves that But from the identity Z+

it follows from (5.9.4) that J.)g occurs exactly 2[1/2] times in u 0 0 and that no other irreducible representation of 9 occurs in u 0 O. The lemma then follows from the statement above concerning the weights of u 0 O. q.e.d. 5.10. Let a E W. Define the subset " "

=

aA_

~ A+

n A+

by putting

•

It then follows at once that

(5.10.1)

ag

=g

- " of W is a bi;"ectionof Wonto the family of all subsets of A+ which satisfy the condition that and its complement C in A+ are both closed under PROOF. Since g is a regular element of 1)' (see (5.9.1» it follows immediately from (5.10.1) that the mapping a . . . . . " is an injection. But since the Now by definition it is obvious that " is closed under complement of " in A+ is equal to aA+ n A+ = a(ICA_) n A+ = w it follows that it, too, is closed under +. Conversely assume that ~ A+ and its complement C in A+ are both closed under Put

+.

+.

+.

Ao

=

U -(C) •

Obviously A = Ao U - Ao is a disjoint union. On the other hand it is straightforward to verify that Ao is closed under +. Hence, as noted above,

235

359

GENERALIZED BOREL-WElL THEOREM

aA_ for some unique a € W. But then obviously

a 2 € WI. Let T1

Remark 5.5, A(n) is stable under

=

T1"

a 1a;;1 and assume T1 € WI. Then by

But this clearly implies

= 0"-1 On the other hand the inverse a 2a l 1 also lies in WI. Thus 0"-1 2 1 which, by Proposition 5.10 implies a 1 = a2 • Thus no two distinct elements of WI lie in the same right coset of WI. Now let T € W be arbitrary. Let 1 = T(A_) n A(m1) and let 2 be the complement of 1 in A(m1). Then 2 = T(A+) n A(m1) so that both 1 and 2 are closed under Now apply Proposition 5.10 to the case where [gl> g1]' the maximal semi-simple ideal of gl' is substituted for g. It follows then that there exists T1 € WI such that (since A(n*) is stable under T1)

+.

'1

=

1 •

Now put a = TIlT. It is then straightforward to verify a(A_) n A(m1) is empty so that a € WI. q.e.d. It is implicit in the proof above that if T = T 1a is the decomposition given by Proposition 5.13 then (5.13.2) is a disjoint union; the components on the right being also the respective intersections of , with A(m1) and A(n). Now for any a € W put (5.13.3)

n(a)

=

Since, obviously, (5.13.4) note that (5.13.5)

number of roots in 0" •

n(a) = n(a- 1)

•

Furthermore if T € Wand T = Ti a is the decomposition given by Proposition 5.13, then it follows from (5.13.2) that (5.13.6)

n(T)

= n(T1) + n(a)

.

REMARK 5.13. Let T € W. We note as a consequence of (5.13.6) that the unique element a € WI in the right coset WIT can be characterized by

238

362

BERTRAM KOSTANT

the statement that n(a) ~ nCr') for all r' e Wr and that equality holds if and only if r' = a. Using (5.13.5) it follows that a similar statement involving the set {a-I}, a e WI, can be made for the left co sets of WI' 5.14. Now for any non-negative integer j put

W(j) = {a e WI n(a) = j} and let

WI(j) = W(j) n WI . Also "let {e~..,}, qJ ~ d(n), be the basis of An' dual to the basis {e..,}, qJ ~ d(n), of An so that by (5.11.1) and (3.2.1) (5.14.1) r;(e~..,) = e_'l> . We can now state THEOREM 5.14. Let u be any Lie subalgebra of 9 which contains the maximal solvable Lie subalgebra {) of 9. Let n be the maximal nilpotent ideal of u (see Proposition 5.3) and let 91 = un u* so that 91 is a reductive (in 9) Lie subalgebra and u = 91 + n is a semi-direct sum (as Lie algebras). Let )., e D and let ).IA be the irreducible representation of 9 on a vector space VA whose highest weight is ).,. Let H(n, VA) be the cohomology group formed with respect to the representation n = ).IA In of n on VA and let lJ be the representation of 91 on H(n, VA) defined as in § 5.7. Now for any [; e DI let H(n, vAy be the space of all classes in H(n, VA) which transform under lJ according to the irreducible representation ).I~ of 91 whose highest weight is [;. Now for any a e W let [;.,. be defined by [;.,. = a(g + ).,) - g .

"*

Then if a e WI one has [;.,. e Dl and for any [; e DIone has H(n, VA)' 0 if and only if [; = [;.,. for some a e WI. Furthermore H(n, VAY.,. is irreducible for all a e WI so that a -> H(n, VAY.,. is a bijection of WI onto the set of all irreducible (under lJ) components of H(n, VA). Moreover degree-wise, for any non-negative integer j Hj(n, VA) = E"'EW11j)H(n, VAY.,. (direct sum) so that for any a e WI, the elements of H(n, VA)'''' are homogeneous of degree n(a). Finally if S"'A e VA is the weight vector for the extremal weight a)., of).lA then the highest weight vector in H(n, VA)'.,. is the cohomology class having e~'l>.,.

®

s"'A

as a representative (harmonic) cocycle.

239

GENERALIZED BOREL-WElL THEOREM

363

*

PROOF. Now by Corollary 5.7 H(n, VA)< 0 if and only if g is a highest weight of an irreducible component of (3 and (5.14.1) Moreover in such a case the multiplicity of ).I; in (3 is the same as its multiplicity in l3. But now the representation (311) of 1) is obviously equivalent to the sub-representation of t (see § 5.11) of 1) defined by the subspace An* Q9 Vof Am* Q9 V. But then by Lemma 5.12 the only weights of (3 which satisfy (5.14.1) are the weights gcr for a e W' and they occur with multiplicity one. Therefore to prove the theorem up to the statement "Moreover ..• ", it suffices only to show that the tT occur as highest weights in the decomposition of (3. But to prove this it is enough to show, for any cp e .1.(m,), a e WI, that gCT + cp is not a weight of (3. Put g = gCT + cpo Then 1g

+ g 12 =

1a(g

+ >v) +

cp 12

=

1g

+ >v 12 +

2(a(g

+

>V), cp)

+

1cp 12 •

But now by Remark 5.13 (3), a(g + >V) e D, so that (a(g + >V), cp) ~ o. But then 1g + g 1 > 1g + >v I. By Lemma 5.12 this implies g is not a weight of t and a fortiori g is not a weight of (3. Now by Lemma 5.12, e'-IP CT Q9 SCTA is the unique (up to scalar multiple) weight vector for the weight gCT of (3. But from above it must be the highest weight vector of an irreducible component of (3. Hence by Theorem 5.7, e'-IPCTQ9sCTA is a harmonic cocycle (element of Ker L~). But then, clearly, its cohomology class is the highest weight vector in H(n, VYCT. Now this class is obviously homogeneous of degreen(a). SinceH(n, VA) End

(C~(K)

® V

I -,)

is the representation defined by taking the tensor product of).lR I n and the trivial representation, and if (3R: 91 -> End C(ll, C~(K) ® VI') is the representation of 91 on the cochain complex C-< = C(u, C~(K)® VI-i) (formed with respect to 7r:R ) defined in the same way as (3 of § 5.7 (except that).ll replaces).l>" I 91) then, more generally for any j, (CJ,~,)O is canonically isomorphic to the space CO,l(X, E~') of all C ~ differential forms of type (O,}) on X with values in E~'. Here (Cl,~')O is the space of all homogeneous elements of degree j in C~, which transform under (3R according to the zero representation of 91' (We say more generally since if j = 0, this statement is identical with the one made above concerning (C~(K) ® V,-7). Moreover if dR: (Cl,~')O -> (CJ 11,~ End

Nu'

be the representation of Uon Nu' formed by takingthej'h exterior product of the representation defined by (7.1.1). Now for any a e U put x~jl(a)

= trace (3t(a) ,

and let

E7=o( -1)J x6 jl(a) .

xo(a) =

One, of course, knows that (7.1.2)

Xo(a)

Now for any subset U'

= det (1 -

~

R(U')

(3~(a») •

U, let R( U') ~ U' be defined by

=

{a e U' I Xo(a)

"* O} •

Note that, by (7.1.2), R( U) is the set of all a e U such that (3~(a) has no non-zero fixed vectors. 7.1. Although we make no use of the fact, itcan be easily shown that if R( U) is not empty then u is necessarily a nilpotent Lie algebra. REMARK

7.2. Now let I):

U-End V

be a representation of U on Vand, for any a e U, let x>(a) = trace I)(a) .

Our intention now is to give a formula for the character homology groups defined by u. Let (3J: U -> End

til involving co-

Nu' ® V

be the tensor product of the representations (3t and character of (3J one obviously has

I).

Thus if

t

Jl

is the

(7.2.1)

Let 'lC = j.) I u. Then we recall that Au' ® V is the underlying space of the cochain complex C(u, V) defined by 'lC. Furthermore if d n is the corresponding coboundary operator, then it follows easily that for any a e U (7.2.2)

on CJ(u, V). Since (7.2.2) holds also for j - 1, (3J induces a representation

249

GENERALIZED BOREL-WElL THEOREM

373

'$J: U --> End HJ(n, V)

of U on the cohomology group HJ(n, V). Now let X(J) be the character of '$J and put

X=

E (-l)JX(J)

x=

E(-l)Jt i ).

•

Similarly put It is then a simple and well known fact (the Euler-Poincare principle) using (7.2.2) that for any a e U

(7.2.3)

x(a)

=

x(a) .

Let Xoequal Xfor the case when the identity representation is substituted for 1). It follows then from (7.2.3) that also

(7.2.4)

Xo(a)

= Xo(a) .

But now taking the alternating sum with the expressions in (7.2.1) as summands one obtains (7.2.5)

X"Xo = X •

We have proved, using (7.2.3), (7.2.4) and (7.2.5). PROPOSITION 7.2. Let 1) be a representation of U on a vector space V and X' be its character. Let n be the Lie algebra of a normal Lie subgroup of U and let X(resp. Xo) be the alternating sum of the characters of the representations 13i (resp. 130 of U on HJ(n, V) (resp. HJ(n». Let R(U) be the set of all a e U which, (see Remark 7.1) under the representation of U on n induced by con}ugacy, correspond to operators on n without non-zero fixed vectors. Then if a e R( U) one has Xo(a) =1= 0 and

X'(a) If a ¢ R( U) one has Xo(a)

= ~(a) . Xo(a)

= x(a) = O.

7.3. Let u e V (see § 5.2). We apply Proposition 7.2 to the case where U is the subgroup of G corresponding to u and n (= un) is the maximal nilpotent ideal of u. Also let 1) = 1)" I Uwhere ~ e D so that V = V". Now if ~ e D1let xi be the character of the representation 1)i of Gl' Then if, as in §7.2, XU) is the character of the representation j3i of U on H(n, V") it follows from Theorem 5.14 that for any a e G1 ~ U, (7.3.1)

250

BERTRAM KOSTA NT

374

where, we recall, ~CT = a(g + :\,) - g and WI(j) is given by (§ 5.14). For any a e W let sga, as usual denote the determinant of n. If n(n) is defined by (5.13.3) it is then well known that (7.3.2)

sgn = (_l)n(CT) •

(In fact sjnce there are obviouslyn(n) root "walls" separating, for example, = -1 for any cp e d). But now Proposition 7.2, (7.3.1) and (7.3.2) yield

g and ng, (7.3.2) follows from the fact that SgTg>

PROPOSITION 7.3. Let:\' e D and let X~ be the character of the ir1'educible representation lJ~ of G. Let GI and n be defined as in § 5.3 and let R(GI ) be the set of all a e GI such that O(a)z = z, for zen implies z = O. Here e denotes the ad}oint representation of G on g. Then for any a € R(G1) ~() a

(7.3.3)

=

X

ECTEwlSg nXf(UHH(a) ECTEWIsgnXf(U)-U(a) ,

where for any ~ e DH Xi is the character of the irreducible representation of GI and WI is given by (5.13.1).

lJ;

7.4. Now consider the special case of Proposition 7.3 where u = 0 so that n = m and GI = H where H ~ G is the (Cartan) subgroup corresponding to 'fJ. In this case DI = Z and WI = W. Furthermore if a e H then writing a = exp x for x € 'fJ one has, for any ~ e Z xi(a)

= e«'x)

•

Moreover R(H) is the set of all elements in H that are regular in G. Multiplying numerator and denominator of (7.3.3) by e(g·3) one obtains, as an immediate corollary to the proposition above, PROPOSITION 7.4. (Weyl's character formula). Let Xl. be the character of the representation lJ h • Let a e H be regular in G. Then writing a = exp x one has ~(a)

X

-

"

sgae',

252

))1

is dominant if A. e D

376

BERTRAM KOSTANT

Now let A be an index set for the equivalence classes of all dominant irreducible representations of H+. Now just as the elements of D index both the classes of dominant representations of H and all representations of G we now observe that A is an index set for the classes of all irreducible representations of G+. That is, to each 0 e A there exists a unique (up to equivalence) irreducible representation such that if and }.if: H+ -> End

VIS

is the representation defined by restricting }.is I HI- to V,s (obviously a stable subspace), then}.if is a dominant irreducible representation of H t- belonging to the equivalence class corresponding to o. Furthermore every irreducible representation }.i of G+ is equivalent to }.is for some, necessarily unique, o e A. The proof of the statements above proceeds in the same way as in the classical situation as soon as one observes that G+ = HIG. where G" is the subgroup of corresponding to g. Now let a e H+ and let a e W. Since W is normal in C we can let a' e W defined by the relation

G:

r(a)a = a'r(a) .

Recalling that by definition CP.,. = some scalar x~(a),

a(~_)

n~

I'

we then observe that for

(7.5.4) Similarly if V; ~ VS is defined in the same way as VIS except that replaces ~I we observe that

a~

t-

(7.5.5) It follows therefore that if H: is the subgroup of H+ defined by

H:

=

{a e H+ Jr(a) commutes with a} ,

then V; is stable under }.isJ H: and hence defines a representation }.i! of H:. Let X! be the character of }.i!. REMARK 7.5. If xf is the character of }.if note that xf determines X! for any a e W. In fact if b(a) e Ge is any element which induces, by conjugation, the transformation a on 1) observe that

253

GENERALIZED BOREL-WElL THEOREM x~(a)

377

= xHb(a)ab(a)-l)

for any a € H:. The element b(a) is needed since a itself does not in general operate on H~-. Finally put (7.5.6)

H:.

for any a € An element a € G+ is called regular if the rank of 8(a) - 1 is minimum in the connected component of G+ containing a. In case gl = 9 this definition is the same as that given by Gantmacher, [8, pp. 112, 119]. Since 8(a) I c = 8(b) I c for a, b € G+ lying in the same connected component we may apply the results of [8] to the case at hand. In particular, it follows then from Theorems 12, 23 and 29 in [8] that every regular element is conjugate to an element in H+ and that a € H+ is regular if and only if the kernel of 8(a) - 1 lies in 1)+, the Lie algebra of H+. But the latter clearly implies that a € H+ is regular if and only if 8(a) has no fixed vectors in m. Thus if we apply the considerations of §7.2 to the case where Uis the normalizer in G+ of the subgroup of G. corresponding to m and n = m, it follows that R(H+) is the set of all elements in H+ that are regular in G I. Applying Proposition 7.2 where ).1=).181 U, we obtain the following generalization of Weyl's character formula. THEOREM 7.5. Let g+ be any reductive Lie algebra and let G t be any Lie group (not necessarily connected) whose Lie algebra is gl-. We may assume that 9 is the maximal semi-simple ideal in g+. Now let H+ be defined by (7.5.1) so that there is a one-one relation between all dominant irreducible representations of H+ (indexed by A) and all irreducible representations of G+. Let 8 € A and let be the character of the irreducible representation ).18 defined above. Let a € H+ be regular in G+ (every regular element of Gt is conjugate to an element in H+) and let Wa be the subgroup of W consisting of all a € W which commute with -r(a) (see (7.5.2». Then where Xf(a) and Xf,8(a) are given respectively by (7.5.4) and (7.5.6) one has

t

x 8(a) =

EuewasgaXf,B(a) EuewasgaXf(a)

PROOF. Define H(m, VB) with respect to the representation 7'C = ).18 I m. By decomposing VB into irreducible components under the action of ).181 g, it follows from Corollary 5.15, that the space of cochains (e'--q,.,.)® consists (except for zero) of non-cobounding cocycles and if (e'--q,.,.) ® V:) denotes the corresponding space of cohomology classes, one has the direct sum

V:

254

378

BERTRAM KOSTANT

H(m, VB) = EUEW(e~~) ® V:) .

We have now only to apply Proposition 7.2, (7.5.4) and (7.5.5).

q.e.d.

8. Application III. Symmetric complex spaces X and a generalization of a theorem of Ehresmann

1. Let u € V and let 91 and n be defined as in § 5.3. We continue with the notation of § 5 except that now it is assumed that A, = O. Thus (3 is a representation of 91 on An' and $ is the induced representation of 91 on H(n).

Now let (3* be the representation of 91 on An defined by restricting e I91 to An. Thus (3* is the representation contragredient to (3. Since (3* obviously commutes with the boundary operator aon An it defines a representation

$*:

9 -. End H*(n)

on 91 on the homology group H*(n). It is of course clear that, with respect to the canonical duality between H*(n) and H(n), $* is just the representation contragredient to $. Applying Theorem 5.14, one then immediately obtains COROLLARY 8.1. Let u € V and let $ * be the representation of 91 on the homology group H*(n) defined above. For any g € -Dl let H*(nY be the set of all elements in H*(n) which transform under $* according to the irreducible representation (with lowest weight g) )..Ii of 91' Then for any a € WI one has g - ag € - Dl and for any g € - Dl one has H*(n)< 1= 0 if and only if g = g - ag for some a € Wl. Furthermore H*(n)g-U g is irreducible for all a € Wl so that a -. HAn)U- ug is a bijection of WI onto the set of all irreducible (under $*) components of H*(n). Moreover, degree-wise, for any non-negative integer j,

Hj(n)

=

EuEWI{j)

g H*(n)g-U ,

so that the elements of H*(n)g-U g are homogeneous of degree n(a). Finally the lowest weight vector of H*(n)g-U g is the homology class having e~" as a representative cycle. 8.2. We consider the cases (u € V) when n is commutative. Let n(u) ~ n be defined as in § 5.4 and for any cp € a let the integer n ..(cp), a € n, be defined also in § 5.4. It is then asserted that n is commutative if and only if for every cp E a(n) (8.2.1) Indeed since

E"ETI(U)

~(n)

n",(cp) ~ 1 .

is precisely the set of all cp

255

€ ~

such that the left hand

GENERALIZED BOREL-WElL THEOREM

379

sum of (8.2.1) is ~ 1, it follows that the condition (8.2.1) implies that 11 is commutative. On the other hand if there exists a root such that the left hand sum of (8.2.1) is ~ 2 then since m is generated by the e.. , a € 11, it follows that there exists 1J € A(n) and a € ll(u) such that 1J + a € A. But Rince 1J, a, 1J + a € A(n), this implies that 11 is not commutative. This proves the assertion. An immediate consequence of this and symmetric space theory is PROPOSITION 8.2. Let u € CU and, as in § 5.2, let X = G/ U so that X is a complex compact homogeneous space. Then X is also a symmetric space in the sense of E. Cartan if and only if n, the maximal nilpotent ideal of It, is commutative. PROOF. It is immediate that the condition (8.2.1)is satisfied if and only if no two elements of l1(u) lie in the same connected component (in the sense of Dynkin) of 11; and for any a e l1(u), one has n ..(1J) ~ 1 for all 1J e A. But then the result follows from the structure theory of complex, compact, symmetric spaces (see e.g. [3, 40, p. 260]). q.e.d. But now if n is commutative, the boundary operator on An is zero. Thus H(n) = An. Hence in the symmetric case Corollary 8.1 yields Corollary 8.2 below describing how An decomposes under the action of gl' Corollary 8.2 contains, as a special case, results of Ehresmann asserting how A11 decomposes when X is symmetric and G is a classical group. We will work out the case when X is the grassmannian in § 8.6. COROLLARY 8.2. Let u € CU. Assume that X = G/ U is a symmetric space. Let n be the maximal nilpotent ideal of u and let 13* be the representation of gl on An obtained by restricting e I gl to An. (Recall that e is the adioint representation of 9 on Ag). Now let WI be the subset of the Weyl group defined as in § 5.13. Then for any a e W\ one has g - ag € - Dl and for any ~ e - DII the irreducible representation ).If of fh occurs in 13* if and only if ~ = g - ag for some a € WI. Furthermore if a € W\ then ).If- j}

(super-triangular matrices) and the Cartan subalgebra f) of !1 so that f) is the set of all diagonal matrices in g. The corresponding roots are then canonically indexed by all pairs i, j = 1,2, ... , m, i =f=. j, where qJ,} e A is given by (8.3.1) for any x e f) and the corresponding root vectors may be chosen so that e",,} =

V

1

2m e'J

where ei ; is the usual matrix unit. REMARK 8.3. The coefficient (l/V 2m) is necessary to insure the relation (5.1.2). It also insures (5.4.2) where f is chosen to be the set of all skewhermitian matrices in g. Since (r be the irreducible representation of au = GS x G' {)n ViL(j" 0 ViL(r 2 ) given by

It is clear then that every irreducible representation of Gs,t is equivalent to J.ir for some r € Zm and in fact r is uniquely chosen if one insists that r' € DS and r 2 € D'. Now the adjoint representation of AS" on n (see 8.5.1) extends in the usual way to a representation fl •. ,: g8,' ~ End

An

()f gs.t on An. We observe that the representation fls., is obtained as an -extension of the representation fl* of g, on An by defining fl •. ,(1"') = o. We wish to decompose the representation fls., into irreducible components. We first observe, however, that if reD' then-K"r € D' and J.>-K"r is equivalent to the representation contragredient to the representation v r of g'. This is clear from the definition of K". It follows easily therefore from (8.5.1) and Remark 8.4 that any irreducible component of flu is ()f the form J.>t' ,-K"r2) where r' € D~ and r 2 € Dt. Now let Q'" be the set of all pairs (p" p2) where p' € P' and p2 € P' is such that J.>t',-K"r 2) occurs in the complete decomposition of fl •. , if r'=rA(p') a.nd r2 = r'(p2). From the remark above we see that every irreducible

262

386

BERTRAM KOSTANT

component of (3s,t is of this form so that (38,t is determined as soon as the elements of Q8.t are known together with the corresponding multiplicities. The following theorem is due to Ehresmann. See [5, § 5]. THEOREM 8.6. Let Qs.t be the set of pairs of partitions defined above describing the decomposition of the representation (3 s,t of gS E9 gt (gil: is the Lie algebra of all k x k complex matrices) on An where n is isomorphic to the space of all s x t complex matrices. Let WI be defined as in § 5.13 so that here WI is the set of permutations n € W satisfying (8.5.2). If n € WI, let pO' € ps be the partition defined. by rS(pO') = (n-l(s) - s, ... , n-l (l) - 1) . Then n(pO') = n(n)

(8.6.1)

where the left and right sides of (8.6.1) are defined respectively as in § 8.4 and by (5.13.3). Furthermore n --> pfT is a bijection of WI onto the set of all partitions p such that m(p)

~

sand m(p)

~

t

where p is the conjugate partition (that is, the set of all partitions whose Young diagram (block representation) "fits" into an s x t rectangle of blocks). Finally Qs,t is the set of all pairs (pfT, pfT) where n runs through W]. Moreover the irreducible representation of g8,t corresponding to any pair (pfT, pfT) occurs with multiplicity one and the representation induced on gl is lJ~l(g-fTg). Moreover the space of the representation consists of homogeneous n(pfT) vectors and a highest weight vector of lJ K1 (g- O' U) is (in any order)

where II denotes exterior multiplication. PROOF. The equality n(pO') = n(n) follows from (8.5.11) and the other statements about pfT follow from Remark 8.5. To prove the theorem therefore we have only to apply Corollary 8.2 and Remark 8.2, and to determine the element of Qs,t corresponding to representation lJ~l(g-O'U) of gl on the subspace (/\ nYI(g-fT g) of N(fT) n. That is we must find the pair (pI, p2) € QU such that (1) p(r S(pl), -1C"rt(p~» = 1C1(g - ng) and (2) n(pl) = n(n) (since (3s,t(Y) must reduce to the scalar n(n) on N(rT)n if y € gS,t is the element such that A 22(y) = 0 and A ll (y) = 18).

263

GENERALIZED BOREL-WElL THEOREM

387

It is easy to see that (1) and (2) define (pt, p2) uniquely. But by (8.5.9), (8.9.10), and the equality (8.6.1), it follows that (pt, p2) = (per, per). The final statement follows from (8.5.11) and Remark 8.2. q.e.d.

REMARK 8.6. Theorem 8.6 lends some insight into the nature of the weight g - ag, at least for the case at hand. The striking thing is that the partitions pI and p2 of the pair (pI, p2) corresponding to the weight g - ag not only determine each other but are related to the extent that -one is the conjugate of the other. Furthermore except for a limitation on .size, the choice of pI can be made arbitrary by choosing a properly. UNIVERSITY OF CALIFORNIA, BERKELEY REFERENCES 1. A. BOREL and F. HIRZEBRUCH, Characteristic classes and homogeneous spaces, I. Amer. J. Math., 80 (1958), 458-538. 2. R. BOTT, Homogeneous vector bundles, Ann. of Math., 66 (1957) 203-248. 3. E. CARTAN, Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math. France, 54 (1926), 214-264. 4. P. CARTIER, Remarks on "Lie algebra cohomology and the generalized Borel-Weil theorem", by B. Kostant, Ann. of Math., 74 (1961), 388-390. 5. C. EHRESMANN, Sur la topologie de certains espaces homogenes, Ann. of Math., 35 (1934), 396-443. 6. H. FREUDENTHAL, Zur Berechnung der Charaktere der halbeinfachen Lieschen Gruppen Neder, Akad. Wetensch. Indag. Math., 57 (1954),369-376. 7. G. HOCHSCHILD and J.-P. SERRE, Cohomology of Lie algebras, Ann. of Math., 57 (1953), 591-603. 8. F. GANTMACHER, Canonical representation of automorphisms of a complex semi-simple Lie group, Mat. Sb., 47 (1939), 101-143. 9. J. L. KOSZUL, Homologie et cohomologie des algebres de Lie, Bull. Soc. Math. France, 78 (1950), 65-127. 10. H. C. WANG, Closed manifolds with homogeneous complex structure, Amer. J. Math .• 76 (1954). 1-32.

