Lie Theory pp 1-211 | Cite as

Nilpotent Orbits in Representation Theory

  • Jens Carsten Jantzen
Part of the Progress in Mathematics book series (PM, volume 228)


The term “nilpotent orbits” in the title is an abbreviation for “orbits consisting of nilpotent elements.” We shall consider here such orbits only for the adjoint action of a reductive algebraic group on its Lie algebra.


Algebraic Group Irreducible Component Parabolic Subgroup Maximal Torus Nilpotent Element 
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  1. [AL]
    D. Alvis, G. Lusztig: On Springer’s correspondence for simple groups of type E n (n = 6, 7, 8), Math. Proc. Comb. Phil. Soc. 92 (1982), 65–78.MathSciNetMATHCrossRefGoogle Scholar
  2. [BC1]
    P. Bala, R. W. Carter: Classes of unipotent elements in simple algebraic groups I, Math. Proc. Comb. Phil. Soc. 79 (1976), 401–425.MathSciNetMATHCrossRefGoogle Scholar
  3. [BC2]
    P. Bala, R. W. Carter: Classes of unipotent elements in simple algebraic groups II, Math. Proc. Comb. Phil. Soc. 80 (1976), 1–18.MathSciNetMATHCrossRefGoogle Scholar
  4. [BV]
    D. Barbasch, D. Vogan: Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra 80 (1983), 350–382.MathSciNetMATHCrossRefGoogle Scholar
  5. [BBD]
    A. A. Beilinson, J. Bernstein, P. Deligne: Faisceaux pervers, Astérisque 100 (1982), 5–171.MathSciNetGoogle Scholar
  6. [BL]
    W. M. Beynon, G. Lusztig: Some numerical results on the characters of exceptional Weyl groups, Math. Proc. Camb. Phil. Soc. 84 (1978), 417–426.MathSciNetMATHCrossRefGoogle Scholar
  7. [BB]
    A. Bialynicki-Birula: Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973) 480–497.MathSciNetMATHCrossRefGoogle Scholar
  8. [Bo]
    A. Borel: Linear Algebraic Groups, 2nd ed. (Graduate Texts in Math. 126), New York etc. 1991 (Springer).Google Scholar
  9. [Bh1]
    W. Borho: Definition einer Dixmier-Abbildung für s [(n, C), Invent. Math. 40 (1977), 143–169.MathSciNetMATHCrossRefGoogle Scholar
  10. [Bh2]
    W. Borho: Zum Induzieren unipotenter Klassen, Abh. Math. Seminar Univ. Hamburg 51 (1981), 1–4.MathSciNetMATHCrossRefGoogle Scholar
  11. [BhB1]
    W. Borho, J.-L. Brylinski: Differential operators on homogeneous spaces I, Irreducibility of the associated variety for annihilators of induced modules, Invent. Math. 69 (1982), 437–476.MathSciNetMATHCrossRefGoogle Scholar
  12. [BhB2]
    W. Borho, J.-L. Brylinski: Differential operators on homogeneous spaces III, Characteristic varieties of Harish Chandra modules and of primitive ideals, Invent. Math. 80 (1985), 1–68.MathSciNetMATHCrossRefGoogle Scholar
  13. [BBM]
    W. Borho, J.-L. Brylinski, R. MacPherson: Nilpotent Orbits, Primitive Ideals, and Characteristic Classes—A Geometric Perspective in Ring Theory (Progress in Math. 78), Boston etc. 1989 (Birkhäuser).Google Scholar
  14. [BK]
    W. Borho, H. Kraft: Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), 61–104.MathSciNetMATHCrossRefGoogle Scholar
  15. [BM]
    W. Borho, R. MacPherson: Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sc. Paris (1) 292 (1981), 707–710.MathSciNetMATHGoogle Scholar
  16. [Bou1]
    N. Bourbaki: Algèbre Commutative, Paris 1961/1962 (Hermann).Google Scholar
  17. [Bou2]
    N. Bourbaki: Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Paris 1968 (Hermann).Google Scholar
  18. [Bou3]
    N. Bourbaki: Groupes et algébres de Lie, Chapitres 7 et 8, Paris 1975 (Hermann).Google Scholar
  19. [BP]
    M. Brion, P. Polo: Generic singularities of certain Schubert varieties, Math. Z 231 (1999), 301–324.MathSciNetMATHCrossRefGoogle Scholar
  20. [Br1]
    B. Broer: Line bundles on the cotangent bundle of the flag variety, Invent. Math. 113 (1993), 1–20.MathSciNetMATHCrossRefGoogle Scholar
  21. [Br2]
    B. Broer: Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety, pp. 1–19 in: J.-L. Brylinski et al. (eds.), Lie Theory and Geometry, In Honor of Bertram Kostant (Progress in Math. 123) Boston etc. 1994 (Birkhäuser).Google Scholar
  22. [Br3]