264

Reprinted from the TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 102, No. 3, March 1962 Pp. 383-408

DIFFERENTIAL FORMS ON REGULAR AFFINE ALGEBRAS BY

G. HOCHSCHlLD, BERTRAM KOSTANT AND ALEX ROSENBERG(I)

1. Introduction. The formal apparatus of the algebra of differential forms appears as a rather special amalgam of multilinear and homological algebra, which has not been satisfactorily absorbed in the general theory of derived functors. It is our main purpose here to identify the exterior algebra of differential forms as a certain canonical graded algebra based on the Tor functor and to obtain the cohomology of differential forms from the Ext functor of a universal algebra of differential operators similar to the universal enveloping algebra of a Lie algebra. Let K be a field, R a commutative K-algebra, TR the R-module of all K-derivations of R, DR the R-module of the formal K-differentials (see §4) on R. It is an immediate consequence of the definitions that T R may be identified with HomR(D R, R). However, in general, DR is not identifiable with HomR(TR, R). The algebra of the formal differentials is the exterior Ralgebra E(D R) built over the R-module DR. The algebra of the differential forms is the R-algebra HomR(E(TR), R), where E(TR) is the exterior R-algebra built over T R and where the product is the usual "shuffle" product of alternating multilinear maps. The point of departure of our investigation lies in the well-known and elementary observation that TR and DR are naturally isomorphic with Ext1-(R, R) and Torr(R, R), respectively, where R6=R@K R. Moreover, both ExtR.(R, R) and TorR·(R, R) can be equipped in a natural fashion with the structure of a graded skew-commutative R-algebra, and there is a natural duality homomorphism h: Exh.(R, R)~HomRCTorR·(R, R), R), which extends the natural isomorphism of TR onto HomR(D R, R). We concentrate our attention chiefly on a regular affine K-algebra R (d. §2), where K is a perfect field. Our first main result is that then the algebra TorR·CR, R) coincides with the algebra E(D R ) of the formal differentials, Exh.CR, R) coincides with E(TR), and the above duality homomorphism h is an isomorphism dualizing into an isomorphism of the algebra ECD R) of the formal differentials onto the algebra HomR(E(TR), R) of the differential forms. In order to identify the cohomology of differential forms with an Ext functor, we construct a universal "algebra of differential operators," VR, Received by the editors May 5, 1961. (1) Written while B. Kostant was partially supported by Contract AF49(638)-79 and A.

Rosenberg by N.S.F. Grant G-9508.

383 B. Kostant, Collected Papers, DOI 10.1007/b94535_14, © Bertram Kostant 2009

265

384

G. HOCHSCHILD, BERTRAM KOSTANT AND ALEX ROSENBERG

[March

which is the universal associative algebra for the representations of the K-Lie algebra TR on R-modules in which the R-module structure and the TR-module structure are tied together in the natural fashion. After establishing a number of results on the structure and representation theory of V R, we show that, under suitable assumptions on the K-algebra R and, in particular, if R is a regular affine K-algebra where K is a perfect field, the cohomology K-algebra derived from the differential forms may be identified with ExtvR(R, R). In §2, we show that the tensor product of two regular affine algebras over a perfect field is a regular ring, and we prove a similar result for tensor products of fields. §§3, 4 and 5 include, besides the proof of the first main result, a study of the formal properties of the Tor and Ext algebras and the pairing between them, for general commutative algebras. In the remainder of this paper, we deal with the universal algebra V R of differential operators. In particular, we prove an analogue of the Poincare-Birkhoff-Witt Theorem, which is needed for obtaining an explicit projective resolution of R as a VR-module. Also, we discuss the homological dimensions connected with V R. We have had advice from M. Rosenlicht on several points of an algebraic geometric nature, and we take this opportunity to express our thanks to him. 2. Regular rings. Let R be a commutative ring and let P be a prime ideal of R. We denote the corresponding ring of quotients by R p • The elements of Rp are the equivalence classes of the pairs (x, y), where x and yare elements of R, and y does not lie in P, and where two pairs (Xl, YI) and (X2' Y2) are called equivalent if there is an element z in R such that z does not lie in P and Z(XlY2 - X2YI) = o. By the Krull dimension of R is meant the largest non-negative integer k (or 00, if there is no largest one) for which there is a chain of prime ideals, with proper inclusions, PoC ... CPkCR. A Noetherian local ring always has finite Krull dimension, and it is called a regular local ring if its maximal ideal can be generated by k elements, where k is the Krull dimension. A commutative Noetherian ring R with identity element is said to be regular if, for every maximal ideal P of R, the corresponding ring of quotients Rp is a regular local ring [2, §4]. It is well known that a regular local ring is an integrally closed integral domain [14, Cor. 1, p. 302]. It follows that a regular integral domain R is integrally closed; for, if X is an element of the field of quotients of R that is integral over R then xER p , for every maximal ideal P of R, which evidently implies that xER. Let K be a field. By an affine K-algebra is meant an integral domain R containing K and finitely ring-generated over K. An affine K-algebra is Noetherian, and its Krull dimension is equal to the transcendence degree of its field of quotients over K, and the same holds for the Krull dimension of everyone of its rings of quotients with respect to maximal ideals [14, Ch.

VII, §7].

266

19621

DIFFERENTIAL FORMS ON REGULAR AFFINE ALGEBRAS

385

THEOREM 2.1(2). Let K be a perfect field, and let Rand S be regular affine K-algebras. Then R®K S is regular.

Proof. Suppose first that R®K S is an integral domain, and let M be one of its maximal ideals. Put Ml = (M(\R) ®K S+R®K (M(\S). Then Ml is an ideal of R®K S that is contained in M, and we have (R®K S)/M1 = (R/(M(\R)) ®K (S/(M(\S»). Now R/(R(\M) and S/(M(\S) are subrings of (R ® K S) / M containing K. Since (R ® K S) / M is a finite algebraic extension field of K, the same is therefore true for R/(M(\R) and S/(M(\S). Since K is perfect, it follows that we have a direct K-algebra decomposition (R®K S)/M1= U+M/M1. Let z be a representative in R®K S of a nonzero element of U. Then z does not belong to M, and zMCM1. Hence it is clear that M(R®K S)M=M1(R®K S)M. Since R is regular, the maximal ideal (M(\R)RMnR of the local ring RMnR is generated by d R elements, where d R is the degree of transcendence of the field of quotients of Rover K. Similarly, (M(\S)SMns is generated by d s elements, where d s is the degree of transcendence of the quotient field of S over K. These dR+d s elements may be regarded as elements of (R®K S)M and evidently generate the ideal M1(R®K S)M. Hence we conclude that the maximal ideal of (R ® K S) M can be generated by d R+ds elements. Since the degree of transcendence of the quotient field of R®K S over K is equal to dR+d s , this means that (R®K S)M is a regular local ring. Thus R®K S is regular. N ow let us consider the general case. Let Q(R) and Q(S) denote the fields of quotients of Rand S. Let KR and KS be the algebraic closures of K in Q(R) and in Q(S), respectively. Since Rand S are integrally closed, we have KR CR and KS CS. Since Q(R) and Q(S) are finitely generated extension fields of K, so are KR and KS. Thus KR and KS are finite algebraic extensions of K. Let M be a maximal ideal of R ® K S. Since K is perfect, we have a direct K-algebra decomposition KR®KKs= U+M1, where M1=M(\(KR®KKS). Hence we have R ®KS

=R

®KR(KR®KKS) ®KSS

=R

®KB U ®KSS

+ M 2,

where the last sum is a direct K-algebra sum, and M2=R®KB M1®KS SCM. Evidently, U may be identified with a subring of the field (R®K S)/M containing K. Hence U is a finite algebraic extension field of K. Identifying KR and KS with their images in U, we may also regard U as a finite algebraic extension field of KR or KS. Since K is perfect, U is generated by a single element over KR or over KS. The minimum polynomial of this element over KB or over KS remains irreducible in Q(R)[x] or in Q(S)[x], because KR is algebraically closed in Q(R) and KS is algebraically closed in Q(S). Hence (") The referee informs us that this result is an immediate consequence of cohomology results obtained by D. K. Harrison in a paper on Commutative algebras and cohomology, to appear in these Transactions.

267

386 G. HOCHSCHILD. BERTRAM KOSTANT AND ALEX ROSENBERG

[March

R ® KR U and U ® KS S are integral domains. Moreover. by the part of the theorem we have already proved, they are regular. Let T denote the field U®KS Q(S). This is a finitely generated extension field of the perfect field KR. Let (tl, ...• t n ) be a separating transcendence base for T over KR, and put To = KR(tl' ... , t n ). We have Q(R) ®KR T = (Q(R) ®KR To) ®To T, and we may identify Q(R) ®KR To with a subring of Q(R) (h •...• t n ), with (tl' ... , t n ) algebraically free over Q(R). Since KR is algebraically closed in Q(R), it follows that KR(h, ... , t n ) is algebraically closed in Q(R)(lI, ... ,tn ) [6, Lemma 2, p. 83]. Now it follows by the argument we made above thatR ®KR Tis an integral domain,so thatR®KR U®KSS is an integral domain. On the other hand, this is the tensor product, relative to the perfect field U, of the regular affine U-algebras R ® KR U and U ® KS S. Hence we may conclude from what we have already proved that R ® KR U ® KS S is regular. Now consider the direct K-algebra decomposition R ®K S

=

R ®KR U ®Ks S

+M

2•

Since M2 C M, the corresponding projection epimorphism R ® K S -R®KR U®KS S sends the complement of M in R®K S onto the complement of Mrl(R®KR U®KS S) in R®KR U®KS S. Moreover, there is an element z in the complement of M such that ZM2= (0). Hence it is clear that the projection epimorphism yields an isomorphism of (R®K S)M onto the local ring over R ® KR U ® KS S that corresponds to the maximal ideal Mrl(R®KR U®KS S). Hence (R®K S)M is a regular local ring. and Theorem 2.1 is proved. THEOREM 2.2. Let K be an arbitrary field, let F be a finitely and separably generated extension field oj K, and let L be an arbitrary field containing K. Then F®K L is a regular ring.

Proof. It is known that the (homological) algebra dimension dim(F), i.e., the projective dimension of F as an F®K F-module is finite; in fact, it is equal to the transcendence degree of F over K [11, Th. 10]. Since dim(F®K L) =dim(F), where F®K L is regarded as an L-algebra [4, Cor. 7.2, p. 177] we have that dim(F®K L) is finite. Since L is a field, this implies that the global homological dimension d(F®KL) is also finite [4. Prop. 7.6, p. 179]. Since F®K L is a commutative Noetherian ring, we have, for every maximal ideal M of F®K L, d«F®K L)M) ~d(F®K L) [4, Ex. 11, p. 142; 1, Th. 1]. Thus each local ring (F®K L)M is of finite global homological dimension. By a well-known result of Serre's [12, Th. 3], this implies that (F®K L)M is a regular local ring. Hence F®K L is a regular ring. Note. Actually. we shall later appeal only to the following special consequence of Theorem 2.2: let F be a finitely separably generated extension field of K; let J be the kernel of the natural epimorphism F®K F-F; then the

268

1962)

DIFFERENTIAL FORMS oN REGULAR AFFINE ALGEBRAS

387

local ring (F®K F)J is regular. This special result can be proved much more easily and directly along the lines of our proof of Theorem 2.1. On the other hand, Theorem 2.1 can be derived more quickly, though less elementarily, from the result of Serre used above. 3. The Tor-algebra for regular rings. Let Rand S be commutative rings with identity elements, and let cp be a unitary ring epimorphism S----'>R. We regard R as a right or left S-module via cp, in the usual way, and we consider TorS(R, R). Since S is commutative, every left S-module may also be regarded as a right S-module, and we shall do so whenever this is convenient. Let H stand for the homology functor on complexes of S-modules, and let U and V be any two S-module complexes. There is an evident canonical homomorphism of H( U) ® s H( V) into H( U ® s V), which gives rise to an algebra structure on TorS(R, R), as follows. Let X be an S-projective resolution of R. With U = V = R ® S X, the canonical homomorphism becomes a homomorphism TorS(R, R) ®s TorS(R, R)

----'>

H«R ®s X) ®s (R ®s X)).

Evidently, (R®sX) ®s(R®sX) maybe identified with (R®sR) ®s(X®sX), and hence with R®s (X®s X). Now X®s X is an S-projective complex over R®s R=R, whence we have the natural homomorphism H(R ®s (X ®s X))

----'>

TorS(R, R).

Composing this with the homomorphism above, we obtain an S-module homomorphism

Tors (R, R) ®s TorS(R, R)

----'>

TorS(R, R).

m

This is the product of [4, p. 211] and it is independent of the choice of the resolution X. Standard arguments on tensor products of complexes and resolutions show that this product is associative and skew-commutative in the sense that a~ = ( -l)pq~a when a is homogeneous of degree p and ~ is homogeneous of degree q. In principle, this product is a product of S-algebras. However, S operates on TorS(R, R) through cp: S----'>R, and we shall accordingly regard TorS(R, R) as an R-algebra. THEOREM 3.1. Let Sand R be Noetherian commutative rings with identity elements, and let cp be a ring epimorphism S----'>R with kernel I. Assume that R is a regular ring and that, jor every maximal ideal M oj S that contains I, the local ring SM is regular. Then TorS(R, R) is finitely generated and projective as an R-module and is naturally isomorphic with the exterior R-algebra constructed over Torf(R, R).

Proof. Let T denote the tensor algebra constructed over the R-module Torf(R, R), let P denote the kernel of the canonical R-algebra homomorphism

269

388 G. HOCHSCHILD, BERTRAM KOSTANT AND ALEX ROSENBERG [March !/t: T-tTorS(R, R), and put Q=TorS(R, R)N(T). Let U denote the 2-sided ideal of T that is generated by the squares of the elements of Torf(R, R). The last assertion of our theorem means that Q = (0) and P = U. The statement Q= (0) is equivalent to the statement RN®R Q= (0), for all maximal ideals N of R. The statement P = U is equivalent to the statement (P + U) / P = (0) and (P+U)/U=(O), or to the statement RN®R (P+ U)/P= (0) and RN®R(P+ U)/ U = (0), for all maximal ideals N of R. This, in turn, is equivalent to the statement that the images of RN ® RP and RN ® RUin RN ® R(P U) coincide with RN®R (P+ U). Since RN is R-flat, these tensor products may be identified with their canonical images in RN®R T; and RN®R P is thereby identified with the kernel of the homomorphism of RN®R T into RN ®R TorS(R, R) that is induced by!/t. Hence it is clear that the statement Q= (0) and P= U is equivalent to the statement that the homomorphism of RN ® R T into RN ® R TorS(R, R) that is induced by!/t is an epimorphism with kernel RN ® R U, for every maximal ideal N of R. Let M be the maximal ideal of S that contains I and is such that M / I = N. Clearly, the epimorphism cf> induces an epimorphism SM-tRN with kernel ISM in the natural fashion. Now let X be an S-projective resolution of R. Since SM is S-flat, SM®S X is then an SM-projective resolution of SM®S R=SM/ISM=RN. Hence we have

+

TorSM(SM/ISM, SM/IS M) = H«SM/ISM) ®SM (SM ®s X)),

On the other hand,

Since RN is R-flat, we have H(RN®R (R®s X)) =RN®R TorS(R, R). Thus RN®R TorS(R, R) is naturally isomorphic with TorSM(SM/IS M, SM/IS M). Similarly, we see that RN ® R T is naturally isomorphic with the tensor algebra constructed over the RN-module TorfM(SM/IS M, SM/IS M). Moreover, it is easily seen that these isomorphisms transport our homomorphism RN ® R T -tRN®R TorS(R, R) into the canonical homomorphism of the tensor algebra over TorfM(SM/IS M, SM/IS M) into TorSM(SM/IS M, SM/IS M). Each Tor~(R, R) is finitely generated as an S-module, and hence also as an R-module. Hence if we show that RN®R Tor~(R, R) is a free RN-module, for every maximal ideal N of R, we shall be able to conclude from a standard result [4, Ex. 11, p. 142] that Tor~(R, R) is a finitely generated projective R-module. In particular, if Torf(R, R) is a finitely generated projective Rmodule, we imbed it as a direct R-module summand in a finitely generated free R-module to show that the exterior algebra constructed over it has nonzero components only up to a certain degree and is a finitely generated projective R-module. From this preparation, it is clear that it suffices to adduce the following

270

1962]

DIFFERENTIAL FORMS ON REGULAR AFFINE ALGEBRAS

389

result (3): let L ( = S M) be a regular local ring and let J ( = ISM) be a prime ideal of L such that the local ring L/ J is regular. Then Torf(L/ J, L/ J) is a finitely generated free L/ J-module, and TorL(L/ J, L/ J) is naturally isomorphic, as an L/ J-algebra, with the exterior algebra constructed over Torf(L/ J, L/ J). To prove this, note first that the assumptions imply that the ideal J can be generated by an L-sequence (aI, ... , aj) of elements of L, i.e., by a system with the property that each ak is not a zero-divisor mod the ideal generated by aI, ... , ak-l [14, Th. 26, p. 303 and Cor. 1, p. 302]. If X is the Koszul resolution of L/ J as an L-module [4, pp. 151-153], constructed with the use of this L-sequence, then X has the structure of an exterior L-algebra over a free L-module of rank j, this algebra structure being compatible with the boundary map, so that it induces the algebra structure on TorL(L/ J, L/ J) via (L/ J) ® LX. Moreover, the boundary map on (L/ J) ® L X is the zero map. Hence it follows immediately that TorfcL/ J, L/ J) is a free L/ J-module of rank j and that TorL(L/ J, L/ J) is the exterior algebra over this module. This completes the proof of Theorem 3.1. 4. Duality between Tor and Ext. Let Rand S be commutative rings with identity elements, and let cp be a ring epimorphism of S onto R. As before, all R-modules are regarded as S-modules via cpo Let X be an S-projective resolution of R, and let A be an R-module. Then Exts(R, A) =H(Homs(X, A». Clearly, we may identify Homs(X, A) with HomR(R®s X, A), so that we may write Exts(R, A) =H(HomR(R®s X, A». Now there is a natural map (a specialization of [4, p. 119, last line]) h: H(HomR(R ®s X, A»

~

HomR(TorS(R, R), A)

defined as follows. Let p be an element of H(HomR(R®s X, A». Then p is represented by an element uEHomR(R®s X, A) that annihilates d(R®s X), where d is the boundary map in the complex R ® s X. Hence, by restriction to the cycles of R®s X, u yields an element of HomR(TorS(R, R), A), and it is seen immediately that this element depends only on p and not on the particular choice of the representative u. Now h(p) is defined to be this element of HomR(TorS(R, R), A). Clearly, h is an R-module homomorphism of Exts(R, A) into HomR(TorS(R, R), A). In degree 0, we have Torg(R, R) =R®s R=R, and Ext~(R, A) = Homs(R, A) = HomR(R, A), and this last identification transports h into the identity map. Thus h is an isomorphism in degree o. Note that Torg(R, R) =R is projective as an R-module, whence the following lemma implies, in particular, that h is an isomorphism also in degree 1. LEMMA 4.1. Let cp: S~R be an epimorphism of commutative rings with identity elements, and regard R-modules as S-modules via cpo Let A be an R-

(3) This is a special case of [13, Th. 4, etc.], which gave the suggestion for our proof of Theorem 3.1.

271

390

G. HOCHSCHILD, BERTRAM KOSTANT AND ALEX ROSENBERG

[March

module, and let k be a positive integer. Assume that Tor~(R, R) is R-projective for aU i < k. Then the map S

i

hi: Exts(R, A) ---+ HomR(Tor. (R, R), A),

obtained by restriction of the map h defined above, is an isomorphism, for all i

~ k.

Proof. Let Zi denote the kernel of d in R®s Xi, and put Bi=d(R®s X i +I), C.=R®s Xi. We have Zo= Co. Suppose that we have already shown, for some i vl8> (wa) , so that SI8>R Sis S-projective whenever R is K-projective. Moreover, S is R-projective as a left or right R-module, so that X is an R-projective resolution of R. Hence H(X 18>R X) = TorR(R, R) and therefore has its components of positive degree equal to 0, so that X 18> R X is still an S-projective resolution of R. For two sided R-modules U and V, regard Homs( U, V) as a two sided R-module such that (r -J) (u) = r -J(u)( = J(r· u» and (J·r)(u) = J(u) ·r( = J(u·r». Now the standard S-module homomorphism

+

1/;: Homs(X, A) 18>R Homs(X, B) ---) Homs(X 18>R X, A 18>R B),

where 1/;(f 18> g)(u 18> v) =J(u) 18>g(v) , induces an S-module homomorphi!5m Exts(R, A) 18> R Exts(R, B) ---) Exts(R, A 18> R B).

This is the product V, as given in [4, Ex. 2, p. 229], and it is independent of the choice of the resolution X. In particular, for A =B =R, this defines the structure of an associative and skew-commutative R-algebra on Exts(R, R). In order to make the algebra structures on TorS(R, R) and Exts(R, R) explicit, we use the following well-known resolution Y of R as an S-module. We put Yo=S and we let cf>: S---)R be the augmentation. Generally, let Y" be

274

1962]

DIFFERENTIAL FORMS ON REGULAR AFFINE ALGEBRAS

393

the tensor product, relative to K, of n+2 copies of R. The S-module structure of Y n is defined so that (a ® b)· (xo ® ... ® Xn+l) = (axo) ® Xl ® ... ® Xn ® (xnHb).

The boundary map don Y is given by n

d(xo ® ... ® Xn+l) =

L (-l) ixo ®

... ® (X;Xi-I-l) ® ... ® Xn+l .

• =0

This complex is not only acyclic but it has actually a right R-module homotopy h, where h(xl ® ... ®x n) = 1 ®XI ® ... ®xn. Since R is K-projective, it follows as above for X ®R X that Y is S-projective. Thus Y is an S-projective resolution of R. The complex Y can be given the structure of an associative skew-commutative S-algebra with respect to which d is an antiderivation, as follows. If Xl, ... , Xp and YI, •.. , yq are elements of R let [Xl, ... , Xp; Yl, •.. , Yq] stand for the sum, in the tensor product over K of p+q copies of R, of all terms of the form ±Zl® ... ®zp+q, where Zir.=Xk for some ordered subset (iI, ... , i p) of (1, ... , P+q), and Zh=Yk for the ordered complement VI, ... ,jq), and where the sign is or - according to whether the permutation (iI, ... ,ip,jl, ... ,jq) of (1, ... ,p+q) is even or odd. Then the product in Y is given by the maps Yp®s Yq~Yp+q that send

+

(xo ® ... ® Xp+l) ®s (Yo ® ... ® YqH)

onto (XoyO) ® [Xl, ... , Xp; Yl, ..• , yq] ® (Yq+lXP+l).

It can be verified directly that this is indeed an associative and skew-commutative product and, if a is homogeneous of degree p and (3 arbitrary, one has d(a{3) = d(a){3+( -l)Pad({3); d. [4, pp. 218-219]. This product evidently induces a product in R®s Y, and hence in TorS(R, R). By the nature of the definition of the product on TorS(R, R), as given earlier in the general case, the product induced from that on Y is the standard product (f) on TorS(R, R). Next we shall define a map of the complex Y into the complex Y®B Y which will serve to make the product on Exts(R, R) explicit. We have (Y®R Y)p= Yr®R Y p- r. As an S-module, each Yr®R Y p- r may be identified with the tensor product, relative to K, of p+3 copies of R, i.e., with Y p +!. With this understanding, we define an S-module homomorphism "Yr: Yp~Yr®R Y p- r such that 'Yr(XO ® ... ® Xp+l) = Xo ® ... ® Xr ® 1 ® Xr+l ® ... ® Xp+l.

L:-o

Now the desired map "Y: Y~Y®R Y is defined so that, for uE Y p, the component of "Y(u) in Yr®R Y p- r is "Yr(U). It is somewhat lengthy, but not difficult, to verify that"Y is compatible with the boundary maps on Y and on

275

394 G. HOCHSCHILD, BERTRAM KOSTANT AND ALEX ROSENBERG

[March

Y®R Y. The product V on Exts(R, R) is induced by the product on Homs( Y, R) induced by 'Y. In particular, with aEHoms( Y p, R) and ~EHoms(Yq, R), we have(5)

(a{3) (xo ® ... ® xpt-a+l)

= a(xo ® .. ® Xp ® 0

= xoa(l ® Xl ® .

0

1),8(1 ®

Xpt-I

®

® Xp ® 1),8(1 ®

•

0

••

XJt+1

®

Xpt-q+l)

® .

0

•

® xpt-q ® l)Xpt-q+l.

Consider a formal differential LxdyED R. It is easily verified that the corresponding element of Tor~(R, R) is represented in R ® s Y 1 by the element LX®s (l®y®l). On the other hand, let rETR(R)=TR (say). Then it is easily seen that its image r*EExt1(R, R) is represented in Homs(YI , R) by the element r', where S'(XO®XI®X2) =XOS(XI)X2. Now let rl, .. Sn be elements of T R , and let ri ... S! denote the product in Exts(R, R) of their canonical images S: in Ext1(R, R). Then si ... r! is represented in Homs(Yn , R) by the product sf . r,:, as induced from the above map 'Y. One sees immediately from the formula written above that 0

,

0

(rt ... 5': )(xo ®

0

0

•

®

Xn+l)

= X051(Xl)

0

0

•

•

tn(Xn)Xn+l.

Now let a E Tor~(R, R), (j E Tor~(R, R), and let us compute h(n ... S;+a)(a(j). Choose representatives aER®s Yp and bER®s Yq of a and ~, respectively. Then a(j is represented in R®s Y p+q by the product abo We obtain h(S:'·· r;+a)(a(j) by applying the element of HomR(R®s Y pH , R) that corresponds naturally to S{ ... S;+q to abo Clearly, the result so obtained is the same as the result one would obtain by performing the shuffling involved in forming ab on the sequence S{ , . 5;+q rather than on the arguments Xi and Yj in the product formula for abo Hence we have 0

0

* h(51'

0

•

* 5pt-q)(a{3)

•

,

* ..• 5t(p»)(a)h(5t(pt-l) * * * = '" L..J U(t)h(5t(l) ... 5t(pt-q»)({3), t

where the summation goes over all those permutations t of (1, .. p+q) for which t(l) < ... n. Hence it is clear from the resolution VR®R E(TR) of R that dVR(R) =d(VR.R)(R) ~n. Now let us consider the same sequence we used at the beginning of our proof of Theorem 8.1: (0)

~

M = HOffiR(R, M)

~

HOffiR(V R ®R EO(T R), M)

~

HOffiR(V R ®R El(T R), M)

~

....

By Theorem 7.1, the resolution VR®R E(TR) of R has an R-homotopy, which evidently induces an R-homotopy of the above dual sequence. On the other hand, we see from Lemma 6.1, as in the proof of Theorem 8.1, that each HomR(VR®R Ep(TR), M) is isomorphic with a direct VR-module summand of a direct sum of VR-modules HomR(V R, M) with the module structure given by (v·h)(u) =h(uv). By [8, Lemma 1], this last VR-module is (VR, R)-injective. Hence the above sequence is a (VR, R)-injective resolution with Rhomotopy of M. Hence it is clear that d( V R, R) ~ n. Now we claim that, for every VQ-module M, ExtvR(R, M) is isomorphic with ExtvQ(Q, M). By Theorem 6.1, V Q is isomorphic with Q®n V R • It is clear from the definition of the algebra structure of Q®R Vn that this isomorphism transports the right Vn-module structure of Q®R VR into the right VR-module structure of VQ obtained from the natural map of V R into V Q. Since Q is R-flat, this implies that, as a right VR-module, VQis Vn-flat. Hence, if X is any VR-projective resolution of R, VQ®v R X is a VQ-projective resolution of VQ®VR R. Using the natural VQ-module structure of Q, we obtain a VQ-module epimorphism VQ®Vn R~Q sending v®r onto v·rEQ. It is easy to verify that this is actually an isomorphism, so that we may identify VQ®VR R with Q, as a VQ-module. Thus VQ®VR X is a VQ-projective resolution of Q. Since HomvQ(VQ®vR X, M) may be identified with Homvn(X, M), this establishes our claim. Now it is clear from Theorem 8.1 (applied to Q) that dVR(R) ~n. We have shown that n~dVR(R) =dWR.R)(R) ~d(VR, R) ~n, so that all but the last statement of Theorem 8.2 is proved. As a by-product of our proof of the existence of an R-homotopy in

288

1962]

DIFFERENTIAL FORMS ON REGULAR AFFINE ALGEBRAS

407

Vn®n E(Tn), we had obtained the result that this complex is R-projective. In particular, VI.: is R-projective as a left R-module; in fact, as an R-module, Vn is isomorphic with G(Vn ) ~S(TR). Similarly, Vn is R-projective as a right R-module. Hence we may apply [9, Th. 1] to conclude that, for every unitary Vn-module N, dVll(N) ~ d(Vll.R)(N)

+ dn(N).

If R is a regular affine K-algebra, we have d(RM) = t, for every maximal ideal M of R, by [2, Ths. 1.9, 1.10]. Since d(R) =maxM(d(R M» [4, Ex. 11, p. 142; 1, Th. 1], we have therefore d(R) =t. Hence the above results give d(Vn) ~d(Vn, R)+d(R) =n+t. This completes the proof of Theorem 8.2. 9. The product for Extvp(P, *). Let K be a field, and let P be a K-algebra which is either an affine K-algebra with Tp P-projective or an arbitrary extension field of K. If A and B are unitary Vp-modules there is a product Extvp(P, A) ®K Extvp(P, B)

~

Extvp(P, A ®P B)

which is defined as follows. Let X be any V p-projective resolution of P. Noting that V p is P-projective, so that X is P-projective, and appealing to Lemma 6.2, we see that the Vp-module X®p X is still Vp-projective. Moreover, since X is also a P-projective resolution of P, we have H(X®p X) = TorP(P, P), whence H(X®p X) has its components of positive degree equal to (0). Hence X®p X is still a Vp-projective resolution of P®p P=P. Hence the natural K-space homomorphism

cf>: Homvp(X, A) ®K Homvp(X, B)

~

Homvp(X ®P X, A ®P B),

where q,(j®g)(u®v) =f(u) ®g(v), induces a product for Extvp(P, *), as indicated above. It is seen as usual that this product is associative and skewcommutative. In order to make this product explicit, we require a map of the complex X into the complex X®pX, when X=Vp®pE(Tp). By imbedding T p as a direct P-module summand in a free P-module, we see easily that there is a P-module homomorphism E"'(Tp)~Ep(Tp) ®P En-p(T p) sending each product of elements of Tp onto

rl ... r.. L, q(t)tl(l) ... tl(p) ® tl(p-tl) ... tt(nh

where the summation goes over all permutations t of (1, ... , n) for which t(l)< ... 0

319

329

LIE GROUP REPRESENTATIONS.

to solve the Dirichlet problem with the sphere as boundary. That is, if f is any continuous function on the sphere one first expands f as a Fourier development of spherical harmonics fm- The sphere is 03) n Rn and the fm are in R (03)). The equality R (03)) = 8 (03)) and the isomorphism H ~ 8 (03)) then yields the extension of fm uniquely as harmonic polynomials hm on X. But this yields the desired extension of f. In Example 1 the conditions (b) and (c) are satisfied for any" regular" element x E X. Our first concern in this paper is to give criteria for (a), (b) and (c) to hold in general. Since our interest is in the continuous case we will assume G is connected (and hence a variety). Thus Example 2 rather than Example 1 serves as a model. Now let P C X be the cone of common zeros defined by the ideal J+8 in 8. Let X* be the dual space to X and let p* C X* be defined in a similar way with the roles of X and X* interchanged. As a criterion to establish (a) and more we prove PROPOSITION 0.1. Assume (1) that J+8 is a prime ideal in 8 and (2) there exists an orbit Oe C P which is dense in P. Then 8 = J 0 H. Furthermore if G is a subgroup of the complex rotation group then H may be taken as the space of all G-harmonic polynomials. Moreover H then coincides with the space spanned by all powers fk where f E P*.

It may be observed that the criterion is satisfied in Example 2.

An element x E X is called quasi-regular if P C C* . ()~. establish (b) is given by

A criterion to

PROPOSITION 0.2. Assume conditions (1) and (2) of Proposition 0.1 are satisfied. Then the G-module epimorphism H ~ 8(03)) is an isomorphism for any quasi-regular element x E X.

It may be observed that in Example 2 every nonzero x E X is quasiregular.

From known facts in algebraic geometry one has the following criterion to insure (c). PROPOSITION 0.3. Let x E X and assume (1) the closure 03) is a normal variety and (2) 0 IlJ - O:c has a codimension of at least 2 in 0 OJ. Then R(O:c) = 8(03)).

It may be observed that the conditions of Proposition 0.3 are satisfied for every x E X in Example 2.

320

330

BERTRAM KOSTANT.

Now assume that X = g is a complex reductive Lie algebra and G is the adjoint group. Here the structure of J is given by a theorem of Chevalley. This aserts that J is a polynomial ring in l (the rank of g) homogeneous generators Ui, i = 1,2,' . " l with deg Ui = mi 1 where the mi are the exponents of g. Now one knows that here P is the set of all nilpotent elements of g ([13J, Theorem 9.1). But then by [13J, Corollary 5.5, P does contain a dense orbit Oe, namely, the set of all principal nilpotent elements in g. Thus to apply Propositions 0.1 and 0.2 one must prove that J+S is a prime ideal. If n = dim g (all dimensions are over C) then one sees easily that n-l is the maximal dimension of any orbit. Let t={xEgldimO",=n-l}. Any regular element x E g belongs to t. But also e E t for any principal nilpotent element. These in fact are extreme cases.

+

+

PROPOSITION 0.4. Let x E g be arbitrary. Write (uniquely) x = y z where y is semi-simple, z is nilpotent and [y, z J = O. Let gY be the centralizer of y in g so that gY is a reductive Lie algebra and z E gY. Then x E t if and only if z is principal nilpotent in gY.

Let x E g. Consider the values (dui) '" of the l-differential forms dUi, i = 1, 2,' . " l, at x. It is known that these covectors are linearly independent whenever x is regular. (One recalls that the product of the positive roots is the determinant of an l X l minor of a certain n X l matrix determined by the duo-) But to prove the primeness of the ideal J+S one needs to know that these covectors are linearly independent if x is a principal nilpotent element. This fact is contained in THEOREM O. 1. Let x E g. Then the (dui) x is linearly independent if and only if x E t. Proposition 0.1 may now be applied. THEOREM 0.2. One has S = J 0 H where H is the space of all Gharmonic polynomials on g. Furthermore H coincides with the space of all polynomials spanned by all powers of nilpotent n linear functionals. U

Since Theorem 0.1 shows also that P is a complete intersection the decomposition S = J @ H when combined with Proposition 5, § 78, in F AC [15J, gives, in the notation of FAC, all the sheaf cohomology groups Hj (P, (fj (m» where P is the projective variety defined by P. Another application of the primeness of J+S in algebraic geometry is THEOREM 0.3. The intersection multiplicity of P, at the origin, with any Cartan subalgebra is w, where w is the order of the Weyl group.

321

LIE GROUP REPRESENTATIONS.

331

Next, Proposition O. 2 is put into effect for all orbits of maximal dimension by 0.4. The set r coincides with the set of all quasi-regular elements in g. (Thus Hand S (0 aJ) are isomorphic as G-modules for any x E r.) THEOREM

As a consequence of Theorems 0.2 and 0.4 one shows that not only is the ideal J+S prime in S but J 1 S is prime for any prime ideal J 1 C J. Furthermore one gets the following characterization of all the invariant prime ideals in S which are generated by elements of J. THEOREM 0.5. Let I C S be any G-invariant prime ideal. Let u C g be the affine variety of zeros of I. Then I is of the form 1= JIS, for J I a prime ideal in J, if and only if u n r is not empty.

Since R ( 0 II!) = S ( 0 II!) in case 0 II! is closed and since 0 II! is closed if x is regular one gets the G-module structure of H by applying Theorem 0.4 and (0.1. 5) for x regular. Thus if D denotes the set of dominant integral forms corresponding to a Oartan subgroup A, so that D indexes all the irreducible representations of G as highest weights, then one has (0.1.6)

where lx = dim VXA is the multiplicity of the zero weight of vx. In order to determine the G-module structure of Sk, the space of homogeneous polynomials on g of degree k, one must know more than (0.1. 6). In fact using the relation S = J ® H what one wants is the multiplicity of vX in Hj = Sj n H for any A and j. As it turns out, for this, one needs R (Oe) = S (Oe) where e is a principal nilpotent element. To show the latter using Proposition O. 3 it is enough to show that P is a normal variety and P-Oe has a co dimension of at least 2 in P. Let @r be the set of all orbits of maximal dimension (n -l). The set @r may be parametrized by CZ in the following way. Let u:

g~Cl

be the morphism given by putting u (x) = (u l (x),' . " Uz (x» Since u reduces to a constant on any orbit it induces a map

for any x E g.

'YJr: @r~Cz.

It is known that u induces a bijection from the set of all orbits consisting of semi-simple elements onto CZ (for completeness a proof of this fact will be given here). Oombining this with Proposition 0.4 one obtains

322

332

BERTRAM KOSTANT. THEOREM

0.6. 'fJt is a bijection.

Thus to each ~ E C! there exists a unique orbit, Ot (~), of dimension n-l which correspond to ~ under 'fJt. Now let pa) = u- 1 (~) for any ~ E C! so that g=UP(~) ~EC'

is a disjoint union. of C!. One proves THEOREM

0.7.

Note that pa) =p and Ota) =0. if For any

~E

~

is the origin

C! one has

P(~) =

Of(~)

so that pa) is a variety of dimension n-l. Moreover P(~) is a complete intersection and Ora) coincides with the set of non-sing1tlar points on P(~). Finally P (~) is a finite union of orbits so that Oil} is a finite union of orbits for any x E g.

Since P(~) is a complete intersection and since its singular locus is the complement (a finite union of orbits) of Of(~) in pa) one would get the normality of P(~) by a theorem of Seidenberg if one knew the dimension of the other orbits in pa) were at most n-l-2. Now it is well known that dim Oil! is even (and hence dimR Orc is a multiple of 4) for any semi-simple element x E g. Less known is the following proposition observed independently by the author, Borel, and (most simply proved by) Kirillov. PROPOSITION

0.5. The dimension of O{ll is even for any x E g.

Combining Theore:m

o. 7

and Proposition

o. 5 one

obtains

THEOREM 0.8. Let ~ E C! be arbitrary. Then P (~) is a normal variety and the co dimension of P(~) -Ot(~) in P(~) is at least 2.

Applying Proposition 0.3 one then has THEOREM 0.9. Let x E t. Then R(O{ll) = 8(0:1). (This implies that all R(OIl}) for xE t are isomorphic as G-modules; even though they are not in general as rings.) Let ~=u(x). Then R(O{ll) (=R(G/Grc» is an affine algebra (even though Orc is not necessarily an affine variety) and P(~) is the variety of all maximal ideals of R (0(II) • Thus the embedding of G/ Grc in g as Oil! is special in that any morphism of G/Gm (or Om) into any affine variety extends uniquely to a morphism of P(~) = Orc into the variety. (In particular

323

333

LIE GROUP REPRESENTATIONS.

this holds for 0 e and {j m = P. ) Finally (using (0. 1. 5) and the equality R(Oa;) = 8(Ox)) one has, for any A ED

dim VAaz=h

(0.1.7)

so that the left side of (0.1. 7) is independent of x E r.

Now let {e_,xo,e} be a principal 8-triple (that is, a "canonical" basis of a principal three dimensional simple Lie sub algebra ) . In particular then e is a principal nilpotent element. Used heavily in the theorems above is the result of [13] which asserts that ge is l-dimensional and has a basis Zi, i = 1, 2,' . " t, such that (0.1.8) where, we recall, the mi are the exponents of g. The main application of this is the following result: Let a be any subspace of g such that (1) g= a [e_, g] is a direct sum and (2) a is stable under ad Xo (e. g. take a = ge). Then if b is the l-plane defined by the translation b = e_ a one has THEOREM 0.10. The variety b is contained in r. Moreover each orbit in (fj r intersects b in one and only one point. Finally the mapping f ~ fib induces an isomorphism of J onto R (b) .

+

+

Remark. If g is the set of all l X l complex matrices then one shows easily that r is the set of all matrices whose characteristic polynomial is equal to their minimal polynomial. An example of the subvariety b is the set of all "companion" matrices. Here the validity of Theorem 0.10 is a wellknown fact in matrix theory. Now since ge = ga e (because ge is commutative) and since (0.1. 7) holds for x = e this suggests a generalization of the notion of exponent. Let V be any finite dimensional G-module with respect to a representation v. If lv is the multiplicity of the zero weight of v then by (0.1. 7) one has dim Va e = lv. It follows therefore that there exists a unique non-decreasing sequence of non-negative integers mi( v), i = 1, 2,' . " lv, such that one has v ( Xo ) Zi =

mi ( v ) Zi

for a basis Zi of Va e • If v is the adjoint representation the m, (v) are the usual exponents. If v=v A we will write m,(A) for m,(v A ) and note (because the highest weight has multiplicity one) that

mj(A) =O(A) for j=l"ll. where

0

(A) is the sum of the coefficients of A relative to the simple roots and

324

334

BERTRAM KOSTANT.

that this highest value occurs with multiplicity one among the generalized exponents m.(A). (This specializes to the familiar relation ml=o(o/) when g is simple and", is the highest root.) The following theorem now gives the G-module structure of Hi and hence Sk for any j and le. THEOREM 0.11. Let A ED be arbitrary and let H (A) be the set of Gharmonic polynomials which transform under G according to vA. Let (by

(0. 1. 6»

IA

="'2:. Hi (A)

H (A)

be a decomposition into irreducible components

i=1

so that Hj(A)C Hnl where nj, j = 1, 2,· .. , lA, is a non-decreasing sequence of integers. Then nj = mj(A) for all j. In particular then le = 0 (A) is the highest degree le such that vA occurs in Hk. Moreover it occurs with multiplicity one for this value of le.

Assume for convenience that g is simple and let tf; E D be the highest root. Let Xi, i = 1, 2,· .. , n be a basis of g. If the Uj E J are chosen properly one sees that

~Uj , i = 1, 2,· . ., n, is a basis of

uXj

H j (tf;).

One notes then that

Theorem 0.11 is a generalization of the result in [13J given by (0.1. 8). H. S. M. Coxeter observed and A. J. Coleman proved in [4J that if W is the Weyl group and '()" E W is the Coxeter-Killing transformation then the eigenvalues of '()" operating on the Cartan sub algebra are e2 11'iffll/8, j = 1, 2,· .. , l, where s is order of fT. Now more generally W operates on the zero weight space of VA for any A E D according (say) to some representation 7rA of W. As a generalization of the Coxeter-Coleman theorem one now has THEOREM

O. 12.

For any A E D the eigenvalues of

7rA

(fT) are e211'iffll(A)/S,

j= 1, 2,· .. , h. 0.2. By applying the Birkhoff-Witt theorem the results above carry over from S to U, the universal enveloping of g (U is obviously a G-module in a natural way). THEOREM 0.13. Let U be the universal enveloping algebra over g and let Z CUbe the center of U. Then U is free as a Z-module (under multiplication) . In fact (0.2.1) U =Z®E

where E is the subspace (and G-submodule) of U spanned by all powers xk for all nilpotent elements X E g. Moreover E is equivalent to H as a G-module so that every irreducible representation of G occurs with finite multiplicity in E (in fact vA occurs lA times in E for any A E D).

325

LIE GROUP REPRESENTATIONS.

335

Let V be a finite dimensional irreducible U-module so that one has a Gmodule algebra epimorphism p: U~End

V

Since p(Z) reduce to the scalars it follows from (0.2.1) that p(E) = End V. Now let Y be any subspace of U. If Y is one-dimensional then it is due to Harish-Chandra that there exists an irreducible U-module V such that p is faithful on Y. This is not true in general if dim Y > 2. However it is true if YCE. O. 14. Let Y C E be any finite dimensional subspace. Then there exists an irreducible U-module V such that p is faithful on Y. THEOREM

I would like to express my thanks to C. Chevalley, M. Rosenlicht and

A. Seidenberg for helpful conversations about questions in algebraic geometry. In particular to Siedenberg for making me aware of his criterion for normality and to Chevalley for simplifying my proof of the primeness of J+S. l. Consequences of the primeness of J+8 and a dense orbit in P. 1. Let X be a n-dimensional vector space over the complex numbers C. Let S* = S* (X) symmetric algebra over X. One knows that S* may be regarded as the algebra of all differential operators 0 on X which may be put in the form

8=}: ail"""in(8~)il. . (8~ )in where the a;,"""in are complex constants and Zl,' " ., Zn are the affine coordinates of X. Let S* = S* (X) (or just S) be the symmetric algebra over the dual space to X. Then S is just the ring of all polynomials on X. In fact we take the point of view that X is an affine variety (over C) and S is its ring of everywhere defined rational functions. The algebra S (resp. S*) is graded in the obvious way and a subspace L C S (resp. L C S*) will be called graded if it is spanned by its homogeneous components LJ =L n Sj (resp. L n Sj). N ow one knows that a non-singular pairing of S* and S into C is established by putting (1.1.1) =of(0) where 8 E S*, f E Sand 8f(0) denotes the value of the function 8f at the origin. In this way Sic is orthogonal to Sj if j =F k and becomes isomorphic to its dual if k = j. It is obvious from (1.1.1) that

(1.1.2)

326

336

BERTRAM KOSTANT.

for any 01, O2 E S* and f E S and hence in particular if by the Taylor expansion

=

(1.1.3)

f E Sm

and x E X then

f(x)

where Ox is the element of S1 (X) :::: X corresponding to x. Now assume that G C Aut X is a connected linear reductive algebraic group, i. e., G is the complexification of a connected compact subgroup of Aut X. We regard G as not only operating on X, but by unique extension, as a group of algebra automorphisms of S* (X) and also as a group of algebra automorphisms of S (X). The action on the latter is also uniquely defined by requiring that

=

(1.1.4)

for all a E G, 8 E S* and f E S. Note by (1.1. 3) that

(a· f) (x)

(1.1.5)

=

f(a- 1 x)

for any xE X, fE Sand aE G. Now let J C S be the graded subring of G-invariant polynomials in S. That is J = {f E S I a' f = f for all a E G}, and let J+ = {f E J I f(0) = O}. We will often be concerned with the homogeneous ideal J+S in S generated by J+. 1. 'Let L is a direct sum.

PROPOSITION

S = J+S

+L

be any graded subspace of S Then

such that

Proof. We must show Sk C J L for all k. This -is obvious if k = 0 since Assume it is true for Si where j < k. But since one clearly has

So C L.

Sk+1 C J+S(k)

+ L where S(k)

k

=

LSi it is then obviously true for Sk+1. Q. E. D. • =0

Weare interested first of all in the question as to when S is free over J, or more specifically, as to when S is a tensor product of J and L. (Choosing L to be G-stable, such a decomposition of S reduces the study of its G-module structure to that of L.) We first observe that the two conditions are equivalent. The expression linear independence (resp. basis) without any reference

327

337

LIE GROUP REPRESENTATIONS.

to a ring always means linear independence (resp. basis) with respect to C. Furthermore, tensor product without reference to a ring means tensor product over C. LEMMA

1.

The following conditions are equivalent,'

1. Let L be as in Proposition 1. Then the map J0L~S

given by f i8) g ~ fg is an isomorphism. 2. S is free over J. 3. Let M C S be any subspace such that M n J+S = (0). Then for elements in M linear independence is equivalent to linear independence over J. Proof. Obviously (1) => (2) since a C basis of L defines a J basis of S. Assume (2) and let~, i=1,2,·· ., define a J basis of S. Let f1,' . ',h be linearly independent elements of M. For j = 1,' .. , le write fj = L G,;je; i

where we may assume i = 1, 2,' .. , p and G,;j E J. To show that the fj are linearly independent over J it clearly suffices to see that the le X p matrix (G,;j) with entries in J is of rank le. If f' denotes the image of f E S in SjJ+S it is clear that f'j=LG,;j(O)e'i' Since the fj are obviously linearly independent it is clear that the C matrix G,;j (0) is of rank le. Hence not all le X le minors of the J matrix (G,;j) can be zero. This proves that the fj are linearly independent over J. It is trivial that linear independence over J implies linear independence. To obtain (1) from (3) we simply put M =L and apply Proposition 1.

Q.E.D. 1. 2.

Now for any x E X let

00;= {yE X

I y=ax, for

A subset 0 C X is said to be an orbit if 0 For any x E X let

=

some aE G}. Oil! for some x EX.

It is clear that Gx is an algebraic (hence a closed, complex Lie subgroup) subgroup of G. Furthermore if

(1.2.1) is the map given by

/3'x (a)

=

ax then /3'Il! induces a bijection

(1.2.2)

328

338

BERTRAM KOSTANT.

Now let U be universal enveloping algebra of the Lie algebra of G. Then since the representation of G on S induces a representation of its Lie algebra on S it is clear that S becomes a U-module by further extension to U. Obviously Sk is a U-submodule for any k. We denote by P . f E S the effect of applying p to f where p E U and f E S. For any subset P C X and f E S let f I P be the restriction of f to P.

Let ff E S, j = 1, 2,' . " k, and let pi, i = 1, 2,' . " be a basis of U. Oonsider the k-column matrix D = (dij) where dij = Pi' ff. (The matrix thus has entries in S and hence, for any x E X, D (x) = (d ij (x» is a C matrix.) LEMMA

2.

Then if x E X the functions fi I O:c on 0 31, j independent if and only if D (x) has ranle le.

=

1, 2,' . " k, are linearly

Proof. Let 0 be the algebra of all holomorphic functions on the complex homogeneous space GIGa:. It is clear that 0 is a module for the Lie algebra of G since the latter defines holomorphic vector fields on GIG3I. Hence 0 also becomes a U-module and in such a way that if a:

S~O

is the homomorphism given by af = f 0 {3:c one has

a (p . f)

(1. 2. 3)

=

P . af

for any f E S. Furthermore if s E GI Ga: denotes the point corresponding to Ga: and g E a then since the Lie algebra of G spans the holomorphic tangent space at s it follows from the Taylor expansion that g vanishes identically on GIG3I if and only if (p' g) (s) = 0 for all p E U. Now assume that 'rank D(x) < k. Then there exists a non-zero vector (c 1 , ' . ',Ck) ECk such that ~dij(X)Cj=O for all i. Thus if f=~CjfjES j

j

one has (p' f) (x) = 0 for all p E U. Thus by (1. 2. 3) (p' af) (s) =0 for all p E U and hence af vanishes identically on GI Ga:; or equivalently flO:c is zero. Hence the fj I O:c are linearly dependent. Conversely assume that the fj I 031 are linearly dependent so that (~cdj) I O:c is zero for a non-zero vector (c1 , ' . " Ck) E Ck. But then a (~cd,) = 0 and hence for i = 1, 2,' J

~

Cjp• . f,(x)

=

(Pi .

a(~

j

i

cifi» (s)

=0 Thus rank D(x)

Q.E.D.

< le.

329

339

LIE GROUP REPRESENTATIONS.

As a corollary we obtain the following criterion for linear independence over J. LEMMA 3. Let fi E S, i = 1, 2,' . " k. Assume the functions fj I 0 are linearly independent for some orbit 0 ex. Then the fi are linearly independent over J.

Proof.

Assume

~

gifj = 0 where gi E J, j

=

1, 2,' . " k.

Let D be the

j

matrix given in Lemma 2 and let x EO. Then by Lemma 2 there exists a k X k minor of D whose determinant et E S is such that et (x) =F O. But then there exists a neighborhood W of x such that et(y) =F 0 for all yEW. Thus D(y) has rank k and hence the fi lOy are linearly independent for all yE W. But since the gi reduce to constants on any orbit Oy it follows from the relation ~ gifi = 0 that the gj vanish identically on W. This implies that the gj j

vanish on X since the gj are polynomials. 1. 3.

Q. E. D.

For any subset Y C X we will let

I(Y)={fES! f!Y=O} be the ideal in S defined by Y. Now let P C X be the cone (since J+ is homogeneous) given by P = {x E X

I f (x) =

0 for all

f E J +}

Since P is defined by the ideal J+S one knows that I (P) is the radical of J+S and that the cone P is irreducible (in the sense of algebraic geometry) and J+S=I(P) if and only if J+S is prime. We now give a criterion for the conditions of Lemma 1 to be satisfied. PROPOSITION 2. Assume (1) J+S is prime and (2) there exists an orbit such that (5 = P. Then the conditions of Lemma 1, § 1. 1, are satisfied. In particular if S = J+S L is a direct sum where L is a graded subspace of S then the map

o

+

(1.3.1)

given by f 0 g ~ fg, is an isomorphism. Proof. Let M C S be any subspace such that M n J+S = (0). Since J+S = I (P) it is clear that if fj E M, j = 1, 2,' . " k, are linearly independent then the fj I P are linearly independent. But this obviously implies that the fj I 0 are linearly independent since (5 = P. But then the fj are linearly independent over J by Lemma 3 and thus the result follows by (1) and (3) of Lemma 1. Q. E. D.

330

340

BERTRAM KOSTANT.

Remark 1. In the proof we have only used the fact that J+8 = I (P) and not that J+8 is prime. However assumption (2) already implies that the cone P is irreducible (recall that G is connected) so that there is no loss in assuming that J+8 is prime. 1. 4. Now assume that B is a symmetric non-singular G-invariant bilinear form on X. Then, as one knows, B induces a unique G-module ring isomorphism (also written B) B: 8*~8

(1.4.1)

of degree zero where O. We wish to show that ozm = O. We may assume that m > k and hence by

331

LIE GROUP REPRESENTATIONS.

(1.1. 2) it suffices to show that (1. 4. 2)

=

0 for all a1 E 8m-k'

341 But by

= = n-l.

To prove that dim P (0 = n - l it sufi1ces to show that dimO..F in R(G/F). (2.1.2)

is a direct sum.

R(G/F)

=

~

>"€D

Furthermore

R>"(G/F)

Proof. The decomposition (2. 1. 2) is obvious since each element of R ( G) is an element of R ( G) and hence generates a finite dimensional sub8pace under the action (left translation) of G. Furthermore it is also obvious that the d>..d>..F functions hij defined in the proposition are in R>"(G/F) and (see (2.1.1)) are linearly independent. To prove the proposition therefore one simply has to show that every element of R>" ( G) invariant under right translation by elements of F is in the span of the hij • Assume that g E R>" ( G) and g . a = g for all a E F. Let %>.. be as in (2. 1. 1) (a basis of R>" ( G)) . Write g = ~ gi/Cji where cij E C defines a matrix and hence, relative to the basis v'j, a linear transformation Q; of V>... It suffices only to show that 1m Q; C V>..F. But the condition on g implies that (v>.. (a) -1)Q; = 0 for all a E F. This proves 1m Q; C V>..F'. Q. E. D. Remark 5. A case of importance for us is the case where F = A is a Cartan subgroup of G. Here V>..A is just the zero weight subspace, corresponding to A, of V>... To make it independent of A we will put h. = d>..A so that h. = multiplicity of the zero weight of VA (2. 1. 3). Remark 6. Since one knows that the multiplicity of any weight p. for v>.. is equal to the multiplicity of - p. for v>" it follows that h. is also the multiplicity of the zero ,weight of v>". 2. 2. Now we wish to apply the considerations of § 2. 1 to the case where F = Gx for any x E X. See § 1. 2. By Proposition 8 any question as to the complete reduction of R (G/Ga;) as a G-module becomes a question in the finite dimensional representation theory of G and how such representations restrict to Gx. Now, as we observed in § 1. 6, the orbit 03] is a subvariety of X. Furthermore the bijection f3x: G/GJJ~ Ox induced by f3'x is an algebraic isomorphism (this follows easily from the transitivity of G together with [3J, Corollary, p. 53 and Corollary 2, p. 90. (See also [lJ, § 2. 2.) Thus if R(Oa;) is regarded as a G-module, using the action of G in Ox, it follows that f3x induces a G-module and ring isomorphism

(2.2.1)

R(G/Gx)

340

~R(O",).

350

BERTRAM KOSTANT.

Now we recall that 8(0a;) is the ring of functions on Oa; obtained by restricting 8 (the ring of polynomials on X) to Ox. Since {3a; is a morphism one obviously has

for any x E X and in fact it is clear that 8 (0 a;) is a G-submodule of R (0 x). Unlike R (Ox) whose G-module structure is completely determined by Proposition 8 because (2.2.1) is a G-module isomorphism, in the general case it seems (to us) to be quite difficult to describe how 8 ( 0 x) decomposes as a G-module. In many instances, however, 8(0a;) =R(Orc) (and hence, in such cases, one knows the G-module structure of 8 (0 JJ) ) • Indeed, in the general case since (j a; is Zariski closed in X one has (2.2.2) Thus (2.2.3)

Remark 7. In the example of Remark 2, § 1. 4, one depends upon the equality 8(0a;) =R(OgJ for a particular x in order to solve the Dirichlet problem in Rn. Indeed let x E Rn where (x, x) = IX> 0 and let f be a conThe problem tinuous function on the sphere 8 n- 1 = 0", n Rn of radius is to extend f as a harmonic function f' defined in the interior of Sn-l. To do this one expands f

Va.

using some limiting process (e. g., L 2 ), as an infinite sum of spherical harmonics fA. That is, here CA E C and h = gA I 8 n - 1 where

However since R (0 a;) = 8 (0 a;) it follows from (1. 5. 3) that there exists a unique harmonic polynomial hA E H on ,Cn such that hA lOa; = gAo One then puts f' = ~ cAh'A AED

where h'A is the restriction of hA to the interior of 8 n - 1 • Now it is not necessarily true, in general, that 8(0a;) =R(Oa;). For example let X be the m 2 dimensional space of all complex m X m matrices and G is the general linear group Gl (m, C) regarded as operating on X by left matrix multiplication. Then if x is the identity matrix Orc is isomorphic to

341

LIE GROUP REPRESENTATIONS.

351

Gl(m,C). But 8(0",) ¥=R(O",) since in particular if f(a) = (deta)-l for aE G then fE R(Ox) but f¢ 8(0",). The equality 8 (0",) = R(O",) in the example of Remark 7 when (x, x) > 0 may be established either using the fact that 0", is closed (see (2.2.3)) or by applying the Stone-Weierstrass theorem to both 8 (0",) and R (0 a;) restricted to Ox n Rn. 'l'hese methods also work more generally in case (x, x) ¥= o. However, they do not apply to 0", where x¥=O and (x,x) =0. Nevertheless it is still true in this case that R(Oa;) =8(Ox). The more powerful tool (and the one that will be required in § 5. 1) needed to establish the equality for this: case is given in the next proposition. For any x E X let Om be the Zariski closed subset of X defined by taking the complement of Ox in Oa;. If we put codim 0", = dim Oa; - dim Oa; then of course one has co dim O:c > 1. An affine variety Y is called normal in case the ring R (Y) is integrally closed in its quotient field. PROPOSITION 9. Let x EX. Assume (1) that Oa; is a normal variety and (2) that codim Oa; > 2. Then

Proof. If Y is any variety let Q(Y) denote the field of all rational functions on Y. In any f E Q(O:c) let 1 denote its image in Q(OlD) under the canonical isomorphism Q( 0 a;) ~ Q( OlD) defined by extension. Now let fE R(Oa;). 'J,'hen obviously IE Q(OID) is defined at every point of Ox. Thus if T is the set of points of Ox where f is not defined then T C Oa;' Since codim Om > 2 one also must have codim T > 2. But now for a normal affine variety Y one knows (see [3], Proposition 2, p. 166 and 10, p. 134. Also Corollary, p. 135), that if g E Q(Y) then either g E R (Y) or the set of points where g is not defined has co dimension 1. Since Oa: is assumed to be normal it follows that the first alternative must hold for 1. That is, 1 is everywhere defined on Oa;. But then 1, as a function on Ox, is the restriction of a polynomial on X to Ox' (See (2.2.2).) But then this is certainly true of f so thatfE8(Ox). Q.E.D. Remark 8. Proposition 9 is stronger than the criterion O:c = Oa: for insuring 8(0",) =R(O:c). In fact if O:c=Oa; (in which case we may take (2) to be trivially satisfied) then Oa; is empty and Oa; is non-singular. But

342

352

BERTRAM KOSTANT.

since non-singularity implies normality the conditions of Proposition 9 are satisfied in case Ox is closed. The proof that Ox is normal for the example of Remark 2 where (x, x) = 0, x oF 0, and d > 3 follows from a result of Seidenberg (see § 5. 1) .

3. The orbit structure for the adjoint representation. 1. Let g be a complex reductive Lie algebra of dimension n. Then g is a Lie algebra direct sum (3.1.1)

g=3+[g,gJ

where 3 is the center of g. The commutator [g, gJ is, as one knows, the maximal semi-simple ideal in g. A subalgebra a egis said to be reductive in g if the adjoint representation of a on g is completely reducible. Such a subalgebra is necessarily reductive (in itself). Let g"', for any x E g, denote the centralizer of x. An element x Egis called semi-simple if ad x is diagonalizable. One knows that gx is reductive in g for any semi-simple element x E g (see e. g. Theorem 'I in [l1J). An element x Egis called nilpotent in case (1) x E [g, gJ and (2) ad x is a nilpotent endomorphism.

Remark 9. If x E a C g where a is reductive in g then x is semi-simple (resp. nilpotent) with respect to a if and only if it is semi-simple (resp. nilpotent) with respect to g. The proof of these statements are immediate consequences of the representation theory of reductive Lie algebras. Now one knows that the most general element x E g may be uniquely written (3.1.2) where y is semi-simple, z is nilpotent and [y, zJ = 0. We will speak of y and z, respectively, as the semi-simple and nilpotent components of x. See [l1J, Theorem 6.

Remark 10. If x E a C g where a is a subalgebra reductive in g then by Hemark 9 the decomposition (3.1. 2) formed in g is the same as the decomposition (3.1. 2) formed in a. In particular given the decomposition (3.1. 2) one should observe that z is not only nilpotent in g but also in the" reductive in g" subalgebra gil. In particular then (3.1.3)

z E [gll, gYJ.

343

LIE GROUP REPRESENTATIONS.

353

Conversely ir y Egis semi-simple and z is nilpotent in gY and one puts x = y z then y and z are respectively the semi-simple and nilpotent components or x.

+

3.2. We wish to apply the considerations or §§ 1 and 2 to the case where X = g and G C Aut g is the adjoint group or g. Thus not only is G a connected algebraic reductive linear group but in ract G is then a semisimple Lie group whose Lie algebra is isomorphic to [g, g]. In this case we observe that the orbit 01/} defined by any x Egis just the set or elements or g that are conjugate to x. Ii a egis any sub algebra then under the adjoint representation a corresponds to a Lie subgroup A C G. Indeed A is the group generated by all exp ad x where x ranges over a. In this way gl/} clearly corresponds to the identity component or the algebraic subgroup Gill. We recall that an element x Egis semi-simple if and only ir x may be embedded in a Cartan subalgebra (C. S.) or g. Equivalently x Egis semisimple ir and only ir gill contains a C. S. or g. The rollowing lemma is known. We will prove it for completeness and also because, as noted in Remark 11 below, the proor may be used to give a more general result.

5. Assume x Egis semi-simple. Then (1) G'" is connected and (2) 03] is closed in g. LEMMA

Proof. We first show Grc is connected. Let bE Gx. Then by Theorem 2, p. 108, in [6], one knows that b may be uniquely written (3.2.1)

b =aexpady

where a EGIS diagonalizable and y Egis nilpotent and a(y) = y. Put Ct = exp t ad x. Then b = CtbCt-1 = (Ctavt-1) exp ad Ct (y). By the uniqueness of the decomposition (3.2.1) it rollows that a= Ctact-1 and Ct(Y) =y. Hence a E Ga; and y E gill. But then b is "connected" to a in Gill by means or the curve a exp s ad y, s E R. Thus we may assume that b is diagonalizable. But now by Theorem 10, p. 117 in [6], ir gb is the Lie subalgebra or all y such that b (y) = y then gb contains a C. S. g or g and i:f g is any C. S. in gb then b=expadz ror some zE g. But now x E gb and since ad x I gb is semi-simple there exists a C. S. g such that x E g C gb. But b = exp ad z ror some z E g. However since g C gill it rollows that b may be joined to the identity in GI/} by a curve; indeed one uses the curve exp t ad z. Hence Gill is connected.

344

354

BERTRAM KOSTANT.

To show that O:c is closed let f) be a Cartan sub algebra such that x E f). By the Iwasawa decomposition we may write G=KMHo where K and Mare connected Lie groups which are, respectively, compact and unipotent (an endomorphism u is called unipotent if u - 1 is nilpotent; a group is called unipotent if all its elements are unipotent) and Ho is an abelian Lie group corresponding to a sub algebra of f). Since x E f) it follows then that x is fixed under Ho. Thus O:c=KMx. We have proved (unpublished) that any orbit of a connected unipotent Lie group is closed. Rosenlicht [14J has generalized this to the case of a field of arbitrary characteristic. Thus we may use the reference [14J to establish that Mx is a closed subset of g. But since Ox is obtained by applying a compact group to a closed set it follows easily that O:c is closed.

Remark 11. Another proof that 0", is closed if x is semi-simple follows from Theorem 4, § 3. 8. In fact one sees there that 0", is closed if and only if x is semi-simple. This observation was also made in [1]. Note however the proof given above, that Ox is closed when x is semi-simple generalizes and shows that the orbit of any zero weight vector for any representation of G is closed. As a consequence of the connectivity of Gx for x semi-simple one has LEMMA 6. Assume x Egis semi-simple. Then gX is stable under Ga; and the restriction of G'" to g'" is the adjoint group of gX.

Proof. It is trivial that gaJ is stable under Gx. Furthermore as we have observed in the beginning of this section the identity component of GJ} corresponds to g'" under the adjoint representation of g and hence its restriction to g:c is the adjoint gmqp of g"'. But Gx is connected by Lemma 5. Q. E. D. 3.3. Now for the case at hand S is just the symmetric algebra S* (g) over the dual space to g. The well known description of the ring of invariants J given below is due to Chevalley. If l is the rank of g then J is generated by l algebraically independent homogeneous polynomials. That is, there exist homogeneous elements u. E J, i = 1,' . " l, such that if C[Y1 , ' • " YzJ denotes the polynomial ring, over C, in l indeterminates and

(3.3.1) is the homomorphism given by p(Y 1 ," "YZ)~P(Ul'" is an isomorphism. Moreover, if we write deg u. = m.

345

',u!) then (3.3.1)

+ 1 then the integers

355

LIE GROUP REPRESENTATIONS. I

rn,!, called the exponents of g, are those special integers such that II (1 ;=1

is the Poincare polynomial of g. Throughout we will assume that the

We will refer to the Ut, i

=

Ui

+t

2ml 1

+ )

are ordered so that

1, 2,' . " l, as the primitive invariants.

Rernark 12. One knows that the primitive invariants and even the l-dimensional space they span is not unique. However, in § 5.4 in connection with G-harmonic polynomials one normalizes the space they span in a natural way. See Remark 26, § 5.4. We now define a mapping u: g~CI

(3.3.2)

by putting U ( x) =

(u1 ( X ) "

• " Ul ( x)

).

It is obvious that U is a morphism. Now let (fj be the set of all obits 0 C g. Since U obviously maps any orbit into a point it is clear that U induces a map

Now if u egis any subset stable under the action of G it is obvious that u is a union of orbits. Let

and we will let '¥Ju be the restriction of '¥J to (fj u' Let ~ be the set of all, semi-simple elements in g. Obviously under G so that we may consider the case where u = £I.

~

is stable

Now it is easy to see that '¥J is not one-one, that is it does not separate all orbits. One observes, however, that not only does '¥J separate the orbits in .5 but also that '¥J9 is a surjection. The following proposition is no doubt known. We prove it for completeness. PROPOSITION

10.

Let.5 be the set of all semi-sirnple elements tn g.

Then the map

induced by

U

(see (3. 3. 2»

is a bijection.

Proposition 10 permits us to parameterize (fj9 by all complex l-tuples. In

346

356

BERTRAM KOSTANT.

order to prove Proposition 10 we need some further notation and Lemma 7 below. Let f) be a Cartan sub algebra of g regarded as fixed once and for all. Let W be the Weyl group of g regarded as operating in f). Let Ll C 8 1 (f) be the set of roots and let Ll+ C Ll be a system of positive roots fixed once and for all. An element x Egis called regular if g'" is a Cartan subalgebra. If x E f) one knows that x is regular if and only if <x,cf»=FO for all cf>E Ll. Now let ulj:

be the restriction of

U

f)~C!

to f).

LEMMA 7. The map ulj is proper. compact set is compact).

(That is, the inverse image of any

Proof. Let 7r: g ~ End V be a faithful completely reducible representation of g and let m = dim V. For any positive number 7c let ric be a positive m-1

number such that for any monic polynomial ym

+ .=0~ CiYi

=

P (Y) in the

indeterminate Y, where Ci EC, one has 1 Ci 1 < 7c, i = 0, 1,' . ., m -1, implies 1 A 1 < ric for any root A of p (Y). In fact, it suffices to take ric = m7c 1. Now let fi E J be the invariant polynomials defined so that

+

(3.3.3)

m-l

det(Y -7r(x»

=

ym

+ ~ fi(X)Yi i=O

for any x E g.

Now there exist unique polynomials

so that fi=Pi(U , ,'" ·,u!). Thus regarding C[Y l ," nomial ring on Cl it follows that

',Y l] as the poly-

(3.3.4) for any x E f). Now let ECC! be any compact set. We wish to show that ulj-l(E) IS compact. Let

7c = sup 1 Pi(~) I. ~EE

i=O,l, .. , ,m-l

It follows therefore from (3. 3. 4) that 1 fi (x) 1 < 7c for all x E Ulj-l (E). Hence if A is a root of (3.3.3) it follows that 1,\ 1 + =

0

for all x E g,8 E 8* and f E 8. Furthermore any x E 9 operates as a derivation of degree 0 of 8 and 8* and hence, by (4.1.1), its action is completely determined by its restriction to 8 1 , But the latter is given by x'8y =8[IC,y]

(4.1.2)

for any y E g. Note that if 8 E 8* is of the form 8 = x . 81 where x E 9 and 81 E 8* then by (4.1.1) ( 4. 1. 3)



=

0 for all f E J.

This criterion for an element 8 E 8* to be orthogonal to J is especially convenient to use when x equalS a certain element Xo E f), now to be defined. Recall that IT C ~+ is the set of simple positive roots. We now put Xo equal to the unique element in f) n [g, g] such that (see [13], § 5. 2)

<xo, ex> =

1 for all ex E IT.

If ep E ~ is arbitrary and the order

(4.1. 4)

o(ep)

0

(ep) of ep is the integer defined by

=~na(ep) a€II

where (4.1. 5)

=

0 ( ep )

and hence (4.1. 7)

[xo, e] =

0 ( cp ) e.

As usual let Z denote the set of all integers. For every integer j E Z let

8*(J) = .

It is obvious that S*U) is a graded subspace of 8* and since

derivation of 8 it follows immediately from (4.1. 7) that 8*=~8*(j) ja

359

Xo

operates as a

369

LIE GROUP REPRESENTATIONS.

is a direct sum and (4.1. 8)

Similarly let gU) be the eigenspace of ad Xo for the eigenvalue j so that g is a direct sum of the gU). Since ad Xo is a derivation of g clearly [g(i), gU)]

(4.1. 9)

C

g(i+j).

The decomposition (3.6.2) is related to

Xo

in the following way.

LEMMA 9. The nilpotent Lie algebras m and m* may be expressed in terms of the eigenspaces gU> of ad Xo as follows,'

Moreover h=g(O)=g"'o (i.e.

Xo

is regular).

Proof. Obvious from (4. 1. 7) and the fact roots cp and negative for negative roots cp. Since S*U) is in the range of the action of by (4.1. 3),

(4.1.10)



=

0 if f E J,

0

(cp) is positive for positive Q. E. D.

Xo

whenever j =1= 0 one has,

aE S*U) where j =1= o.

In the obvious way the symmetric algebra S*(u) over any subspace u C g may be regarded as a subalgebra of S*. Let b be the maximal solvable Lie subalgebra of g given by the direct sum ( 4.1.11)

(resp. put b* = m*

+ f).

One knows that if g = gl (d, C) then f (x) depends only on the diagonal entries of x in case x is a triangular matrix and f E J. More generally one has

+

PROPOSITION 17. Let x E b* so that x = y v where y E f) and v E m*. Then for any fE J one has f(x) =f(y). In particular

u(x) =u(y) where u is the map (3.3.2). Proof· Since J is graded we may assume f E Jk. Then, by (1.1. 3), k!f(x) = (om)\ f> = (Oy ov)\ f> = «Oy)" 0, f> = k!f(y) where,


-k

Now if (4.1.10)

f EJ

(4.6.7)

it follows, since

Zj

E g(ln j ), that by (1. 1. 2), (4. 1. 8) and

=0

for all 0 E 8*(p) where P =1= - mj; in particular for all p

>-

mj.

But now if k = mi in (4. 6.6) then the sum there is over all p where p>-mi. Hence if m;<mj, so that -m;>-mj, the sum in (4.6.6) is over all p, where p > - mj. Thus, by (4.6.7), (4.6.8)

+

for all x E e_ b whenever mi < mj. We now assert that this implies (4.6.9)

+

for any x E e_ '6 whenever mi < mj. Indeed replace f by Ui and divide by mil in (4. 6. 8) . Recalling that deg OZj Ui = mi the left side of (4. 6. 8) becomes the left side of (4. 6. 9) by (1. 1. 3) . On the other hand by (1. 1. 2) the right side of (4.6.8) becomes the right side of (4.6.9) by (4.6.2). (Recall that 8* is commutative.) This proves (4.6.3). But now if mi < mj then the right side of (4.6.9) vanishes by Lemma 14. Hence one obtains (4.6.4). Q. E. D. We can now show that the Jacobian matrix of functions OZjUi \ b of the map u b takes triangular form and reduces to non-zero constants along the diagonal.

370

380

BERTRAM KOSTANT.

6. There exists a unique basis zj, j 1, 2,' . " l,

THEOREM

that for i

=

(4.6.10)

=

1, 2,' . " l, of oe+ such

gi(zj)=8ij.

Furthermore the basis satisfies the condition of Theorem 5. That is for all j. Furthermore

Zj

E g(ffl l )

(4.6.11) so that not only is b transversal but in fact

(4.6.12) and hence (see § 4. 5)

d(b) =C.

(4.6.13)

Proof. An integer le will be called an exponent if le = mi for some i. Let E be the set of exponents and for any le E E let P k C {1, 2,' . " l} be the set of all i such that mi = le. Now, for any le E E put

It then follows from Lemma 15 that det OzJUi is a constant on e_ in fact

+ V and

+

But since b C e_ V and since b is transversal (Lemma 13) this constant can not be zero. Thus bk¥=O for any leEE. That is, the matrix g.. (Zj), i,jEP k , is non-singular and this holds for any le E E. It follows immediately then from Lemma 14 that a unique basis zJ of ge+ exists so that (4.6.10) is satisfied. It is also clear from L~mma 14 that the Zj necessarily satisfy the condition of Theorem 5. Since b C e_ V the remaining statements follow from Lemma 15. Q. E.D.

+

4. 7. We will assume from here on that the basis Zj of Oe+ is given by Theorem 6. Now let Sj E S (b) be the coordinate functions on b corresponding to the Zj. That is, Sj is such that x = e_ + ~ Sj (x) Zj. We have already noted that S(b) =C[Sl,' . ',sz] (see (4.5.1)). In notational simplicity let

(instead of u b ) denote the restriction of u to b. Thus for any x E b t(x)

=

(t1(x),' . " tz(x))

371

381

LIE GROL"P REPRESENTATIONS.

where

ti =

Ui

I b.

It follows therefore from (4. 5. 2) that

,,= Oti

(4.7.1)

uS;

OZJUi

I b.

Now if u is an arbitrary k-plane in 9 let J

(4.7.2)

~S(u)

be the ring homomorphism obtained by restricting an invariant polynomial to u. Now in general one could hardly expect (4. 7. 2) to be an isomorphism. Indeed if (4. 7. 2) is an epimorphism one must have k > l and if (4. 7.2) is a monomorphism one must have k < l (since the Ui are algebraically independent) . Hence the possibility could only exist if k = l. If u is a Cartan sub algebra the one knows that (4. 7. 2) is a monomorphism and the image is the space of Weyl group invariants. Hence in such a case (4.7.2) is an isomorphism only when 9 is abelian. On the other hand when u = b we have, in general, the following corollary of Theorem 6 THEOREM

7.

If u = b then (4. 7. 2) is an isomorphism. Moreover ill e

map (4.7.3) obtained by restricting U to b is an algebraic isomorphism so that t 1 , · define a global coordinate system on b.

.

.,

tl

Furthermore the relationship between the ti and the linear coordinates S; on b is as follows: For i = 1, 2,· .. , l, there exists polynomials Pi and qi in i - I variables without constant term such that (4.7.4) and (4.7.5) Proof. To prove the theorem observe that it suffices only to prove (4. 7. 4). Indeed using (4. 7.4) we can solve for Si obtaining (4. 7. 5) . It is then immediate that t is one-one, onto and is in fact a biregular birational map. Since the ti generate the image of J in S (b) it is then also obvious that (4. 7. 2 ) is an isomorphism. But now (4.7.4) is immediate from (4.6.11) and (4.7.1). Finally by definition of the coordinate system Si one has Si ( e_) = 0 for all i. On the other hand ti ( e_) = 0 for all i since t ( e_) = U ( e_) = 0 (recall that e_ is nilpotent). Thus the Pi and qi have no constant term. Q. E. D.

372

382

BERTRAM KOSTANT.

Any orbit 0 of semi-simple elements (i. e., 0 E (h) intersects ~ in a finite number but in general more than one point. We now find that any orbit 0 of maximal dimension (i. e. 0 E @t) intersects 0 in one and only one point. THEOREM

8.

One has 0 C r.

j1'urihwrrnore if

(4.7.6)

is the map given by x --?o Om then (4. 7. 6) is a bijection. That is, no two distinct elements of 0 are conjugate and every element in r is conjugate to one and only one element in 0.

+

Proof. Since 0 C e_ fJ one has 0 C r by Lemma 10, § 4. 2. But now if we compose (4.7.6) with the bijection T)t (Theorem 2, §3.5) we obtain the bijection t (see Theorem 7). Hence (4.7.6) must be a bijection. Q. E. D. We can now obtain the following characterization of the set r. THEOREM 9. Let x E g. Then x E r if and only if (dui) x, i are linearly independent.

=

1, 2, ... , l,

Proof. By (4. 6. 12) the matrix (OZ/Ui) (x) is of rank l for any x E 0. Thus (du.) m, i = 1,· .. , l, are linearly independent for any x E 0. But then by Theorem 9 and conjugation the same is true for any x E r. Now let x E g but where x ~ r. We must prove that the (dui).;, i = 1, 2,· . ., l, are linearly dependent. Assume first that x E's (that is, x is semi-simple). Then x is not regular so that gm contains a Cartan subalgebra as a proper subalgebra. It follows therefore that if u is the center of gm and lm = dim u one has lo; < l. Furthermore it is also clear that It is the set of l1xed vectors for the action of Gm on g (recall that Gm is connected. See Lemma 5, § 3. 2) . Thus there exists a non-abelian simple component gl of g of rank, say ll' such that in the notation of § 2. 1 d 2.

Proof. Let J' be any ideal in J. It is immediate that J'S is the image of J' ® H under the isomorphism (4. 8. 4) and hence one has

(4.10.6)

J'S

n J =J'.

Now assume that J' is a radical ideal (an ideal equal to its own radical) in J. We will show that J'S is a radical ideal in S. Let J' * be the radical ideal in S (CZ) corresponding to J' under the isomorphism J ~ S (C!) where f ~ f* and let U C C! be the Zariski closed set of all t E Cl at which J' * vanishes. It is obvious that if u is the Zariski closed set, in g, of all x E 9 at which J'S vanishes then U = u- 1 (U) or (4.10.7)

u= U pa)· ~E

U

To prove J'S is a radical ideal it suffices to show that if f E S is assumed to vanish on U then f E J'S. By Theorem 11, § 4.8, we can write f = ~ fihi where fi E J, hi E H and the hi are linearly independent. Let t E U. Then since the fi reduce to constants on P it follows from the isomorphism (4.8.6) and (3.8.7) that since f vanishes on pet) the fi also vanish on pa). Thus the fi are in J' by the nullstellensatz and hence f E J'S so that J'S is a radical ideal. Now let U E 'U so that JU is prime in J. Put J' = JU so that, from above, JUS is a radical ideal in S. To prove that JUS is prime it suffices now only to show that u is irreducible.

a)

380

390

BERTRAM: KOST.ANT.

Let f E S and let U (f) be the set of all H U such that f I P (0 is not zero. Obviously flu is not zero if and only if U (f) is not empty. We first show that in such a case U (f) contains a non-empty Zariski open subset of U. Indeed assume U (f) is not empty and ~ E U (f) . Then f I Ot (~) is not zero by (3. 8. 7). Hence there exists a E G such that (a· f) I Ot (~) n b is not zero, by Theorem 8, § 4. 7, where b is defined as in (4. 5. 6) . Thus (a· f) I b n b. Using the does not vanish on a Zariski subset of b containing Or isomorphism (4. 7. 3) it follows that U (a· f) contains a non-empty Zariski open subset of U. But clearly U (a· f) = U (f). Hence U (f) contains such a subset.

un

Now let fi E S, i = 1, 2, be arbitrary except that fi I u is not zero. To show u is irreducible we must show that fd2 I u is not zero. From above it follows that U (fi) contain a non-empty Zariski open subset of U. But since U is irreducible it follows that U (fl) n U (f2) is not empty. But then fd2 I P (~) is not zero in case ~ E U (h) n U (f2) since pa) is irreducible by Theorem 3, § 3. 8. Thus u is irreducible and hence JUS is prime. The relation (4.10.1) is just (4.10.7). then (4.10.3) follows from (3.8.4).

Furthermore if u = u(JuS)

Moreover, using (4.10.1), it is immediate that the map, given by (4.10.2), from 'U into the set of all Zariski closed G-stable subvarieties u of g such that u n r is not empty is injective. Now assume that u is such a subvariety. We will show that It is in the image of the map defined by (4.10.2). Let the set U C cz be defined by putting U = U (u n r). Since u is Zariski irreducible and u n r is Zariski dense in u it follows that U is 'Zariski irreducible. On the other hand by Theorem 8, § 4.7, it is clear that U corresponds to u n b under the isomorphism (4. 7. 3). But since u n b is Zariski closed in b it follows that U is Zariski closed in C z. Hence U E 'U. But U is Zariski dense in u (u). But this implies u (u) = U since U is Zariski closed. Thus u C u-1 (U). But u- 1 (U) is clearly in the Zariski closure of unr since the relation (4.10.3) obviously holds. Thus u=u-1(U) or u=u(JuS). Now obviously (dUi*)~' i = 1, 2,· . ., l, are linearly independent at any point ~E cz. Since (df)a: is in the span of the (dUi)g; for any fE J and xE g it follows from Theorem 9, § 4. 7, that (df) g; = 0 if and only if (df*) ~ = 0 for any f E J and x E r, where ~ = U (x). It follows in particular that if U E 'U and x E u n r where u = u (JuS) then the dimension ra: of the space spanned the (df) g; for all f E JU is the same as the dimension r~ of the space spanned by all (df*) ~ where f E JU and ~ = U(x) . If r is the co dimension of

381

391

LIE GROUP REPRESENTATIONS.

U in CZ and Us is the set of simple points of U then by the Zariski criterion for all ~E U and r~=r if and only if ~E US. Thus rl1J

for any p. E Z it is then clear from (4.1. 6) that 0 (p.) is always an integer. Now since every weight of II is clearly necessarily an element of Z it follows therefore that V = ~ V(k) k€Z

is a direct sum where value k. Obviously

V(k)

is the eigenspace of II(Xo) belonging to the eigen-

(5.3.1) Now e_ be the principal nilpotent element defined as in § 4. 2. For notational simplicity write F = vo e-. Since Ge_ is connected (Proposition 14, § 3. 6) one also has (5.3.2)

F=Kerll(ge-)

and by (5.1.1) dimF=lv where lv is the dimension of the zero weight of v. Now since Xo lies in the normalizer of ge_ it follows from (5. 3. 2) that F is stable under v (xo). But then we observe that there exists a unique sequence of integers mi (II), i = 1, 2,· . ., lv, where m1(V) < ... <mlv(lI)

such that F has a basis

Vi,

i = 1, 2,· . ., l., where

(5.3.3) Remark 25. By applying the inner automorphism which carries e+ into e_ and Xo into - Xo note that we would get the same integers mj (II) by using e+ into instead of e_ and dropping the minus in (5.3. 3). Observe then that the mi(v) generalizes the notion of exponents. Indeed if II is the adjoint representation then lv = l and, by Theorem 5, § 4. 4, mi(V) = mi since F = ge_. Since F C Ker v (e_) it follows from the representation theory of a three dimensional simple Lie algebra (e. g. see [13], § 2. 5) that for any i

(5.3.4)

385

LIE GROUP REPRESENTATIONS.

395

Now, as in Remark 22, § 4. 8, identify D with the (subset of Z) set of all dominant integral (with respect to G) forms on f) so that any A ED is the highest weight of vA. Note then that - A is the lowest weight of VA. When V = VA and v = VA we will write FA for F and miCA) for mi(VA). The miCA), i=1,2,·· ·,lA, will be called the generalized exponents of g (corresponding to A). See Remark 25. Now let VA EVA be the lowest weight (- A) vector. Then since VA, as one knows, is a cyclic module with respect to the universal enveloping algebra of m with VA as cyclic vector it follows that

(V ) (-k)

_

A

-

{O

if k> O(A) (VA) if k=O(A).

Since VA (-o(A» is obviously contained in FA. (One uses the relation ge_ C m* a mentioned in the proof of Theorem 5, § 4. 4). It follows then that (5.3.5) miCA) <mIA(A) =O(A)

+

for 1 ..) is obviously isomorphic to the tensor product J ® H (A) by Theorem 11, § 4. 8, it is clear that such information is needed if one is to determine the formal power series 00

qA ( t)

=

"2:. dim S (A) k

tk

k=O

and, as a consequence, the multiplicity of vA in Sk for any k.

386

396

BERTRAM KOSTANT.

The following theorem asserts that the ni('\) are exactly the generalized exponents mi('\). THEOREM 17. For any ,\ ED and i = 1, 2,· .. , lA, one has ni('\) = mi('\) so that Hi (,\) C Sm,(A). In particular 7c = 0 (,\) is the maximum degree such that H(,\)k#O. Furthermore H(,\)k is irreducible for this value of le. That is,

Moreover the formal power series

qA (t)

(5.4.2)

qA (t)

may be given by

= dA _i7~1_ _ __

II (l-tml) i=l

where d A = dim VA. Proof. Let V'i' i = 1, 2,· . ,h, be a basis of FA such that V'i EVA (-m,(A». Now let c E C* be arbitrary. Let r E C be such that e- r = c and let a E G be defined by putting a = exp r ad Xo. It is then clear that (5.4.3) JIA (a) v'i = Cm,(Alv'i. Also note that (5.4.4) Now by Corollary 14, § 5. 2, there exists a basis Yi, i

=

1, 2,· . ., h, of

Homa(VA,H) such that But now by (5.4.4-5) and (5.2.2) Olle getR the equation (5.4.5)

cm,(A)we_ (Yi)

=

Wce_ (Yi).

Substituting in (5. 2. 1) this implies that for any v E VA

(Yi(V» (ce_)

=

Cm,(A)Yi(V) (e_).

But then, conjugating by G (and using (1. 1. 5)

it also follows that

Yi ( v) (cy) = Cm,(A)Yi ( v) (y)

for all yEO e_. But then since (4. 8. 6) is an isomorphism for x = e_ it follows easily by choosing c, for example, to be positive that

Yi(VA) C Sm,(A). On the other hand by definition of the Yi

387

397

LIE GROUP REPRESENTATIONS.

is a direct sum of irreducible G-modules. By uniqueness of the nt(A) it then follows that ni(A) = mi(A) for i= 1,' .. , h. The second and third statements of the theorem follow immediately from (5.3.5). The equation (5.4.2) is an immediate consequence of the obvious fact that the isomorphism (4.8.4) induces an isomorphism of J®H(A) onto 8 (A). Q. E. D. Remark 26. We observe here that Theorem 17 is a generalization of Theorem 5, § 4. 4, asserting that mi(v) = mi where v is the adjoint representation. Indeed let U be the subspace of all U E J+ such that f for some ~' E D 1 . ) 17. For any ~ E D1 and A ED let nJ\ in End V 1 ~ (regarded as a G-module). Then

LEMMA

of

vJ\

a)

denote the multiplicity

(6.2.5)

and (for fixed A) the equality holds for

~

sufficiently large.

Proof. Since (6.2.4) is an epimorphism the inequality (6.2.5) is an immediate consequence of Theorem 21, § 6.1. However a much simpler and more direct proof of the inequality may be given using § 4. 1 in [12 J. Now identifying V 1(@ V1.~ with End V1~ and regarding G-modules as G1 -modules it follows immediately from Schur's lemma, upon forming the triple tensor product VJ\@V1~@V1.~' that nJ\(~) is also the multiplicity of V1~ in VJ\ @ V1( We refer now to [12J, § 4. 4, for the definition as to when A is totally subordinate to~. By Theorem 5.1, (3) in [12J A is totally subordinate to ~ for ~ sufficiently large. :{3ut now by (6) in this theorem (where JL = 0, A2 = A, A1 =~) the multiplicity of V1~ in VJ\ @ V1~ is lx whenever A is totally subordinate to~. Hence nJ\ (~) = lJ\ whenever A is totally subordinate to ~ or when ~ is Q. E. D. sufficiently large. Harish-Chandra proved in [8J that if Y C U is anyone-dimensional subspace there exists ~ E D1 such that V1~ is faithful on Y. This is not true in general for higher dimensional subspaces. For example if p E Z and q E U where q ¥= 0 and p ¢ U o then q and pq span a two dimensional space in U but its image under V1~' for any ~ E D 1 , is at most a one dimensional space. We now observe, however, that the generalization is true provided that Y C E. THEOREM 23. Let Y C E be any finite dimensional subspace. Then the irreducible representation V1~ is faithful on Y for all ~ E D1 sufficiently large.

Proof·

Since Y is finite dimensional there exists k such that Y C E k •

394

404

BERTRAM KOSTANT.

N ow let 0 CD denote the set of all A E D such that vA occurs with positive multiplicity in E k • Since E is finite dimensional it is obvious that 0 is a finite set. N ow since Dl is a directed set it follows that equality holds in (6.2.5) for all A E 0 and all ~ sufficiently large. But then by Theorem 21, § 6.1, the map (6.2.4) is an isomorphism also for all AE C and ~ E Dl sufficiently large. Thus Vl~ is faithful on Ek and hence on Y for all ~ sufficiently large. Q. E. D.

REFERENCES.

[I] A. Borel and Harish-Chandra, "Arithmetic subgroups of algebraic groups," Annals of Mathematics, vol. 75 (1962), pp. 485-535. [2] C. Chevalley, "Invariants of finite groups generated by reflections," American Journal of Mathematics, vol. 77 (1955), pp. 778-782. [3] - - - , Fondements de la geometrie algebrique, Course notes at the Institut Henri Poincare, Paris, 1958. [4] A. J. Coleman, "The Betti numbers of the simple Lie groups," Canadian Journal of Mathematics, vol. 10 (1958), pp. 349-356. [5] E. B. Dynkin, "Semi-simple subalgebras of semi-simple Lie algebras," American Mathematical Society Translations, ser. 2, vol. 6 (1957), pp. I Il-244. [6] F. Gantmacher, "Canonical representation of automorphisms of a complex semisimple Lie group," Matematii5eskii Sbornik, vol. 47 (1939), pp. 104-146. [7] Harish-Chandra, "On a lemma of Bruhat," Journal de Mathematiques Pures et Appliquees, vol. 9, 315 (1956), pp. 203'-210. [8] - - - , "On representations of Lie algebras," Annals of Mathematics, vol. 50 ( 1949), pp. 900-915. [9] G. Hochschild and G. D. Mostow, "Representations and representative functions on Lie groups, III," Annals of Mathematics (2), vol. 70 (1959), pp. 85-100. [10] S_ Helgason, "Some results in invariant theory," Bulletin of the American Mathematical Society, vol. 68 (1962), pp. 367-371. [II] N. Jacobson, "CompJetely reducible Lie algebras of linear transformations," Proceedings of the American Mathematical Society, vol. 2 (1951), pp. 105-133. [12] B. Kostant, "A formula for the multiplicity of a weight," Transactions of the American Mathematical Society, vol. 9 (1959), pp. 53-73. [13] - - - , "The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group," American Journal of Mathematics, vol. 81 (1959), pp. 973-1032. [14] M. Rosenlicht, "On quotient varieties and the affine embedding of certain homogeneous spaces," Transactions of the American Mathematical Society, vol. 101 (1961), pp. 2Il-223. [15] J. P. Serre, "Faisceaux algebriques cohere;It," Annals of Mathematics, vol. 61 1955), pp. 197-278. [16] A. Seidenberg, "The hyperplane sections of normal varieties," Transactions of the American Mathematical Society, vol. 64 (1950), pp. 357-386. [17] R. Steinberg, "Invariants of finite reflection groups," Canadian Journal of M athematics, vol. 12 (1960), pp. 616·618. [18] O. Zariski and P. Samuel, Commutative Algebra, vol. I, van Nostrand Company, Princeton, 1958. r19] - - - , Commutative Algebra, vol. II, van Nostrand Company, Princeton, 1960.

395

ANNALS OF MATHEMATICS

Vol. 77, No. I, January, 1963

Printed in Japan

LIE ALGEBRA COHOMOLOGY AND GENERALIZED SCHUBERT CELLS By BERTRAM KOSTANT

(Received January 5, 1962)

1. Introduction

1.1. This paper is referred to as Part II. Part I is [4]. The numerical I used as a reference will refer to that paper. A third and final part, Clifford algebras and the intersection of Schubert cycles is also planned. In a word let X be any compact algebraic homogeneous space of positive Euler characteristic. We solve here the problem of § 1, Part I, by constructing on X closed invariant differential forms whose cohomology classes are dual to the Schubert homology classes. These differential forms are defined using irreducible representations of the isotropy group on the homology of a nilpotent Lie algebra (although long suspected, the results are new even in the case of the grassmannian). The method introduces a new type of laplacian. In more detail let 9 be a complex semi-simple Lie algebra and let u be any Lie subalgebra which contains a (fixed) maximal solvable Lie subalgebra of 9. If n is the maximal nilpotent ideal of u the cohomology group, H(n, V), where V is any 9 (and hence n) module, was determined in Part 1. In fact H(n, V) is a 91-module where 91 = u n u* (the *-operation on 9 is defined relative to a fixed compact form f of 9) and the irreducible components were shown to be in a natural one-one correspondence with a subset W1 of the Weyl group Wof 9. See I, Theorem 5.14. (Since u = 91 + n is a Lie algebra semi-direct sum, one could just as well have replaced 91 byujn.) In Part II we will only need the result for the special caseH(n), that is, where V is the trivial module. In fact it is somewhat more convenient to dualize and deal with homology, H*(n), instead of cohomology for n. Now let G and U be a Lie group and subgroup corresponding to 9 and u. If X = Gj U then X is complex compact algebraic homogeneous space and every such space of positive Euler characteristic is of this form. One knows (Chevalley-Borel) that X may be written as a disjoint union

X-u uew1 Vrr where Vrr is the orbit of N (the unipotent subgroup corresponding to n) on X defined by a. Furthermore the Vrr , called Schubert cells (generalizing 72

B. Kostant, Collected Papers, DOI 10.1007/b94535_18 , © Bertram Kostant 2009

396

LIE ALGEBRA COHOMOLOGY

73

terminology from the case where X is the grassmannian) are homeomorphic to cells and define a basis of the homology group of X, and hence define a dual basis x", a E W\ of H(X, C). The problem, posed in § 1 of Part I, is how to relate H(X, C) with the gcmodule HAn) or, more specifically, how to relate, for any a E W\ the class x" E H(X, C) with the irreducible component H*(n) H(g, gl) . Theorem 4.5 then asserts that d and a are disjoint. Thus this proves, for one thing, that H(C, d) and H(C, a) are isomorphic, and in fact (using Ker S, S = da + ad) establishes an isomorphism (1.1.3)

H(C, a)

---->

H(C,d) .

One therefore recovers, in a purely algebraic way, all the known results on the cohomology group H(X, C). That is

*

if p q if p = q (See I, §5.15) . But, more than this, disjointness implies that in each class s E H( C, d) there exists one and only one cocycle 8 E s such that a8 = 0; namely, that (harmonic) cocycle such that 8 E Ker S. Now let s"', (J E WI, be the basis of H(C, d) defined as the image of h", (J E WI, under the map (1.1.3), and let 8" E s'" be the harmonic representative. If oJ" is the K-invariant differential form corresponding to 8" under the map (1.1.1), it is obvious that s" corresponds to [w"'] under (1.1.2). Thus to HAn)'" we have associated a class [w,,] E H(X, C), and a distinguished representative differential form W".

For the proof of disjointness another result is needed. Theorem 4.3 asserts that H(n, 1m an)gl = 0 .

398

LIE ALGEBRA COHOMOLOGY

75

Here an E EndAn is the boundary operator of C*(n) and Iman C;;;;; An is an n-module with respect to the adjoint representation of n on An. The whole cochain complex C(n, 1m an) is easily seen to be a 9cmodule. The relation to the Schubert classes is established by Theorem 6.15. This asserts that up to scalar multiple, [(0""] equals x"". The exact scalar )..,fT will be determined in Part III. In the present paper, however, we give an integral formula for the scalar )..,fT. An essential role in the proof of Theorem 6.15 is the fact that as"" = O. Another application of this property is that if Y = G/ B is the generalized flag manifold and Y - X is the projection defined by inclusion B - U, then harmonic forms on Y, in our sense, go into harmonic forms on X. This is not true for the usual definition of harmonic forms. Other applications of the property as fT = 0 will be made in Part III. Theorem 5.6 gives an explicit formula for computing sfT for any a E WI. Thus, together with Theorem 6.15 and the knowledge of the scalar )..,fT, this formula constructs closed differential forms whose cohomology classes define the dual basis to the Schubert homology classes. 1.2. The definition of t, and the operators d, a and S in the paper are different from that indicated above. They are defined in more general terms. In fact in our definition, t has nothing to do with g. It is simply a complex Lie algebra with a real form tR which itself has an underlying complex structure. It also has a hermitian structure. However to prove the disjointness of d and a we have to assume that t is essentially like grin g. In point of fact we eventually assume it to be the space of complex 1-covectors at the origin of X. The Lie algebra structure on t is then motivated by the fact that tR is in a natural way (real) isomorphic with n. This isomorphism, incidentally, is independent of the choice of the compact form f of 9. In the introduction to Part I, it was remarked that the non-zero eigenvalues of the laplace L" of I, Theorem 5.7, will be needed in Part II. They are in fact used here in Theorem 5.6. However their main use will be in Part III.

2. A family of operators defined by a Lie algebra with a hermitian structure

2.1. We adopt the following conventions. Assume that V is a vector space over C, the complex numbers. Unless called real, a subspace of V will always mean a complex subspace. In case V is graded

399

76

BERTRAM KOSTANT

a subspace Vl ~ V will be said to be graded if it is graded by the intersections V! = VJ n Vl. In case V is, in addition, bi-graded

V = '" L...Jp,q

vp,q

,

the bi-grading will always be consistent with respect to the grading. That is

and a subspace V l ~ V will be said to be bi-graded if it is bi-graded by the intersection Vr q = Vp,q n Vl. In case V l and V 2 are graded (resp. bi-graded) vector spaces, a homomorphism A: V l ---> V 2 will be said to be of degree j (resp. of bi-degree if (s,

t»

for all i (resp. A: VI p q ---> Vr' Ht for all p, q). If V l and V 2 are complex vector spaces, a linear mapping A: V l ---> V2 , unless called real or R-linear, will always mean a complex linear mapping. For convenience we will write End V for End c V. 2.2. If a real vector space is complexified, one defines in a natural way an operation of conjugation on the complexification. If the real vector space is a real Lie algebra, then the complexification carries a Lie algebra structure. However, if the real Lie algebra itself carries a complex structure, then one knows that its complexification decomposes into a direct sum of two commuting ideals, each the conjugate of the other. We will be concerned here with a family of operators on an exterior algebra that arises from this situation when the real Lie algebra carries a positive definite real bilinear form; or, equivalently, when one of the ideals carries a hermitian positive definite inner product. No connection is assumed to exist between the Lie algebra structure and the bilinear form. Most significant for us is the operator St (see (2.8.2).). Let t be a finite dimensional complex vector space. For any U EAt let s(u) E EndAt be the operator of left exterior multiplication by u on At. Thus s(u)v = ul\v for any u,v EAt. A real linear form tR of t is a real subspace tR such that t = tR + itR is a real direct sum. Assume such a space is given once and for all. It is clear then that At = ARtR

+ iARtR

is a real direct sum. Besides being a real linear form of A t, it is clear

400

LIE ALGEBRA COHOMOLOGY

77

that ARtR is a real subalgebra of At. One introduces an operation of conjugation in A t by putting

u

+ iv =

u - iv

for any u, v E A RtR. Conjugation is a conjugate linear automorphism of order 2 and degree zero (with respect to the grading of At) of At. It ind uces a similar operation on End At where if A E End A t then A E End At is defined by Au=Au

where u E At is arbitrary. The fact that conjugation on At preserves multiplication may then be expressed by the relation (2.2.1)

s(u) = s(u)

for any u EAt. If p ~ t is a subset, then :P ~ t is defined by conjugating all the elements in p. It is obvious that :P is a subspace whenever p is a subspace.

2.3. Now assume that j E EndAt is an automorphism of degree zero (and hence determined by its restriction to t) such that (1) j" = -1, where 1 is the identity operator on At, and (2) tR is stable under j. It is clear then that t=a+a is a direct sum where (if i =

vi -1)

0=

{xEtijx

=

ix}.

By putting

A'Mt

= (APa)/\(Aqa) ,

we observe that one thus defines a bi-grading of At. It is also clear that

(2.3.1) 2.4. If V is a complex vector space, then as in Part I, {V} denotes a positive hermitian inner product on V and (V) is a bilinear form on V. About {V}, we always assume that for any j, g E V, {f, g} = {g,j} ,

and for any A. E C, A.{j, g}

= {:\,j,

g} = {j, :\g} .

We now assume that {a} is defined. We then observe that there is a

401

78

BERTRAM KOSTANT

unique {t} such that a1-

(2.4.1)

=a

(the orthogonal complement being taken in t); and for any x, yEt, {x, y}

(2.4.2)

= {y, x}

;

and {a} is the restriction of {t} to a. (We could have defined {t} by starting with a positive definite bilinear form on tR with respect to which j ItR is an orthogonal transformation.) But now {t} defines, in the usual way, a positive definite hermitian inner product {At} on At. That is, with respect to {At}, if p if p

(2.4.3)

*q =

q

REMARK 2.4. It is immediate from (2.4.2) and (2.4.3) that (2.4.2) holds more generally for all x, YEA t. Also observe that (2.4.1) generalizes to AMt and AP',q't are orthogonal

(2.4.4) if (p, q)

* (p', q').

2.5. Now define a bilinear form (At)a on At by putting (u, v)a = {u, v}

for any u, v E At. It is obvious that (At)a is non-singular and (by Remark 2.4) symmetric. We note also that (At)a agrees with {At} on ARtR and hence, in particular, is positive definite there. Now if A E EndAt, denote by AS E EndAt the transpose of A with respect to (At)a, and by A* E EndAt the adjoint of A with respect to {At}. PROPOSITION 2.5. The three operations A ----> A, AS and A* on EndAt are related in the following way: The operations commute with each other, and the composite of any two is the third. That is, these operations together with identity operation on End A t form a group, and this group is isomorphic to the Klein 4-group. PROOF. Since each is of order 2 it suffices only to show that

A* = AS for any A E EndAt. Let u, v E At. Then {Au, v} = (Au, V)B = (u, ABv)B = {u, A8V} = {u, ABv}. But {Au, v} = {u, A*v}. Now let 0 E EndAt be the automorphism of At defined by

402

LIE ALGEBRA COHOMOLOGY au

=

79

(-l)Ju

for any integer j and u E A Jt. Let A E EndAr. If we put At = aA5a ,

(2.5.1)

then At E End A t is the transpose of A with respect to the symmetric bilinear form (At) defined by (2.5.2)

(u, v) = (u, av)s = {u" av} .

REMARK 2.5. Since (At) is non-singular it defines an algebra isomorphism (2.5.3) of At onto its dual and if A E EndAt, then At corrasponds under (2.5.3) to the usual transpose of A formed on At'. We observe that by (2.3.1) and (2.4.4) (2.5.4)

(u, v)

=

(u, v)s

=0

for any u E A p,qt and v E AP',q't unless (p, q) = (q, p). One defines t(u) E EndAt for any u E At by (2.5.5)

t(u)

=

c(u)t ,

and we recall that (see Remark 2.5) t(x) is a derivation of degree -1 for x E t and that (2.5.6)

c(x)t(y)

+ t(y)c(x) = (x, y)l

for any x, yEt. Here 1 E EndAt is the identity operator. 2.6. Now assume that 0 is a (complex) Lie algebra. We then see that there exists a unique Lie algebra structure on t such that (2.6.1)

[x, y]

= [x, 17]

for all x, yEt (which implies that tR is a real Lie subalgebra of t) and (2.6.2)

[0, a]

=

0

so that 0 and a are ideals in t. (Equivalently we could have assumed that tR is a real Lie algebra such that j ItR commutes with the adjoint representation and hence defines a complex structure on tR and that t is the complexification of t R.) Let at E EndAt be the boundary operator of the chain complex C*(r). Thus a~ = 0, and ar(x/\y) = [x, y] for any x, yEt. It is then obvious from (2.6.1) that

403

80

BERTRAM KOSTANT

(2.6.3) Now since ar anti-commutes witn 0, we can define br E EndAt by putting t

(2.6.4)

br = -ar =

S

ax

and note that, by Remark 2.5, br corresponds under (2.4.4) to the coboundary operator of C *(t), and hence br is a derivation of degree 1. By (2.6.3), (2.6.4), and Proposition 2.5, one has (2.6.5) br = ar* , and hence a positive semi-definite hermitian operator Lr E End A t of degree zero is defined by putting

+ brar .

Lr = arb r

(2.6.6)

By (2.6.5) the operator Lr is the laplacian defined by at and (see I, Remark 2.3) its kernel is isomorphic to H*(t).

a; and hence

2.7. For any x E t, let n(x) E EndAt be defined by (2.7.1)

n(x) = c(x)ar

+ arc(x)

.

Then one knows that n: t

----->

EndAt

is the adjoint representation of t on At (see e.g. [5]) . Now let nt: t

----->

EndAt

be the representation defined by putting (2.7.2) for any x E t. Since nt(x) is clearly of degree zero, we can now define a new bracket relation [x, y]~ on t by putting (2.7.3) for any x, yEt. We recall (see I, § 3.6) that any linear mapping t -----> A 2t is the restriction of a unique derivation of degree 1 of At. Since [x, y]" is clearly alternating in x and y, we therefore observe that there exists a unique derivation d r E EndAt of degree 1 of At such that (2.7.4)

(drZ, x/\y) = -(z, [x, y],,)

for all x, y, Z E t. Now let er E EndAt be the derivation of degree 1 of At defined by

404

LIE ALGEBRA COHOMOLOGY (2.7.5)

dt

=

bt

81

+ Ct .

One now finds that Ct is given explicitly by PROPOSITION 2.7. Let Xi be any basis of t, and let y j be the basis of t defined so that (x., yj) = O.i. Then (2.7.6)

Ct

=

EAxi)n(y,) .

PROOF. Let et E EndAt be the operator given by the right side of (2.7.6). Since n(y.) is a derivation of degree zero, it follows immediately that et is a derivation of degree 1. Therefore it suffices only to prove that CtZ = etZ for all Z E t or that (2.7.7) for all x, yEt. But now by (2.6.4) and the definition of Ct, it follows that -cf(x 1\ y)

=

-ei(xl\y)

= (E,nt(Yi)c(X,»)Xl\y = nt(x)y - nt(y)x ,

nt(x)y - nt(y)x .

On the other hand q.e.d.

This proves (2.7.7). 2.8. One of our principal concerns in this paper will be with the operator St E EndAt (or rather its restriction, S, to the subspace C ~ At defined in § 3.9) of degree zero defined by (2.8.1) By (2.7.5) and (2.6.5) it follows that (2.8.2)

St = L t

+ Et

where (2.8.3) The operator E t is given explicitly by PROPOSITION 2.8. Let Xi and Yj be as in Proposition 2.7. Then Et

=

Etn(xi)n(Yj) .

PROOF. We have only to apply Proposition 2.7, the fact that n(y.) commutes with at (follows from (2.7.1», (2.7.1) itself, and of course (2.8.3). 2.9. In this section we give an example (to be used later) of a space t having the structure assumed in §§2.2-2.6. Proposition 2.9 gives an explicit expression for the operator d r for this case.

405

82

BERTRAM KOSTANT

We recall some of the notation and definitions of Part I. First of all, 9 denotes the complex semi-simple Lie algebra considered in Part I, {) is a fixed maximal solvable Lie subalgebra of g, and V is the family of all Lie algebras u such that {)~u~g.

The Cart an-Killing form on 9 is denoted by (g), and its extension to Ag is denoted by (Ag). Given any subspace $ ~ g, the polar (see I, § 4.1) of !3 in 9 with respect to (g) is denoted by !30 • If u E V then one knows (I, Proposition 5.3) that n = U O is the maximal nilpotent ideal in u. A real compact form f of 9 has been fixed once and for all. With respect to t, a *-operation and a positive definite hermitian inner product {Ag} have been defined on Ag. We recall that if gl = un u* for u E V then

+n

u = gl

is a Lie algebra semi-direct sum (with n as ideal) and

(2.9.1)

+ n + n*

9 = gl

is an orthogonal (linear) direct sum. Now put

(2.9.2)

tx = n

+ n*

.

In this section t will equal t x. We will now put structure on Xx so that the assumptions §§ 2.2-2.6 are satisfied. The subscript X is used to distinguish this example from the general case. A real linear form Xx.R is defined by putting

(2.9.3)

tX.R

= Xx n f

,

and conjugation in A tx is defined relative to this real form in the same way as in § 2.2. Recalling the definition of the *-operation in A 9 (see I, § 3.3) note that it follows, for any U E A xx, that

(2.9.4) where Ox E EndAtx is defined in a similar way to O. Obviously then n*

= it.

Now let ix E EndAtx be the automorphism defined so that and - i on n*. Then, here n

ix

= i on n

=0.

Now let {n} be the restriction of {Ag} to n. It follows that if {A xx} is defined as in §2.4, then {Axx} is just the restriction of {Ag} to Axx.

406

LIE ALGEBRA COHOMOLOGY

83

Furthermore recalling (2.5.2), (2.9.4), and (I, (3.3.1», it also follows that (A xx) is just the restriction of (Ag) to A xx. We recall (see §2.6) that Xx is made into a Lie algebra by a Lie algebra structure on n = Q. For the latter we use the structure induced on n as a Lie subalgebra of g. Thus if x, y E X and x = e1 + f1 and y = e2 + f2 where ei , fi E n, i = 1, 2, then denoting the bracket in Xx with the subscript X one has (2.9.5) where the brackets on the right are the usual brackets in g. Let .p be any subspace in g. For any x, y E g let [x, yh be the component of [x, y] in.p according to the decomposition g = .p + .pl.. We now observe LEMMA 2.9.1. Let x, y E Xx. T hen if n is given as in § 2.7 where x = Xx and .p denotes either n or tt one has if x, y E 'p nt(x)y = {O [x, y]P if x E l3 and y E 'p • PROOF. Let

Z

E Xx. By definition

(2.9.6) If x, Y E .p then this expression vanishes since 'p is an ideal in Xx and (A xx) is totally singular on.p. This proves that nt(x)y = O. In case x E:P and y E.p, then nt(x)y can have no component in l3 since (2.9.6) vanishes if Z E.p (because [:P, .p]x = 0). To find its component in .p, it is enough to let Z E j). But then [x, z]x = [x, z]. Hence nt(x)y = [x, y]p by the invariance of the Cartan-Killing form. q.e.d. As an immediate consequence of Lemma 2.9.1 we now observe that if x E:P and y E.p, where .p is either n or tt, then

(2.9.7) Now consider the bracket relation (see (2.7.3» [x, y]" for the case X= XX. LEMMA 2.9.2. Let x, y E Xx. Then [x, y]"

= [x, Y]rx •

PROOF. By definition [x, y]" = [x, Y]x + nt(x)y - nt(y)x. Writing x and y as in (2.9.5) it follows from Lemma 2.9.1 and (2.9.7) that nt(x)y - nt(y)x =

[ell h]r x + L{;., e2]rx' Adding this to (2.9.5) proves the lemma. q.e.d. We recall some further notation of Part I. By definition 'Y E EndAg is the boundary operator for the chain complex GAg) and c = -'Yt so that C E EndAg corresponds to the coboundary operator of G*(g) under the

407

84

BERTRAM KOSTANT

isomorphism Ag -> Ag' induced by (Ag). Now let

r: Ag -> Arx be the orthogonal projection of Ag onto A rx. Lemma 2.9.2 may then be expressed by the relation (2.9.8) for any x, y

r'Y(X/\Y) E

= [x,

y)'"

rx. One may then prove

PROPOSITION 2.9. In the case r d rx

= rx one has = rc

on Arx .

PROOF. This follows immediately from the definition of drx (see (2.7.4», (2.9.8) and the fact that rc is a derivation of degree 1 in Arx (see I, Ftemark 3.8). q.e.d. 2.10. We continue in this section with the example of §2.9. As in Part I, let () denote the adjoint representation of 9 on Ag. Now observe that Arx is stable under (}(x) for all x E gl' Thus Arx becomes a gl-module with respect to the representation (2.10.1)

flrx: gl -> EndAtx

where flrx is the sub-representation of () Igl defined by A t x. The following proposition asserts that the given structure on A rx is "invariant" under flr x' PROPOSITION 2.10. Let x E gl then one has (a) [flrx(x), ix] = 0 , (b) [fltx(x), arx] = 0 , (c) fl r/ x )* = flrx(x*) , (d) flrx(x) = -flrx(x*) .

PROOF. Clearly (a) and (b) follow from the fact that gl lies in the nomalizer of both nand n. The relation (c) follows immediately from I, (3.9.7). Finally (d) follows obviously from the fact that f is a real Lie subalgebra of g. q.e.d. 3. Lie algebra cohomology defined by the adjoint representation

3.1. The operator d r is more easily understood by writing it as a sum d r = d; + d;' where d; and d;' are respectively of bi-degree (1,0) and (0,1). Proposition 3.1 asserts that such a decomposition (necessarily unique)

408

LIE ALGEBRA COHOMOLOGY

85

exists. It will be shown later that d~' (and also d;) is essentially the coboundary operator associated with the co chain complex defined by Aa, regarded as an a module (using the adjoint representation). We begin with PROPOSITION 3.1. Let e E EndAt be any derivation of degree 1 such that (a) e = e, and (b) (3.1.1) e: a - a/\t ,

then e may be uniquely written (3.1.2)

e = e'

+ e" ,

where e', e" E EndAt are, respectively, of bi-degree (1, 0) and (0, 1). Furthermore e' and e" are derivations (of degree 1) and e" = ?

(3.1.3)

PROOF. Since e = e it follows from (3.1.1) that

e: a-t/\o; therefore since e is a derivation,

e: A p,qt -

A P+1,qt

+A

p,q+1 t

for any p, q. But this clearly implies the existence of a unique decomposition (3.1.2). Since e is a derivation, it is obvious that e' and e" are derivations of degree 1. Now if e1 E EndAt is of bi-degree (1,0) it follows immediately from (2.3.1) that e1 is of bi-degree (0, 1). Since e = e = e' + e" it follows therefore from the uniqueness of (3.1.2), that (3.1.3) holds. q.e.d. We now observe that btl Cr and d r satisfy the conditions of Proposition 3.1. LEMMA 3.1. Let e = br , cr or dr. Then e = e and (3.1.4) e:a-a/\t. PROOF. By Proposition 2.5, (2.6.3) and (2.6.4), it follows that lir = br. Now if we conjugate the equation (2.7.1), it follows from (2.2.1) and (2.6.3) that (3.1.5)

1r(x)

=

1r(x)

for any x E t. But now it follows easily from (2.4.2) that (3.1.6)

(x, y)

for all x, yEt. Conjugating from (3.1.5) and (3.1.6) that

Cr ,

=

(x, jj)

as given by (2.7.6), it follows therefore

er = ero

409

86

BERTRAM KOSTANT

This of course implies that dr = dr. We have only to prove (3.1.4). The orthocomplement of a /\ t in A 2t is A20 and the orthocomplement of a in t is o. Since ar = to prove (3.1.4) for e = bn it is enough therefore, to observe that ar maps A 20 into a. But this is obvious since a is a Lie subalgebra of t. If e = Cr. then (3.1.4) is obvious from Proposition 2.7 since a is an ideal in t. This of course proves that (3.1.4) holds also if e = dr. q.e.d.

b:,

c;,

3.2. By Lemma 3.1 we can define derivations of degree 1, b;, d~ and and d;' in accordance with Proposition 3.1. The former are of bidegree (1,0) and the latter are bi-degree (0,1). By (2.7.5) it is obvious that

b;', c;',

(3.2.1)

d't = b'r

+ c'r

and (3.2.2)

d"r

=

b"r

+ c"t '

PROPOSITION 3.2. Let fi be a basis of a, and let {fj be the basis of a defined so that (fi' {fj) = Oij. Then (3.2.3)

c;' = EtC({fi)n-(fi)

and

(3.2.4) PROOF. Let e" and e' be respectively the operators given by the right side of (3.2.3) and (3.2.4). Now recall Proposition 2.7. By letting Yk be the basis fi' {fj of t, it follows from Proposition 2.7 that cr = e' + e". But since a and a are ideals in t it is obvious that e' and elf are respectively of bi-degree (1,0) and (0, 1). Hence e' = c~ and elf = 0;'. q.e.d. Now define operators E EndAt by putting

a;, a;'

(3.2.5)

a; =

-(b~'Y

and

a"t = -(b')t t

•

By (2.5.4) it is obvious that a; and a~' are respectively of bi-degree ( -1,0) and (0, -1). Furthermore by (2.6.4) one has (3.2.6)

at =

a; + a;' . a;

By the uniqueness of this decomposition, note that and given by the property that for any U E Aa, v E Aa one has (3.2.7)

410

ar'

may be

LIE ALGEBRA COHOMOLOGY

87

Note then that (3.2.8) 3.3. Now define L~, E: and S: and also L~', E;', and S:' in the same way as Lx, En and Sx except that all the operators are primed in one case and double primed in the other case. Thus (3.3.1)

S'x =L'x +E'x

and (3.3.2)

S" = L'X X

+ E'x '

and we note that all the operators in (3.3.1) and (3.3.2) are of bi-degree (0,0). Matters are considerably simplified by PROPOSITION 3.3.1. One has 1

E'x = E" x = -Ex. 2 PROOF. Let g E a. By (3.2.7) it is obvious that multiplication (c(g)8) by g. Hence one has

c(g)a;

(3.3.3)

+ a;c(g) =

a; commutes with right

0.

But now obviously for all x E t since 7r(x) is of bi-degree (0, 0) and it commutes with by (3.2.3) one has (3.3.4)

ax'

Thus

c"a' x x + a'xc" x = 0 •

Hence (3.3.5)

E;' = c~'ar

+ al;' = 'E 7r (g;)7r(fi) j

by (3.2.3). Similarly, by conjugating (3.3.4) one obtains (3.3.6)

c~a;'

+ ar'c; = 0

since

f1r

(3.3.7) by Proposition 2.5. Thus

=

a·'r

E; = c;ax + arcr and hence

(3.3.8) by (3.2.4). But also since cr

= c; + c;',

411

one has

Er

= Ex' + E;'.

On the

BERTRAM KOSTANT

88

other hand comparing (3.3.5) and (3.3.8), it follows that x(f.) commutes with XUii). q.e.d. Next one has

E; = E;' since

LEMMA 3.3. Let er be either br , Cr or dr. Then a'relfr + e"a' r r = 0

and a"e' r r

+ e'a" r r = 0 •

PROOF. In case er = Cr these are just the relations (3.3.4) and (3.3.6). If er = br the first relation is obvious from (3.2.7) and (3.2.8). Conjugating gives the second relation. In case er = d t the result follows from (3.2.1) and (3.2.2). q.e.d. As an immediate corollary one obtains PROPOSITION 3.3.2. One has

L r = L'r

+ L"r

and S r = S r '+ SIt r in particular Lr and Sr are operators of bi-degree (0, 0). PROOF. Immediate from Lemma 3.3 and the definitions of the concerned operators. q.e.d.

80

REMARK 3.3. Let ...Arc End A t be the set of the 21 operators ar, bt , Cn dn and Sr' primed, and double primed. We observe here that ...Ar lies in the smallest subalgebra of End A t that contains d;' and is closed under the operations A -> At, A. Lro E r ,

a;

3.4. Let aa E EndAa be the boundary operator for the chain complex C*(a) (recall that a is a Lie subalgebra of t) and let ba, E EndAa', the negative transpose of ar' be the coboundary operator of the cochain complex C(a). If the vector space V is an a-module with respect to a representation Xo:

a -> End V

then we may take V Q9 A a' as the underlying vector space for the cochain complex C(a, V). (For notational convenience here the order of the factors is the reverse of that given in I, §3.1.) Furthermore we recall the coboundary operator bo for this complex may be written (3.4.1)

412

89

LIE ALGEBRA COHOMOLOGY

where (3.4.2) and (3.4.3)

g:

where fi is a basis of a, and is the dual basis of a'. See T, §3.1 and § 3.12. We are particularly interested in the case where V = A a and (henceforth shall assume that)

no: a -> EndAa is the adjoint representation of a on Aa. Now let

YJa': Aa -> Aa' be the linear map defined so that Aa ® Aa'

(3.4.3) given by (3.4.4)

YJ(u/\ v)

= u ® YJa'V

for U E A a, v E A a is also a (linear) isomorphism. But obviously (3.4.3) sets up an algebra isomorphism EndAt -> End(Aa ® Aa') .

(3.4.5)

Now bo, b1 and b2 are elements of the right side of (3.4.5). What they correspond to in End A t is given in PROPOSITION 3.4. Let Va be the restriction of V to isomorphism (3.4.5) one has

a; -> aa ® 1 (b) b;' -> Va ® b~ = (va (c) c;' -> (va ® 1)b2

Aa. Under the

(a)

® 1)b1

and consequently (3.4.6) PROOF.

Clearly (a) is obvious from (3.2.7), and the fact that

413

90

BERTRAM KOSTANT

aa

= atl Aa

.

Also (b) is similarly obvious from (3.2.8) and the definition of bt • But now (c) is also true by (3.2.3) and (3.4.3) and the obvious fact that under(3.4.5) (3.4.7) for any f Ea. (Note that oa comes in since S(gi) means left multiplication by y•. ) Finally one obtains (3.4.6) by (3.2.2) and (3.4.1). q.e.d. REMARK 3.4. Since oa ® 1 clearly commutes with bo, we note that as a. consequence of Proposition 3.4 one has (d~')2 = O. Similarly (d~)2 = 0 by (3.1.3). However d~ 0, in general, as the example in § 2.9 shows. See Proposition 2.9.

*"

3.5. Let {Aa} be the restriction of {Ax} to Aa. Now v conjugate linear isomorphism Aa ----> Aa' where

---->

v' defines a.

{u, v} =

Ker L

is the isomorphism, and V'L.S is of degree 0 where

V'L.S = (V'a.L)-l°V'a.s . Let (3.7.11)

P: C ---Ker L

by the projection operator of C onto Ker L that vanishes on 1m L. Then the mapping V'L.8 is given by LEMMA 3.7. One has

V'L.8 = PIKer S . Furthermore V'L.8 is oj bi-degree (0, 0) in case both Land S are oj bidegree (0, 0). PROOF. Let x E Ker S. Put y = Px E Ker L. To show that y = V'L.8X, it suffices only to show that x and y define the same class in H(C, 8). But x - Y E 1m L n Ker a = 1m a. Hence x and y do define the same class in

H(C, a). If Land S are of bi-degree (0, 0), then V'L.S is of bi-degree (0, 0) since P is clearly of bi-degree (0, 0) and Ker S is bigraded. q.e.d. 3.8 Our final structure assumptions is that v is some Lie algebra, At is a v-module and certain conditions are satisfied. More specifically assume that v is a Lie algebra with a *-operation (x --> x* is a conjugate linear endomorphism of order 2 and [x, y]* = [y*, x*] for any x, y E tJ) and that

417

94

BERTRAM KOSTANT

(3.8.1) is a representation such that fJt(x) is a derivation of degree 0 for all x E V. Furthermore it is assumed that the v-module structure on A't defined by (3.8.1) is compatible with the given structure on A't to the extent that (3.8.2), (3.8.3), (3.8.4) and (3.8.5) below are satisfied:

=0

(3.8.2)

[fJt(x), j]

(3.8.3)

[fJt(x), ar] = 0

(3.8.4) (3.8.5)

fJt(x)* fJr(x)

= fJb;*) = -fJix*)

for all x E V. PROPOSITION 3.8. The representation fJr is completely reducible. Moreover for any x E v, fJX<x) is ofbi-degree (0,0) and all the (21) operators in Jlr (see Remark 3.3) commute with fJix). PROOF. Condition (3.8.4) insures complete reducibility. In fact by (3.8.4) one may use orthocomplements for v-stable complements. Obviously (3.8.2) implies that fJr(x) is of bi-degree (0, 0). Now let y E 't, and put z = fJX<x)y. Now by (2.7.1) and (3.8.3), it is clear that (3.8.6)

[fJr(x), x(y)]

=

x(z) •

But now by (3.8.4), (3.8.5), Proposition 2.5, and (2.5.1), it follows that (3.8.7)

(fJr(x»)'

=

-fJt(x) •

But then taking the negative transpose of (3.8.6) one obtains (3.8.8)

[fJt(x), x'(y)] = x'(z) •

It follows immediately then from (3.8.3), (3.8.7) and (3.8.8), and the definition of dn that

(3.8.9) But since fJt(x) is of bi-degree (0, 0), it follows from (3.8.9) and (3.8.3) that fJix) commutes with a~ and d~'. But then recalling Remark 3.3, it follows from (3.8.4), (3.8.5) and (3.8.7) that fJt(x) commutes with all the operators in Jlt • q.e.d. 3.9. If a vector space F is a v-module, we denote by F'D the subspace of all elements in F that are annihilated by all x E V. Once and for all we put

C = (A't)'D .

418

LIE ALGEBRA COHOMOLOGY

95

Now it is obvious from Proposition 3.8, that Cis bi-graded. Let {C} and (C) be respectively the restrictions of {At} and (At) to C. Since C is clearly stable under conjugation, it is immediate that (C) is non-singular. Thus the operations A -> At, A *, A and AS are definable in End C in the same way as in EndAt. Furthermore if EnduAt is the subalgebra of all v-endomorphisms on A t, and (3.9.1)

EnduAt -> End C

is the homomorphism defined by A -> A Ic, it is obvious that the operations above commute with (3.9.1). But now Jir ~ EnduAt by Proposition 3.8. We adopt the following notational convention. Recall that all the operators in Jir were notated with the subscript t. We will now denote its image in End C, under the map (3.9.1), by dropping the subscript t. Thus if er E Ji r , then by definition e E End C is defined by (3.9.2)

e

= eriC.

Since the orthocomplement of C in A t is clearly stable under en note that for all er E Jir (3.9.3) 1m e = 1m er n C . Concerning questions of homology we adopt the notation of § 3.6. Our immediate concern is with H(C, 8). 3.10. Assume that the finite dimensional vector space F is a v-module with respect to a completely reducible representation fJ of v. Now let e be a b-endomorphism of F such that e2 = 0 (a b-differential operator) and let H(F) be the homology space defined with respect to e. Then one knows that fJ induces a representation ~: v -> End H(F) ,

and that ~ is also completely reducible. More generally let Fl ~ F2 ~ F be b-submodules of F that are stable under e. Then e induces a differential operator on FN FP and a b-differential operator on the v-module F2/ Fl. The natural map of the former into the latter clearly induces a map (3.10.1) LEMMA 3.10. The map (3.10.1) is an isomorphism. PROOF. Since fJ is completely reducible it is clear the map inducing (3.10.1) defines an isomorphism F 2°/FlU -> (F2! Fl)U • Put D = F 2 /Fl • We must show that H(Db) = H(D)u.

419

96

BERTRAM KOSTANT

By the complete reducibility of f3, we can write uniquely D = Db + Do where the latter is a v-module direct sum. But then it is obvious that the differential operator leaves both summands stable. But then clearly H(Db) = (H(D»b. q.e.d. Now if u E An and v E Acr, it is obvious that ar(ul\v) = arul\v

+ oul\arv

.

If F = At, f3 = f3r and e = ar, it follows therefore by the Kiinneth formula (and an obvious identification) that

(3.10.2) so that ~r is a representation (3.10.3) In fact if f3o, and f3a are the sub-representations of f3r defined by Aa, it is clear (up to an obvious equivalence) that (3.10.4)

f3r = f3o, @ 1

An and

+ 1 @ f3a

and

~r = ~o, @ 1 + 1 @ ~a

.

As an application of Lemma 3.10 we obtain PROPOSITION 3.10. This is a natural identification (3.10.5)

Furthermore H(C, a) is bi-graded and (3.10.6) PROOF. Since C = Fb we have only to identify by means of the isomorphism (3.10.1) where F2 = F, Fl = 0 and H(F) is given by (3.10.2). (For the complete reducibility of f3r see Proposition 3.8.) The relation (3.10.6) is obvious by definition of HP,q(C, a). Since H(C, a) is clearly a direct sum of these spaces it is, by definition, bi-graded. q.e.d. Since b = a* (see (2.6.5» it follows that b and a are disjoint, and L = ba + ab is the corresponding laplacian. But since L is of bi-degree (0, 0) (see Proposition 3.3.2) it follows that Ker L is bi-graded and that furthermore the isomorphism (3.10.6) defined in § 3.6, is, by Proposition 3.7, of bi-degree (0, 0).

420

97

LIE ALGEBRA COHOMOLOGY 4. The disjointness of d and 8

4.1. The structure assumptions in §§2 and 3 put no restrictions whatsoever on the nature of the Lie algebra a (the Lie algebra '0 could just have well been zero). It will be one of the main consequences of this section that, under certain conditions, d" and 8" are disjoint. When, furthermore, it is assumed that d 2 = 0, it will be shown that d and 8 are disjoint as well. But then this establishes a natural isomorphism ('ta.a, see (3.7.9» between H(C, d) and (H*(a) ® H*(a»b = H(C, 8), (see Proposition 3.10). Since in our applications H(C, d) is the complex cohomology of the algebraic homogeneous space GI U on one hand and '0 = g10 and a is isomorphic to n on the other hand, this establishes the relation between the cohomology of GI U and H*(n), as a gcmodule mentioned in the introduction of Part I. Using the results of Part I, which determined the structure of H*(n) as a gl-module, one obtains, for example, the "strange" relation (1.1.1) of Part I. The easy half of the disjointness of d" and 8" (Proposition 4.1.1) will be a consequence of

Condition AI. The Lie algebra a is nilpotent (4.1.1). If.

Aa =

(4.1.2)

~

£..JI=O

V.•

is an orthogonal finite direct sum decomposition of

Aa let

(4.1.3) so that its orthocomplement V(tl in

Aa is given by

(4.1.4) LEMMA 4.1.1. If Condition Al is satisfied, there exists a decomposition (4.1.2) such that for all k (a) for any fE a n(f): V k

->

V(tl ,

(b) V k is stable under the representation (3t of '0. PROOF. Let Vo = O. Assume Vi' i ~ j, has been defined, and VUl is

stable under n(f). Put

E j = {space spanned by n(f)u, all f

E

a, u

E

V(]l}

so that E j ~ V(]l. But, if V(]l =1= 0, then E j =1= VGl since a is nilpotent. Put Vi+1 equal to the orthocomplement of E j in V Gl. This defines a decomposition (4.1.2) and (a) of Lemma 4.1.1 is satisfied. By Proposition 3.8 one

421

98

BERTRAM KOSTANT

knows that An is stable under (Jr' Now, however, by (3.8.6) the subspaces V(t) are also o-submodules. Taking orthocomplements and using (3.8.4), it follows finally that the V k are themselves stable under (Jr. q.e.d. By (b) in Lemma 4.1.1 and (3.10.4), it is clear that V k ® (A a) is stable under (Jr' If we then define (4.1.5) it is obvious that (4.1.6) is an orthogonal direct sum decomposition of C. Let C(l EndAa' be the representation contragredient to f3 a• Recalling the definition of

423

100

BERTRAM KOSTANT

the isomorphism 1)a': A Q-+ Aa', if then follows immediately from (3.8.7) and Proposition 3.8 that 1)a' is in fact a b-module isomorphism. Furthermore if Aa ® Aa' is regarded as a b-module with respect to the representation fJ defined by (4.3.2)

fJ

=

fJa

® 1 + 1 ® fJa'

,

then one has LEMMA 4.3 The map 1):

At -+ Aa ® Aa'

defined in § 3.4 is a b-module isomorphism. Moreover the co boundary operator bo of the complex C(a, Aa) is a b-endomorphism of Aa ® Aa'. PROOF. Since 1)a' is a b-module isomorphism, the first statement follows by comparing (4.3.2) and (3.10.4). The second statement follows from (3.4.6) and the fact that d~' is a b-endomorphism of At. (See Proposition 3.8.) q.e.d. Now if Vl ~ Aa is an a-submodule and a b-submodule(with respect tofJa) then not only is C(a, Vl) defined, but also the underlying space Fl of C(a, Vl) is a b-module. Moreover since bo is a b-endomorphism, H(Fl ) = H(a, Vl) is also a b-module. We are now particularly interested in the case where V l = 1m aa. Since aa is both an a- and a b-endomorphism of Aa, it is clear that 1m aa and Ker a.:t are each both an a- and a b-submodule of Aa. The following is our first theorem. THEOREM 4.3. Assume that Conditirms Al and A2 are satisfied. (See § 4.1 and 4.2). Then (4.3.3)

H(a, 1m act)\)

=0.

PROOF. We first observe that if Vl ~ V2 ~ A a are both a- and b-submodules then where Fl and F2 are as in (4.3.1), the isomorphism (3.10.1), together with (4.3.1) becomes an isomorphism (4.3.4) where, it may be understood, the underlying space V2/ Vl ® Aa' of C(a, V 2/ Vl) is a b-module by taking the tensor product of fJo.' with the representation on V 2 / V l induced by fJ a• Next since a is nilpotent (Condition A l ) there exists an orthogonal direct sum decomposition 1m aa = E,=o Vi such that, in the notation of (4.1.3) and (4.1.4), (a) and (b) of Lemma

424

101

LIE ALGEBRA COHOMOLOGY

4.1.1 hold. The proof is the exact same as Lemma 4.1.1 with 1m (Ja replacing Aa. But then the subspaces Vlt) of 1m (Ja are both a- and v-submodules of A a. Hence if Fk = Vlt) @ A a', one has by (4.3.4) isomorphisms (4.3.5) But now the Fib are a filtration of Fob there is an isomorphism

= (1m (Ja @ Aa,)b,

and by (4.3.4)

(4.3.6) To prove the theorem, it suffices therefore to prove that H(FoD) = O. But to prove this, it suffices to show that the El terms of the spectral sequence ·defined by the filtration above are identically zero. But the El terms are made up of the left side of (4.3.5) for all k. But then by the isomorphism (4.3.5) it suffices to prove that the right side of (4.3.5) is zero for all k. But now V~-lJ Vlt) is a trivial a-module by (a) of Lemma 4.1.1, and as an v-module it is isomorphic to V k • Thus one has an isomorphism (4.3.7)

H(a, VILIJ Vtt))D -

(Vk @ H(a))b .

It suffices therefore to prove that the right side of (4.3.7) vanishes for all k. Now clearly as subspaces of Aa, Ker ba,

= Ker La' + 1m ba,

is a v-module direct sum decomposition where La' (see §3.5) is the laplacian defined by ba, and its adjoint. But then Ker La' is equivalent to H(a) as a v-module. It therefore suffices to show that (Vk @ Ker La,)b = 0 for all k. Assume not. Then there exists a k and U E (Vk @ Ker La,)b such that U =1= O. But now (4.3.8)

(L a @ l)u =1= 0

since Ker(L a@l) = Ker L a@ Aa' and V k On the other hand obviously

n Ker La = 0, because

V k ~ Im(Ja.

(4.3.9) Now let v EAt be defined by putting v = r;-lu. Then clearly v E C, and by Lemma 3.5, (4.3.8) and (4.3.9), one has L'u =1= 0 and L"u = O. This contradicts the equality L' = L" of Condition A 2. q.e.d. REMARK 4.3. For the proof of Theorem 4.3, note that Condition A2 could have weakened to the assumption that only Ker L' = Ker L". Now in accordance with the notation of § 3.6, H(C1 , e) is defined if C1 is a bi-graded subspace of C that is stable under e where e E EndC is of degree ± 1 and e2 = O.

425

BERTRAM KOSTANT

102

We now wish to consider the case where e = d". Since a' anti-commutes withd" (Lemma 3.3) and a' isofbi-degree( -1,0), it follows that H(lma',d") is defined. As a corollary to Theorem 4.3 (and in fact equivalent to it) we have COROLLARY 4.3. Assume that conditions Al and A2 hold. Then H(lm a', d") = 0 .

(4.3.10)

PROOF. Now by Theorem 4.3 and (4.3.6) one has that H(FoO) = 0 where Fo = ImaaQ9 Aa', where we recall the coboundary operator on Foo is bolFoo. But now by (a) of Proposition 3.4, the space Fo corresponds to 1m a~ ~ At, and hence Foo corresponds to 1m a~ n C under the map r;. But now by (3.9.3) this implies that (4.3.11)

is an isomorphism. But d~' corresponds to (oQ Q91)b o under the map (3.4.5) according to (3.4.6). But obviously H(FoO) also vanishes in case (oa Q91)b o is substituted for boo Hence by (4.3.11) one has H(lm a', d") = O. q.e.d. 4.4. If e E End C is an in § 3.6 and C1 ~ C2~ Care bi-graded subspaces stable under e, then let H(C 2{C1 , e) be the homology group defined by C2{C1 and the differential operator on C2{C1 induced bye. Now H(Ker a', d") is defined as well as H(lm a', d"). Let (4.4.1)

H(Ker a', d")

---+

H(C, d")

be the map (of bi-degree (0, 0), see Remark 3.6) induced by the injection Ker a' ----> C. LEMMA 4.4.1. Assume that Conditions Al and A2 hold. Then the map (4.4.1) is an isomorphism. PROOF. It is clear that a': C{Ker a'

---->

1m a'

is an isomorphism. But since a' anti-commutes with d". it follows immediately from Corollary 4.3 that H(C{Ker a', d") = 0 .

(4.4.2)

Now consider the exact sequence

o

---->

Ker a' ----> C ----> CfKer a' ----> 0 .

This induces, on the level of homology, an exact sequence H(C{Ker a', d")

---->

H(Ker a', d")

---->

H(C, d")

---->

H(C{Ker a', d") ,

where the map in the middle is just (4.4.1). Since the ends vanish by

426

LIE ALGEBRA COHOMOLOGY (4.4.2), this proves the lemma. Next we need

103

q.e.d.

LEMMA 4.4.2. Assume Conditions Al and A2 hold. Then (4.4.3)

Z(Ker a', d") n 1m L = B(Ker a', d") .

PROOF. Let u lie in the left side of (4.4.3). Then since 1m L = 1m L' (by Proposition 4.2), one has

u E Ker a' n 1m L' = 1m a' . But d"u = O. Thus u E Z(lm a', d"). But then by Theorem 4.3, one has u = d"v where v E 1m a' ~ Ker a'. Hence u E B(Ker a', d") so that the left side of (4.4.3) is contained in the right side. Now conversely assume that u E B(Ker a', d"). We have only to prove that u E 1m L. In order to prove this we will first prove that e"; Ker a' -> 1m a'

(4.4.4)

.

To prove this, by (3.2.3) and (3.9.3), we have only to show that

n(f); Ker a~ -> 1m a~

(4.4.5)

for all f Ea. But (4.4.5) is obvious from the relation

.s(f)a;

(4.4.6)

+ a;.s(f) =

n(f)

which one obtains from (2.7.1) and the conjugate of (3.3.3). Hence (4.4.4) is established. But now we assert that d"; Ker a' -> 1m L .

(4.4.7) Indeed d" = b" (4.4.8)

+ e".

Hence by (4.4.4) d"; Ker a'

->

1m b"

+ 1m a' .

But 1mb" ~ ImL" and lma' ~ ImL'. But ImL" = ImL' = ImL, by Proposition 4.2. Thus (4.4.8) implies (4.4.7). But now by assumption u = d"v where v E Ker a'. Thus u Elm L by (4.4.7). q.e.d. Finally we can prove THEOREM 4.4. Assume that Conditions Al and A2 hold. Then the operators d" and a" are dis}oint. Similarly d' and a' are dis}oint. PROOF. Using conjugation, it is enough to prove only that d" and a" are disjoint. Recalling Proposition 4.1.1, we have only to prove that a"d"u = 0 implies d"u = 0 for any u E C. Indeed assume that a"d"u = O. Put v = d"u. Then obviously a"v = d"v = O. Hence S"v = O. But S" = S' by Proposition 4.2. Thus S'v = O. But then by Proposition 4.1.2,

427

104

BERTRAM KOSTANT

one has a'v = O. That is v E Kera'. That is v E Z(Kera', d"). But now v = d"u. That is v defines the zero class in H(C, d"). But then by Lemma 4.4.1, it must have already defined the zero class in H(Ker a', d"). Thus v E B(Kera', d"). But then byLemma4.4.2,itfollowsthatv E ImL = ImL" (by Proposition 4.2). But now recall that a"v = 0 (by assumption). Thus

v E 1m L" n Ker a" = 1m a"

.

That is, there exists w such that v = a"w. But d"v = O. Hence d"v = d"a"w = O. But now by Proposition 4.1.1, this implies that a"w = v = d"u = O. q.e.d. We may substitute a for a' and a" in Theorem 4.4. COROLLARY 4.4.1. Assume that Conditions Al and A2 hold, then d" and a are disjoint. 8imilarly d' and a are disjoint. PROOF. We first observe that by Lemma 3.3 and Proposition 4.2, (1/2)8 is given by either

~8 = d'a

(4.4.9)

2

+ ad'

or

= d"a + ad" .

(4.4.10)

On the other hand by Theorem 4.4, d" and a" are disjoint and hence (1/2)8, their anti-commutator (see Proposition 4.2) is the laplacian they define. A similar statement holds for d' and a'. But then by property (3.7.4) of the laplacian (4.4.11)

1m d'

+ 1m d" + 1m a' + 1m a" =

1m 8

and hence (4.4.12)

1m d

+ 1m a ~ 1m 8

.

Now let u E C, and assume d"au = o. Put v = au so that by (4.4.12) v E 1m 8. But now d"v = av = O. Hence v E Ker 8 by (4.4.10). But since (1/2)8 is the laplacian defined by d" and a", one has Ker 8

n 1m 8 = 0 .

Thus v = o. A similar argument, using 1m d" ~ 1m 8, (see 4.4.11) shows that ad"u = 0 implies d"u = O. Thus d" and a are disjoint. Conjugation shows that d' and a are likewise disjoint. q.e.d. REMARK 4.4.1. To obtain the disjointness of d" and a we used only two properties, (1) the anti-commutator of d" and a is a non-zero multiple of 8, and

428

LIE ALGEBRA COHOMOLOGY

105

(2) 1m d" and 1m a are contained in 1m S. One consequence of Corollary 4.4.1 is that the homology groups H(C, d") and H(C, a) are isomorphic.

COROLLARY 4.4.2. Assume Conditirms Al and A2 hold so that by Corollary 4.4.1, d" and a are disjoint. Then the map 'ta.d" defined as in §3.7 (see (3.7.9» is an isomorphism (4.4.13)

'ta.d": H(C, d")

-+

(H*(a) 0 HAaW

of bi-degree (0, 0). PROOF. We have only to use Propositions 3.7, 3.10 and the fact that S is of bi-degree (0, 0) (see Proposition 3.3.2). q.e.d. REMARK 4.4.2. It is immediate from (4.3.4) and (3.4.6) that H(C, d") is isomorphic to H(a, Aa)'o. It follows therefore by Corollary (4.4.13) that if Conditions Al and A2 hold, there is an isomorphism (4.4.14) However this fact can be obtained directly from Theorem 4.3. An argument similar to the one used in the proof of Lemma 4.4.1 shows directly that (4.4.15) H(a, Aa)'o = (H*(a) 0 H(a»)'o , where the tensor product on the right is a v-module with respect to the representation fJ a 01 + 10 fJ a,. 4.5. In the cases that interest us d 2 = 0, and we will be particularly concerned with H(C, d). We therefore consider the case where Condition Aa: d 2 = 0 is satisfied. One finds that not only is d disjoint from a but also from a' and a". THEOREM 4.5. Assume that Conditions All A2 and Aa hold. (See §§4.1 and 4.2.) Then where ae denotes a, a' or a", one has that d and ae are disjoint and the corresponding laplacian is given by (4.5.1)

dae

+ aed = {~/2)S

if ae = a' or a" if ae = a .

Furthermore H(C, d) is bi-graded, and if the map 'ta.d is defined as in §3.7 (see (3.7.9» then 'ta.d is an isomorphism (4.5.2)

'ta.d: H(C, d)

-+

(H*(a) 0 HAaW

of bi-degree (0, 0). PROOF. The proof of disjointness follows exactly as in Corollary 4.4.1. See Remark 4.4.1. With regard to the latter, (1) is satisfied by (4.5.1),

429

106

BERTRAM KOSTANT

which follows immediately from Lemma 3.3. Moreover (2) is satisfied by (4.4.12). Since S is of bi-degree (0, 0) by Proposition 3.3.2, the last statement follows from Propositions 3.7 and 3.10. q.e.d. 4.6. We shall assume in this section that Conditions AlJ A2 and A3 are satisfied. Note then that the notation of §3.7 (concerning laplacians and the "Hodge decomposition") apply here. We can therefore use freely here all the relations of that section. Now if e = 8 or d and u E Z(C, e) we will let [u]. E H(C, e) be the class determined by u. Now let s E H(C, d). Among the various cocycles which represent s, we will be particularly interested in the unique such (harmonic) cocycle s lying in Ker S. In the notation of §3.7 (4.6.1)

'Vra.s(s)

=s.

Largely because 8s = 0 the harmonic representative will later be seen to enjoy properties making it useful for a number of applications. In our considerations the problem will thereupon arise, how does one construct s "knowing" s. By "knowing" we really mean knowing its image (4.6.2)

'Vra.is)

=

h

in (H*(a) ® H*(a»b under the isomorphism (4.5.2). It should be pointed out that the right side of (4.5.2) will be explicitly known by the results in Part I. But now, because of the relatively simple nature of L, knowing h, one immediately determines the unique cycle 'Vra.~(h)

(4.6.3)

=h

in Ker L such that [hla = h. Thus the problem effectively is, givenh E Ker L, find the unique element s E Ker S such that (4.6.4) (see (3.7.10) for the definition of the isomorphism 'VrL.S). A formula for computing s in terms of h will be given by Theorem 4.6. Now by Lemma 3.7, one has h = Ps where P, we now observe (using the fact that L is a hermitian operator) is the orthogonal projection of Con Ker L. Thus if u E C is defined by (4.6.5) then u (4.6.6)

s=h+u, E

1m L. That is u = Qs

430

LIE ALGEBRA COHOMOLOGY

107

if Q E End C is defined by (4.6.7)

P

+ Q=

1

(and hence is the orthogonal projection of C on 1m L). Now let Lo E End C be the (inverse of L on 1m L) the unique operator which vanishes on Ker L and satisfies (4.6.8) Finally where we recall S = L

+ E,

(4.6.9)

= -LoE.

R

put

REMARK 4.6. The operator E has been given by (3.3.8), and in our applications Lo is known by the results of Part 1. Thus R is given explicitly in our applications (see § 5.6). We now observe LEMMA 4.6. The operator R is nilpotent. PROOF. Let the subspaces Ck ~ C be given by (4.1.5) and Lemma 4.1.1. Because of (4.1.6), it suffices to show that for all k (4.6.10) using the notation of §4.1. But now by Lemma 4.1.2 (b), it is obvious that C k is stable under L" = (1/2)L. Thus since (4.1.6) is an orthogonal decomposition, it follows that Ck is also stable under Lo. Thus to prove (4.6.10) one need only show that E maps Ck into C(tl. But this is obvious from Lemma 4.1.1 and the formula (3.3.8) after one recalls that E" = (1/2)E (see Proposition 3.3.1). q.e.d. Now let p be a positive integer such that RP+l = o. We then have THEOREM 4.6. IJs,hand u are as in (4.6.5) where s = 'Vrs.ih),h E Ker L. (and hence s E Ker S), then (4.6.11)

s

=

(1 - R)-lh

=

E~=oRjh

and (4.6.12)

u = E~=lRjh .

PROOF. Let sland u 1 be defined respectively by the right side of (4.6.11) and (4.6.12). Thus SI = h + u 1 • It is obvious from (4.6.9) and (4.6.12) that U 1 E 1m Lo = 1m L. Thus PSI = h. To prove s = S1> it suffices therefore to prove only that SI E Ker S since 'VrL.8 = PI Ker S is an isomorpism. Next observe that (4.6.13)

[R,8]

=0.

Indeed 8 commutes with Land S since the latter are laplacians defined using 8. Thus 8 commutes with E and also Lo. This proves (4.6.13). But

431

108

BERTRAM KOSTANT

now h E Ker L ~ Ker a. But then by (4.6.13) one clearly has also 81 E Ker a. But by the properties of the laplacian, one knows that 8 maps Ker a into 1m a. Thus to prove that 881 = 0, it suffices to show that L oS8 1

=0

since Lo is non-singular on 1m a c 1m L. But since 8

= L +E

LoS = Q - R = (1- R) - P But then by (4.6.11) L oS8 1 8 1 = 8. q.e.d.

= h - P8 = O. Thus I

by (4.6.7) . 81 E

Ker 8 and hence

5. The case where a = n 5.1. It will be convenient at this time to recall and collect some further (see § 2.9) notation from Part 1. The Cartan subalgebra ~ is fixed once and for all by putting ~ = 0 n 0* (see §2.9 and 1, §§5.1-4). The discrete group Z ~ ~' denotes the spaces of integral linear forms on l) and DI ~ Z is the set of dominant integral linear forms relative to gl (and 0). We recall that G is the group corresponding to g, and G1 is the subgroup corresponding to gl' All the representations of gi considered here induce representations of G1 • If F is any such gcmodule and ~ E Z we denote by F" the subspace of all vectors in F which transform according to the irreducible representation vi of gl' It is clear of course that F" = F"" for any a E WI' the Weyl group of G1 • Now A ~ Z is the set of roots, A+ is the set of positive roots (A+ = A(O» and A(n) ~ A+ is the subset corresponding to n. Now to each a E W, the Weyl group of G, one associates a subset =

\~ 27C~

if fJ

*- a

if fJ

=a .

But if x has the property indicated by (5.8.12) then clearly for any cP E ~+ . This together with (5.7.9) and the fact that cP E ~+, n",(cp) cp E ~(n) (see I, Proposition 5.4) proves (5.8.5). q.e.d.

*-

0, implies

5.8. The above proof together with Corollary 5.3.4 gives new information about c-cocycles in A tv. In fact if U E A l·Jty, then REMARK

eu

=

0 implies

u = 0 {U = ex,

if}

*- 1

x E 1) if} = 1 .

Indeed eu = 0 implies U E Cy by (5.7.15). But then U E KerSy by (5.8.8). However Hl.J( C, d) = 0 for} *- 1 by Corollary 5.3.4. Hence U = 0 if } *- 1. If } = 1 we have only to apply (5.8.7).

446

LIE ALGEBRA COHOMOLOGY

123

5.9. Assume in this section that n = m (Case 2 of § 5.6). Now let CU, and let no = (uo)o. It is clear from I, Proposition 5.4, that a(n o) and its complement in a+ are closed under + and hence by I, Proposition 5.10, there exists a unique a E W such that U oE

(5.9.1) The following proposition gives s" for a satisfying (5.9.1). PROPOSITION 5.9. If a E W is such that (5.9.1) holds then s" = h" 1 )"(")

= ( -. 27r~

fq, /\f-q, • "

"

PROOF. By (5.7.12) with no substituted for n, it follows that 7r (f'P) fq, ba(a)-l·o induces a holomorphic diffeomorphism (6.3.4)

Mer-l ---> Ver .

PROOF. The fact that (6.3.3) is a disjoint union is an immediate consequence of Proposition 6.1. Now let a E WI. We first show that

UnM: =

(6.3.5)

(1).

Indeed let 1 * a E Mer*. Then {}(a) * 1 since {}(M:) is a unipotent (hence simply connected) group. Now if x E f;J is regular, then clearly {}(a)x = x + e where 0 e E m:. But since m: n u ~ n* n u = 0, this implies that a ¢ U which proves (6.3.5). We next show that

*

(6.3.6) Indeed where

~_

=

-~+,

a( -_

WE

ZJ(X). Then

{ 0

PROOF. If d *- 2n(a), the result is obvious. Hence we may assume d = 2n(a). Now it follows easily from (6.3.4) that, since l.i commutes with the action of G, it defines by restriction a diffeomorphism (6.6.4)

452

129

LIE ALGEBRA COHOMOLOGY

(the subscripts X and Yare introduced into VO" to distinguish the case X from Y). One then obtains the first relation of (6.6.3) by using (6.5.4). Now assume that a ¢ WI. By I, Proposition 5.13, we can then uniquely write a-I = all!, where a l E WI and!' E WI. But now (6.1.4) defines by restriction a surjection But (6.6.5) by I, (5.13.6), (see also I, Remark 5.13), and (6.3.9) since !'"* 1. But (6.6.5) and (6.5.4) clearly imply the second relation of (6.6.3). q.e.d. As an immediate corollary one has PROPOSITION 6.6. If).i is the map (6.6.2), then ).i*(xO")

(6.6.6)

= yO"

for any a E WI. REMARK 6.6. One effect of Proposition 6.6, is that for most cohomological questions one need consider only the case Y. 6.7. In this and the next section regard X as a real manifold, G as a real Lie group, and 9 a real Lie algebra. Let tR be the tangent space to X at o. Let Tl be the R-linear epimorphism (6.7.1) defined so that Tl(X), x at 0, and let

E

g, is the tangent vector to the curve exprx, r

E

R,

(6.7.2) be the homomorphism which extends Tl. LEMMA 6.7. Let tX,R ~ 9 be the subspace defined by (2.9.3). Then the restriction of T defines an isomorphism (6.7.3) PROOF. Let (6.7.4) so that fl is the set of all x E u such that x* = -x. Since gl follows immediately that fl is a real form of g10 that is

453

= un u*, it

130

BERTRAM KOSTANT

(6.7.5) and hence (6.7.6)

= fl + tX.R

f

is a direct sum. But then by dimension one must have (6.7.7) Since u is the kernel of T\ it follows from (6.7.4) and (6.7.7) that TIl f is an epimorphism with kernel fl. Hence TIl tX.R is an isomorphism by (6.7.6). Thus (6.7.3) is an isomorphism. q.e.d. 6.8. Now once and for all fix tors at o. That is

to be the space of all complex 1-covec-

t

(6.8.1) Next let (6.8.2)

AI: Atx -> At

be the isomorphism (by Lemma 6.7) defined so that (6.8.3)



=

(u, v)

for any U E Atx and v E ARtX.R. Now, using AI' carryover the structure from tx to t so that t satisfies the conditions of §§ 2.2-6, and § 3.8 with b = gl. But then AI obviously satisfies the conditions of § 5.2, and hence the results of § 5 apply. Now let (6.8.4) be linear isotropy representation on the space of all covectors at o. If a E U, and v E g, note then by definition of T

YEt,

(6.8.5)

da

where cp(a) is left multiplication of M by a.

PROOF. If f(a) is the integrand on the right side of (6.12.3) then clearly f(a)a = w by (6.12.1). But then the lemma follows from (6.12.2). q.e.d. 6.13. Recall that q (6.13.1)

= if. Now put 1)q = q n 1)

and Oq

= 1)q + m .

These subspaces are clearly real Lie subalgebras of g, and moreover if Hq and Bq are the corresponding subgroups, one knows that Bq is an Iwasawa subgroup of G; (6.13.2)

is a semi-direct product, and (6.13.3)

G = KBq

is an Iwasawa decomposition of G. Now for a E G, let k(a) E K and b(a) E Bq be the unique elements such

459

136

BERTRAM KOSTANT

that (6.13.4)

a

=

k(a)b(a) .

Now since b(a)-l E Bq ~ U obviously Pr(b(a)-l) is defined for any a E G. PROPOSITION 6.13. Let a E WI, and let u

E Z2 n l<TI(C, d) so that W U is a

closed K-invariant differential 2n(a) form on X. Then

(6.13.5) where rV:

is the map defined by putting a(a) = a·o, it follows from (6.3.5) that a is a diffeomorphism. But now by (6.2.6) and Proposition 6.3, it also follows that the action of a(a) on X induces a diffeomorphism But since, as one knows, a(a) can be chosen to be in K, it follows from the K-invariance of W U and (6.5.4) that

< [WU] , X = Now if w that

r

Jv~

WU •

= a*(wU) then since a is a diffeomorphism, it follows from (6.12.3)

(6.13.8) where cp(a) is left multiplication of M: bya. But by definition of a* and T, it is obvious that (6.13.9) where p(a) denotes the diffeomorphism of X defined by the action of a. Now a·o

by (6.13.4) since b(a)

E

= k(a)·o

U. But then by definition of wU , one has

(wU)a.o

=

p(k(a»).u .

460

LIE ALGEBRA COHOMOLOGY

But since p(a)-lp(k(a»

=

137

p(b(a)-l) by (6.13.4) it follows from (6.13.9) that

Im at

,

since 11:(Y) has this property (by (2.7.1» for all yEt. Exponentiating, this clearly implies pt(a)v - v E Im at' q.e.d. 6.15. If hE Hq (see § 6.13) and homomorphism defined so that for h

~E

Z, we will let h --> h< E R be the

= expx, x E 1}q.

LEMMA 6.15. Let a E Wl and hE Hq then (6.15.1)

where r" E At is defined by (5.5.7) and ~" = g - ago PROOF. Let x E u and YEa. We assert that (6.15.2) Indeed let v E g, and let [x, v]u E u and [x, v]n* E n* be defined as in the proof of Proposition 6.14.1. Then since T([x, v]u) = 0, one has by (6.8.8) (which of course holds for any x E u) (6.15.3) Now let

~

(resp

~) ~

=


466

143

LIE ALGEBRA COHOMOLOGY

O(m(a»)e",

= e...

But then since O(k(a» is unitary with respect to {A g} one has

1h(ayrT 12 =

{O(a)e"" O(a)e",}

by (6.16.4). Inverting one obtains (6.16.3). Now if c = s + it E C then a = a(s, t) = expce_"" and hence

O(a)e", = e..

+ [ce_""

e..] + 2..[ce_"" [ce_"" e",]] 2

(the Taylor series stops with 3-terms since O(e_",)3e.. (5.1.3)

= 0). Thus by I,

(6.16.5) where x.. E 1) is the root normal corresponding to a. But since the three vectors on the right side of (6.16.5) are orthogonal to each other, and since {x"" x",} = <x.. , a) = (a, a), one has by (6.16.3)

f(a)

Since 1c 12

=

S2

PROPOSITION

+ t2,

(1 + IcI2(a, a) + 1:1'(a, a)2r 2 = (1 + 1~12(a, a)r . =

this proves (6.16.2).

1

(q.e.d.)

6.16. If a = '/:'"" a E II, then ).,rT

=

(g, a)-l

where as usual g is one half the sum of the positive roots. PROOF. By (6.15.11)

But then by (6.16.1) and Lemma 6.16, (6.16.6) But since the integrand is a radial function, one converts to integration over (0, co) by introducing the factor 2nrdr where r2 = S2 + t 2 • Thus

,< ~ 2 [ (1 + ¥r.), ~ (a~ a) .

467

144

BERTRAM KOSTANT

But (g, a) = «a, a){2). See I, (5.9.1). REMARK

(6.16.7)

6.16. For any

(J

E

q.e.d.

W\ put 1 -T -1- SCT SC ",CT

and let sf = [sna. The sf are a more natural basis of H(C, d) than SCT since, by Theorem 6.15, they correspond exactly to the integral classes x CT under the map 'Vrx,a. In case (J = T"" a E IJ(u), one has, by Proposition 6.16, sf= (g, a )SCT, and hence (6.16.8) by (5.8.5). This will be needed in Part III to determine

",CT

for all

(J.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY REFERENCES 1. C. CHEVALLEY and S. ElLENBERG, Cohomology theory of Lie groups and Lie algebras Trans. Amer. Math. Soc., 63 (1948), 85-124. 2. C. EHRESMANN, Sur la topologie de certains espaces homogenes, Ann. of Math., 35 (1934), 396-443. 3. HARISH-CHANDRA, On a lemma of Bruhat, J. Math. Pures et Appl. (9), 315 (1956), 203-210. 4. B. KOSTANT, Lie algebra cohomology and the generalized Borel- Weil theorem, Ann. of Math., 74 (1961), 329-387. 5. J. L. KOSZUL, Homologie et cohomologie des algebres de Lie, Bull. Soc. Math. France, 78 (1950), 65-127. 6. P. LELONG, Integration sur un ensemble analytique complexe, Bull. Soc. Math. France, 85 (1957), 239-262. 7. G. DE RHAM, Seminars on analytic functions, Princeton, 1957, vol. 1, 54-64. 8. M. ROSENLICHT, On quotient varieties a.nd the affine embedding of certain homogeneous spaces, Trans. Amer. Math. Soc., 101 (1961), 211-223. 9. A. WElL, VariHes Kahleriennes, Hermann, Paris, 1958.

468

Topology Vol. 3, Suppl. 2, pp. 147-159. Pergamon Press, 1965. Printed in Great Britain

EIGENV ALUES OF A LAPLACIAN AND COMMUTATIVE LIE SUBALGEBRAS BERTRAM KOSTANTt

(Received 29 August 1963) §1. INTRODUCTION

(1.1). IF K

IS a compact semi-simple Lie group and 9 is the complexification of its Lie algebra then one knows that the algebra Q of (Maurer-Cartan) complex-valued left invariant differential forms may be naturally identified with the exterior algebra Ag. Also, one knows then that Ag is stable under the Laplacian defined with respect to the canonical Riemannian metric on K.

Let L be the restriction of the Laplacian to Ag. It is then well known that the minimal eigenvalue of L is 0 and the corresponding eigenspace is the space of all harmonic differential forms on K. What we wish to consider here is the maximal eigenvalue mk of L on Akg and the corresponding eigenspace Mk ~ Ng. Now on the other hand let Ak ~ Akg be the subspace spanned by all non-zero decomposable elements in Akg (an element u E Akg is decomposable if u = Zl/\ ... /\Zk where Zj E g) whose corresponding subspace in 9 (the subspace corresponding to u is the one spanned by the Zj) is a commutative Lie subalgebra of g. If p = max dim a

where a runs over all commutative Lie subalgebras of 9 then obviously Ak # 0 if and only if 1 ::; k ::; p. The determination of the integer p was first made by Malcev [4]. In the case of the classical groups, like the dimension, p is a quadratic expression in the rank of K. For the exceptional groups G2 , F 4 , E 6 , E7 and E8 one has, respectively, p = 3, 9, 16, 27 and 36. The two matters brought up above are related by the following (proved here as Theorem (5), §4.4). THEOREM.

If mk

is the maximal eigenvalue of the Laplacian L on Akg then one always

has k mk = (¢1' ... , ¢k) be any set of k positive (distinct) roots. Let elf) E Akg be the decomposable vector eq,l A ... A eq,k (multiplied in some order) and let alf) = Lq,elf)(eq,) be the corresponding subspace in g. Then one always has the inequality Ilg + ¢1 + ... + ¢k11 2 -11911 2 :::; k (4.5.3)

and the equality holds if and only ifalf) is a commutative ideal of6 (see (4.1.1.)). Moreover every commutative ideal in 6 is (uniquely) of this form.

480

159

EIGENVALUES OF LAPLACIAN AND COMMUTATIVE LIE SUBALGEBRAS

Now let q denote the number of commutative ideals in 6 of dimension k and let 1' .,. , q be the corresponding subsets of k positive roots such that equality holds in (4.5.3). For convenience put Wi = e,. Then the Wi are highest weight vectors in Ak and (up to scalar multiples) are the only highest weight vectors in A k • That is, VWI ' ••• , VWq are the only simple K-submodules in A k , or what is the same thing, Vw, is inequivalent to Vwjfor i =I- j and

(4.5.4) is a unique decomposition of Ak as a direct sum of simple K-modules. In fact vary then the V w, are the only simple K-submodules of A = LkAk'

if we

let k

Proof Let w = e' Then clearly {w, w} = 1 and w is a weight vector belonging to u = CPl + ... + CPk' But since F = 2L on Ng one gets the inequality (4.5.3) by (3.1.3)

and Corollary (Ll). Furthermore by Theorem (5) the equality holds if and only if 1 :-:;; k :-:;; p and J1. is a g-maximal weight (see §3.2). But since w is decomposable and also 0 £;; b this is the case, by Proposition 2, (6) and Theorem (5) if and only if 0 is a commutative ideal in 6. But now assuming 1 :-:;; k :-:;; p (or else there is nothing more to prove) so that Ak = Mb then the Wi of Proposition (4) are decomposable vectors corresponding to 6-normal subspaces of 9 which are also commutative Lie subalgebras (by Theorem (5)). We will have proved (4.5.4) therefore (using Proposition (4)) if we can show that every such subspace equals 0, for some 1 :-:;; i :-:;; q. This proves in particular that every commutative ideal of 6 is also of this form. But this is immediate from (4.5.1) and Remark (5) since one must have 0 n q = 0 for any such subspace o. (Indeed if 0 =I- x Eon q there exists cP E Ll+ such that [x, e",l = cp(x)e", where cp(x) =I- O. But then e", E 0 which contradicts the fact that a is commutative.) Finally Vw , is not equivalent to VWj for i =I- j, even assuming the k for Wi and Wj are possibly different, by Theorem (7) since a, and aj are both contained in m. Q.E.D. Combining parts of Theorems (5) and (7) and using the terminology of Proposition (2) one immediately has (7.1). Let 1 :-:;; k :-:;; p. Then in the notation ofProposition (2) where V =Akg so that M = Mk one has D(M) is the set of all AE Z such that A may be written A = CPl + CP2 + ... + CPk where CPl, ... , CPk are k distinct positive roots satisfying Ilg + CPl + ... + CPkll 2 - IIgl1 2 = k. Moreover the multiplicity of v;. in Mis 1 for any such A.. COROLLARY

REFERENCES 1. A. BOREL: Groupes lineaires algebriques, Ann. Math., Princeton 64 (1956), 20-82. 2. B. KOSTANT: Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. Math, Princeton 74 (1961), 329-387. 3. J. L. KOSZUL: Homologie et cohomologie des algebres de Lie, Bull. Soc. Math. Fr. 78 (1950),65-627. 4. A. I. MALCEV: Commutative subalgebras of semi-simple Lie algebras, Izv. Akad. Nauk SSR, Ser. Mat. 9 (1945),291-300 (Russian); Translation No. 40, Series 1, American Mathematical Society (English).

Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.

481

Reprinted from the Proceedings of the United States-Japan Seminar in Differential Geometry Kyoto, Japan, 1965 Nippon Hyoronsha, Co., Ltd. Tokyo, Japan

Orbits, Symplectic Structures and Representation Theory Bertram

KOSTANT

We introduce a general approach to unitary representations for all Lie groups. An underlying feature is a study of sympletic manifolds X2n (i. e. there exists a closed non-singular 2-form on X). If [w] e H2(X, R) is an integral class there is an associated affinely connected Hermitian line bundle L over X which is unique if X is simply connected. Given a complex involutory totally singular distribution F" on X there is an associated cohomology theory HeLF) for the sheaf L.F of local sections of L which are constant along F. There then exists a Lie subalgebra a of the Lie algebra (under Poisson bracket) C of all smooth functions on X which naturally operates on HeLF)' For an element of C to operate on H(LF) in the case when X is a cotangent space is the inverse operation of taking the symbol of a differential operator. In general it is analogous to what the physicists call quantizing a function. Next one gives a complete classification of all sympletic homogeneous spaces for all Lie groups. They are related to orbits in the dual of the Lie algebra. The correspondence with an orbit maps the Lie algebra of the group into a (see above) and yeilds a representation of the group on H(L.F). The theory thus obtained embraces the Borel-Weil theory for compact groups, the Kirilov theory for nilpotent groups and Harish-Chandra-Gelfand theory for semi-simple groups.

Massachusetts Institute of Technology

71

B. Kostant, Collected Papers, DOI 10.1007/b94535_20, © Bertram Kostant 2009

482

Reprinted from Proceedings of Symposia in Pure Mathematics Volume 9 Algebraic Groups and Discontinuous Subgroups Copyright by the American Mathematical Society, 1966

Groups Over Z BY

BERTRAM KOSTANT 1. Preliminaries.

1.1. Let C be a commutative ring with 1. Let A be a co algebra over C with diagonal map d: A ~ A ®c A (it is assumed A has a counit B : A ~ C) and let R be an algebra over C with multiplication m: R ®c R ~ R (it is assumed R has a unit p : C ~ R). Then one knows that HomdA, R) has the structure of an algebra over C with unit where if f, g E HomdA, R) the product f * g E HomdA, R) is defined by f*g = m c(f@ g) cd. That is, one has a commutative diagram

In particular if we put R = C the dual A' = HomdA, C) has the structure of an algebra. Now assume that A is a Hopf algebra (A is an algebra and coalgebra such that d and B are homomorphisms and Bp is the identity on C). By an antipode on A we mean an element (necessarily unique if it exists) S E HomdA, A) such that I * S = S * I = B where I is the identity on A and * is as above with A taken for R. From now on Hopf algebra means Hopf algebra with antipode. 1.2. Now assume A is a Hopf algebra over C and R is any commutative C-algebra. Then if GR

=

U E HomdA, R)I f

is an algebra homomorphism}

one sees immediately that GR is a group under

* where

for any f

E

GR , a E A.

Thus one has a functor R ~ GR from all commutative algebras over R into groups and the functor is represented by A. Now if C is the set of integers Z then we may drop the word algebra so that R ~ GR is a functor from all commutative rings R to groups. 90 B. Kostant, Collected Papers, DOI 10.1007/b94535_21, © Bertram Kostant 2009

483

GROUPS OVER

Z

91

EXAMPLE. If A = Z[X ij , liD], i,j, = 1,2"", n, where the Xij are indeterminates and D = det(Xij), then A is a Hopf algebra over Z where

dX ij =

L X ik ® X kj , k

so that dD = D ® D. Also c(Xij) = 0 and s(X i ) = (_l)i+ j cofactor XjdD. Here G R = GI(n, R) for any commutative ring R. In the example above if one replaces A by its quotient with respect to the ideal generated by D - 1 then one obtains G R = Sl(n, R) for any commutative ring R. More generally for any semisimple Lie group G we will define a Hopf algebra Z(G) over Z with the following properties: (1) Z( G) is a finitely generated commutative integral domain; (2) for any field k k(G)

= Z(G) ®z k

is an affine algebra defining a semisimple algebraic group over k which is split over k, and is of the same type as G; (3) Q(G) defines Gover Q, where Q is the field of rational numbers. 1.3. From now on C = Z. Let B be a Hopf algebra over Z. An ideal Is; B will be said to be of finite type if BII is a finitely generated free Z-module. If I and I' are of finite type then the kernel I 1\ I' of the composed map

B~B ® B~BII ® BII' is again clearly of finite type defining an operation on the set of all such ideals. A family F of ideals of finite type will be said to be admissible if (1) nIEFI = (0); (2) s(I) E F for all I E F; (3) F is closed under I\. Now given such a family put AF

= {J E Hom(B, Z)IflI = 0 for some IE F}.

It is immediate then that AF has the structure of a Hopf algebra over Z. The multiplication in AF is defined as the transpose of the diagonal map in B. (It exists since F is closed under 1\.) The diagonal map in AF is defined as the transpose of the multiplication in B. (It exists since each f E AF vanishes on an ideal of finite type in B.) The antipode is simply the transpose of the antipode in B. (It exists since F is closed under s.) 1.4. Now let G be a complex semisimple Lie group and let 9 be its Lie algebra. Let V be the universal enveloping algebra of 9 so that V is a Hopf algebra over C

where

dx=x®l+l®x

484

BERTRAM KOSTANT

92

for any x E g. Also c; is given by c;(x) = 0 for any x E 9 and s is the anti-automorphism of U defined by s(x) = - x for any x E g. We will now define a Hopf algebra B over Z where B s U. The family of ideals F will be defined by G and one puts Z(G)

=

AF •

2. The definition and structure of B. Let 1) be a Cartan sub algebra of 9 and let d be the corresponding set of roots. Chevalley has shown (see [1]) the existence of a set of root vectors e"" cP E d, such that if cP, 1/1, cP + 1/1 E d then

[e"" e",l = ±re",+", where r E Z + (the set of nonnegative integers) is the minimum integer such that (ad e_",Ye", = 0 and if h", = [e"" e_",] then

cP(h",) = 2. We fix the e", as above and put gz equal to the Z span of all the e", and h", for cP E d. We recall some facts from [1] which, in fact, are easy to check. Let d+ be a system of positive roots and let II = (1J(1, ... , 1J(1) be the corresponding set of simple roots. Put hi = hai , i = 1,2,···, I, for simplicity. Then one has PROPOSITION 1. The elements h 1 , · · · , hI together with all e"" cP E d form a free Z-basis of gz. REMARK 1. Proposition 1 is of course only really a statement about the Z-span of the h", and the statement is of course well known. Now it is clear that gz is a Lie algebra over Z. Somewhat less obvious is the following fact of [1]: PROPOSITION 2. gz is stable under (aded>t/n! for any cPEd and nEZ+. REMARK 2. If h, e and f is a basis of the Lie algebra of SI(2, C) where [h, e] = 2e, = - 2f and (e,f) = h then Proposition 2 in essence reduces to the following fact: If Vb· .. ,Vk is a basis of an irreducible SI(2, C) module consisting of h-eigenvectors such that [h,f]

then the Z-span of the Vj is stable under em/m! and r/n! for all n, mE Z +. We now define B to be the algebra generated over Z by all elements e~/n! E U for all cPEI1 and nEZ+. 2.2. To prove that B is a Hopf algebra over Z with suitable properties we shall need some multiplication relations in U.

485

93

GROUPS OVER Z If h, e E 9 where [h, e] = Ae for some scalar }. then one easily establishes

(2.1.1) for any mE Z + and polynomial P E C[X]. N ow if U E U is arbitrary and mE Z + put Cum . =

u(u - 1)··· (u - m m.,

+

1)

.

Somewhat less trivial than (2.1.1) is the following useful relation among the generators of the Lie algebra of SI(2, C). LEMMA 1. Let h,e,jEg where [h,e] = 2e, [h,j] = -2fand [e,j] = h. Then for any n,mEZ+ one has k

f n- j

em-j

L (n _ J..),Ch-m-n+2j,j(m _")' J.

j=O

where k is the minimum of n and m. PROOF. One first of all proves directly from the bracket relation that

Lemma 1 is then just an exercise using (2.1.1), the relation above, and induction on m. 2.3. A sequence of C-linear independent elements to u(n) E U, n = 0,1,2,' . " where u(O) = 1, is called a sequence of divided powers in case du(n) =

m

L

u(j) ® u(n- j)

j=O

for all n. It is clear of course that the Z-space of the u(n) is a coalgebra over Z. EXAMPLE. If X E 9 and U p, then put Ak = 0. Otherwise let Ak be the span of all the 1-dimensional subspaces ∧k a where a is any k-dimensional abelian

511

512

Kostant’s Comments on Papers in Volume I

subalgebra of g. Let m k be the maximal eigenvalue of Cas| ∧k g. The following theorem is proved in paper #19. Theorem. For any k one has m k ≤ k.

(19.1)

Furthermore one has equality in (19.1) if and only if k ≤ p. Moreover in such a case the corresponding eigenspace for Cas | ∧k g is Ak . Finally if 0 = u ∈ ∧g is of the form u = x1 ∧ · · · ∧ xk for xi ∈ g, then u ∈ Ak if and only if the xi mutually commute. Let h be a Cartan subalgebra of g and let  ⊂ h∗ be the set of roots for (g, h). For all roots ϕ let corresponding root vectors eϕ be chosen. Also choose a system of positive roots + thereby defining a Borel subalgebra b containing h. Let n be the nilradical of b. Simply order + and if  ⊂ + we write  = {ϕ1 , . . . , ϕk }

(19.2)

in increasing order. Let e ⊂ ∧k g be defined by putting e = eϕ1 ∧ · · · ∧ eϕk .

(19.3)

Any ideal v of b which is contained in n defines a subset  ⊂ + of the form (19.2) and necessarily k  v= C eϕi . (19.4) i=1

In such a case the G-submodule spanned by G · e of ∧k is irreducible and C e is the highest weight space. Thus  =

k 

ϕi

(19.5)

i=1

is the highest weight. Write  = (v). Given two such ideals v1 , v2 , we show in this paper that v1 = v2 ⇐⇒ 1  = 2  (19.6) where for i = 1, 2, we have put i = (vi ). In particular, distinct such ideals define inequivalent irreducible representations. Now let C be the set of all abelian ideals a in b and let C(k) be the set of abelian ideals of dimension k. One has a ⊂ n for any a ∈ C so that the cardinality of C is

512

Kostant’s Comments on Papers in Volume I

513

finite by (19.4). Now for a ∈ C let Aa be the G-module generated by e(a) . Then if a ∈ C(k), (19.7) Aa is an irreducible G-submodule of Ak . ∞ Put A = k=0 Ak . The following result in paper #19 places the set of abelian ideals in b at center stage. Theorem A. A is a multiplicity-free G-module. Furthermore  A= Aa

(19.8)

a∈C

is the unique complete reduction of A as a sum of irreducible G-modules. Degreewise, for any k ∈ Z+ ,  Ak = Aa (19.9) a∈C (k)

is the unique complete reduction of Ak as a sum of irreducible G-modules. The abelian ideals in b are characterized in paper # 19 by the following result. Theorem B. Let  ⊂ + . Let k be the cardinality of  and let the notation be as in (19.2). Then (with the usual present-day definition of ρ and the usual norm in weight-lattice) one has |ρ + ϕ1 + · · · + ϕk |2 − |ρ|2 ≤ k

(19.10)

and equality occurs in (19.10) if and only if  is of the form  = (a) for some a ∈ C(k). Interest in the subject matter of paper # 19 was considerably stimulated by the subsequent discovery, due to Dale Peterson, of the following striking result: Card C = 2

(19.11)

where  = rank g. Another surprise in Peterson’s proof of (19.11) was the role played by the affine Weyl group. A considerable clarification of (19.11) was provided by P. Cellini and P. Papi. If V is the fundamental alcove in the Weyl chamber, then 2 V is a union of 2 alcoves. Cellini and Papi established a natural bijection of C with these 2 alcoves. The dimension of an abelian ideal associated to an alcove is the number of walls separating the alcove from the fundamental alcove. Beautiful results of D. Panyushev related the maximal abelian ideals with the long roots in + . R. Suter showed that Peterson’s result can be deduced from Theorem B above.

513

514

Kostant’s Comments on Papers in Volume I

Paper # 19 became the basis of later results and conferences. In particular, it has been used by Etingof–Kac and Kumar in the solution of the Cachazo–Douglas– Seiberg–Witten conjecture on the structure of conformal algebras.

20. Orbits, Symplectic Structures and Representation Theory, Proc. U.S.-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965, p. 71. In the early 60s I became interested in Hamiltonian mechanics and its symplectic manifold and Poisson bracket underlying structure. I also thought it was quite mysterious and marvelous that physicists in quantizing classical mechanics converted scalar functions (classical observables) on phase space in some fashion or other to operators on Hilbert space. Particularly striking in this process was that the classical observables were functions of position and momentum, q’s and p’s, whereas the elements in the Hilbert space were “functions” on half the variables (e.g., the q’s or the p’s). It seemed to me it would be very interesting to be able to make this process rigorous. The ideas I developed during the early 60s to do this are now referred to as geometric quantization of Kostant–Souriau theory. It was a fortunate time to think about these matters. For one thing there was the Borel–Weil theorem, and growing out of Hirzebruch’s Riemann–Roch theorem, line bundles and Chern classes were very much in the air. Bott had proved his generalization of the Borel–Weil theorem. There were also new constructions of unitary representation of Lie groups: Kirillov’s complete treatment for nilpotent groups and Gelfand and Harish-Chandra’s construction of such representations for semisimple groups using parabolic induction. The spark which ignited geometric quantization for me was Kirillov’s observation that there is a nonsingular alternating 2-form on Lie group coadjoint orbits. Symplectic manifolds as an object of study were not in vogue at that time, but I soon realized that this 2 form is indeed symplectic and that coadjoint orbits yield a vast supply of symplectic homogeneous spaces. Much more than that I came to the realization that what the physicists were doing and the above construction of representations are in fact manifestations of the same idea. The space C ∞ (X ) of smooth functions on a symplectic manifold X is a Lie algebra under Poisson bracket, and as such is a central extension of the Lie algebra Ham (X ) of Hamiltonian vector fields on X , thereby giving rise to a Lie algebra exact sequence 0 −→ C −→ C ∞ (X ) −→ Ham(X ) −→ 0.

(20.1)

The point of departure, in quantization, was the critical recognition that the symplectic 2-form, ω — constrained only by an integrality condition for the corresponding de Rham class [ω] — should be regarded as the curvature of a line bundle L, with connection, over X . I then found that the Lie algebra C ∞ (X ) operates, via what I

514

Kostant’s Comments on Papers in Volume I

515

called prequantization, on the space (L) of smooth sections of L. So functions become operators. Moreover, in the spirit of the Heisenberg uncertainty principle, under prequantization, the constant function operates as a nonzero scalar operator so that, unlike in classical mechanics, the action does not descend (see (20.1)) to Ham(X ). If a Lie group G with Lie algebra g operates symplectically on X in such a fashion that the action induces a homomorphism σ : g → Ham(X ) (this is always the case if X is simply connected) then one says that X is a Hamiltonian G-space if σ lifts to a homomorphism σ  : X → C ∞ (X ). I introduced this terminology but restricted my considerations to the case where G operated transitively on X . It has since become standardized terminology but without the assumption of homogeneity. If X is a Hamiltonian G-space, then the points of X define linear functionals on g giving rise to a map µ : X → g∗ (20.2) now well known as the moment or momentum map. In the homogeneous case (20.2) is a covering of a coadjoint orbit and using this, one of early results was a classification of all symplectic homogeneous spaces for G. For example, if g is semisimple, then the most general symplectic homogeneous space is a covering of a coadjoint orbit. In case G is also compact, then the coajoint orbits are themelves simply-connected so that one obtains a generalization of a theorem of H. C. Wang on the classification of all compact K¨ahler homogeneous spaces for G. To carry out geometric quantization one requires some additional structures, the main one involving a choice of what I called a polarization F of (X, ω). This is a choice of a complex involutory distribution of half the dimension of X whose “leaves” (in a complex sense) are Lagrangian (e.g., a K¨ahler structure). This “explains” the choice of half the variables in constructing the Hilbert space of states for physicists and parabolic induction in representation theory. The term polarization has been widely accepted and is now in common usage. Another ingredient required for geometric quantization (in order to obtain a Hilbert space structure) was the introduction of what I called half-forms. Given a polarization F, and inspired by the Bott–Borel–Weil theorem, one is led to introduce the sheaf S of germs of local sections of L which are constant along the leaves of F and then to consider the sheaf cohomology H (X, S). If X is an integral coadjoint orbit of G and F is invariant under the action of G, then G operates on H (X, S). Although there are many unresolved questions there are still a large number of examples where irreducible unitary representations of G can be extracted from this action. Except for half-forms, I gave a course at MIT in 1965 on the above subject. Notes of these lectures by N. Iwahori were widely distributed. See J. Wolf, Bull. AMS Vol 75 (1969) and Repr´esentations des groupes de Lie r´esolubles, P. Bernat et

515

516

Kostant’s Comments on Papers in Volume I

al, Dunod, vol. 4 (1972) for a reference to these notes. I spoke about this subject at a 1965 conference in Differential Geometry in Kyoto, Japan. An all too brief outline, paper # 20, appears in the 1965 proceedings of this conference published by Nippon Hyoronsha Co. I also presented the material above as Phillips lecturer at Haverford college in 1965. I finally published some of the material in Vol. 170 of the Lecture Notes in Mathematics, Springer, 1970. 21. Groups Over Z, Proc. Symposia in Pure Math., 9 (1966), 90–98. I became interested in the theory of Hopf algebras in the early 60s. I was mainly inspired by a paper of Milnor and Moore. They proved a theorem which asserted that a “connected cocommutative Hopf algebra H over a field of characteristic zero is the universal enveloping algebra U (g) of the Lie algebra of primitive elements in H .” If one discards connectedness, then H may contain elements g with augmentation value 1 such that δ(g) = g ⊗ g where δ is the diagonal homomorphism. I called such elements group-like since the set of elements form a group. This terminology has been adopted and become standardized terminology in Hopf algebra theory. I then went on to prove that the most general cocommutative Hopf algebra H over, say C, is the smash product H = C[G] # U (g)

(21.1)

where C[G] is the group algebra over the group G of group-like elements in H and g is the Lie algebra of primitive elements in H . I did not publish the theorem but it appears in a well-known (and by now classic) book, Hopf Algebras, written by one of my students at that time, Moss Sweedler. See the introduction in Hopf Algebras for the proper citation of this theorem. If H is a Hopf algebra, let H  be the space of those linear functionals on H which vanish on an ideal of finite codimension in H . We will say that H is dualizable if H  is nonsingularly paired to H . In such a case H  is a dualizable Hopf algebra and H ⊂ H  .

(21.2)

But now (21.1) and (21.2) provide a possible algebraic device for constructing a group G associated to a Lie algebra g without appealing to the usual Lie theoretic machinery, i.e., G ⊂ H  for the case, where under suitable conditions, H = U (g). This was part of the motivation which led to paper #21. In more detail Chevalley in his famous Tohoku paper introduced a group G(F), where F is any field, “modeled” after a complex simple Lie group. If g is a complex simple Lie algebra, Chevalley found a lattice gZ in g and root vectors eϕ ∈ gZ with the property that gZ was stable 1 under n! (ad eϕ )n for any n ∈ Z+ and any root ϕ ∈  where  is the set of roots

516

Kostant’s Comments on Papers in Volume I

517

with respect to a Cartan subalgebra h. Tensor product by F replaces gZ by g F and introduces F-parameter groups eϕ (t), where t ∈ F, with an automorphism action on g F . G(F) is the group generated by these F-parameter groups. The key objective of paper # 21 was to do the above in a Hopf algebra context so that hopefully we would get the affine ring of the desired algebraic group as a Hopf dual, construct the hyperalgebra at the identity, and find G(F) in the double dual. The first problem was to replace C by Z and construct a Z-form UZ (g). We defined UZ (g) as the algebra over Z in U (g) generated by all elements of the form 1 n n! eϕ for n ∈ Z+ and ϕ ∈ . Let + be a choice of positive roots and order, + = {ϕ1 , . . . , ϕr } so that if ϕ j − ϕi is a sum of positive roots, then j > i. Let  = rank g and if the set of simple roots  = {α1 , . . . , α }, let h i = [eαi , e−αi ]. For N , M ∈ Zr+ and K ∈ Z+ put     m1 eϕm r h1 h  e ϕ1 b(N , K , M) = ··· ··· r ··· n1! n r ! k1 k m 1 ! mr ! n1 e−ϕ 1

nr e−ϕ r

(21.3)

where N = {n 1 , . . . , nr }, K = {k1 , . . . , k } and M = {m 1 , . . . , m r }. Let d = dim g. The main theorem of Theorem in paper # 21 asserts the following. Theorem 1. The elements b(N , K , M) for (N , K , M) ∈ Zd+ are a Z-basis of UZ (g) and also a (PBW) C-basis of U (g) so that U (g) = C ⊗Z UZ (g).

(21.4)

Furthermore the Hopf structure on U (g) induces a Z-Hopf structure on UZ (g). In fact the b(N , K , M) are a d-multisequence of divided powers. In addition if V is a finite-dimensional U (g)-module, then UZ stabilizes a Z-lattice VZ in V . Moreover VZ is the sum of its intersections with the weight spaces in V . The Z-algebra UZ (g) has been referred to as the Kostant Z-form of U (g) and is well known in Lie theory. The last statement in Theorem 1 above implies that UZ (g) has a Hopf dual H  , where the definition of the latter is modified so that Z replaces C. If A is any commutative ring, then H A = H  ⊗Z A has the structure of a Hopf algebra and the group-like elements G(A) in its dual define a functor, A → G(A), from commutative rings to groups. In case A is an algebraically closed field, I had hoped at some later point to show that G(A) was the Chevalley group, modeled on G, and associated to A, and that H A is the affine ring of G(A). However I did not succeed in doing this. An unsolved problem for me was even to show that H A is Noetherian. (Theorem 3, attributed to Chevalley, in paper # 21 should be ignored since it is a misunderstanding on my part of a statement of Chevalley.). However the result is true and was proved by George Lusztig. See

517

518

Kostant’s Comments on Papers in Volume I

his paper entitled “Study of a Z-form of the coordinate ring of a reductive group”, Jour. AMS, March 31, 2008, posted online. Lusztig also establishes that this Hopf algebra approach to Chevalley theory generalizes to the quantum case.

518