    B. Broer: Normal nilpotent varieties in F 4, J. Algebra 207 (1998), 427–448.MathSciNetMATHCrossRefGoogle Scholar
  23. [Br4]
    B. Broer: Decomposition varieties in semisimple Lie algebras, Can. J. Math. 50 (1998), 929–971.MathSciNetMATHCrossRefGoogle Scholar
  24. [BG]
    K. A. Brown, I. Gordon: The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras, Math. Z. 238 (2001), 733–779.MathSciNetMATHCrossRefGoogle Scholar
  25. [Bw]
    K. S. Brown: Cohomology of Groups (Graduate Texts in Mathematics 87), New York etc. 1982 (Springer).Google Scholar
  26. [Ca]
    R. W. Carter: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters (Pure and Applied Math.), Chichester etc. 1985 (Wiley & Sons).Google Scholar
  27. [Ch]
    J.-T. Chang: Asymptotic and characteristic cycles for representations of complex groups, Compositio math. 88 (1993), 265–283.MathSciNetMATHGoogle Scholar
  28. [CG]
    N. Chriss, V. Ginzburg: Representation Theory and Complex Geometry, Boston etc. 1997 (Birkhäuser).Google Scholar
  29. [CM]
    D. H. Collingwood, W. M. McGovern: Nilpotent Orbits in Semisimple Lie Algebras, New York 1993 (Van Nostrand).Google Scholar
  30. [CR]
    C. W. Curtis, I. Reiner: Methods of Representation Theory, Vol. I (Pure and Applied Math.) New York etc. 1981 (Wiley).Google Scholar
  31. [DLP]
    C. De Concini, G. Lusztig, C. Procesi: Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), 15–34.MathSciNetMATHCrossRefGoogle Scholar
  32. [De]
    P. Deligne: Séminaire de Géométrie Algébrique du Bois Marie 41/2 (Lecture Notes in Math. 569, Berlin etc. 1977 (Springer).Google Scholar
  33. [DL]
    P. Deligne, G. Lusztig: Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), 103–161.MathSciNetMATHCrossRefGoogle Scholar
  34. [Dm]
    M. Demazure: Désingularisation des variétés de Schubert généralisées, Ann. scient. Éc. Norm. Sup. (4) 7 (1974), 53–88.MathSciNetMATHGoogle Scholar
  35. [DG]
    M. Demazure, P. Gabriel: Groupes Algébriques I, Paris/Amsterdam 1970 (Masson/Noth-Holland).Google Scholar
  36. [Di]
    J. Dixmier: Algébres enveloppantes, Paris etc. 1974 (Gauthier-Villars).Google Scholar
  37. [Do1]
    S. Donkin: On conjugating representations and adjoint representations of semisimple groups, Invent. Math. 91 (1988), 137–145.MathSciNetMATHCrossRefGoogle Scholar
  38. [Do2]
    S. Donkin: The normality of closures of conjugacy classes of matrices, Invent. Math. 101 (1990), 717–736.MathSciNetMATHCrossRefGoogle Scholar
  39. [Ei]
    D. Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry (Graduate Texts in Math. 150, New York etc. 1995 (Springer).Google Scholar
  40. [El]
    G. B. Elkington: Centralizers of unipotent elements in semisimple algebraic groups, J. Algebra 23 (1972), 137–163.MathSciNetMATHCrossRefGoogle Scholar
  41. [Fo]
    J. Fogarty: Fixed point schemes, Amer. Math. J. 95 (1973), 35–51.MathSciNetMATHCrossRefGoogle Scholar
  42. [Fr]
    H. Freudenthal: Elementatteilertheorie der komplexen ortogonalen und symplektischen Gruppen, Indag. Math. 14 (1952), 199–201.MathSciNetGoogle Scholar
  43. [FP]
    E. M. Friedlander, B. J. Parshall: Cohomology of Lie algebras and algebraic groups, Amer. J. Math. 108 (1986), 235–253.MathSciNetMATHCrossRefGoogle Scholar
  44. [Fu]
    W. Fulton: Intersection Theory (Ergebnisse der Math. etc. 3) 2, Berlin etc. 1984 (Springer).Google Scholar
  45. [Ge]
    M. Gerstenhaber: Dominance over the classical groups, Ann. of Math. (2) 74 (1961), 532–569.MathSciNetMATHCrossRefGoogle Scholar
  46. [Gr]
    W. A. Graham: Functions on the universal cover of the principal unipotent orbit, Invent. Math. 108 (1992), 15–27.MathSciNetMATHCrossRefGoogle Scholar
  47. [Ha]
    R. Hartshorne: Algebraic Geometry (Graduate Texts in Math. 52), New York etc. 1977 (Springer).Google Scholar
  48. [He1]
    W. H. Hesselink: Singularities in the nilpotent scheme of a classical group, Trans. Amer. Math. Soc. 222 (1976), 1–32.MathSciNetMATHCrossRefGoogle Scholar
  49. [He2]
    W. H. Hesselink: Polarizations in the classical groups, Math. Z 160 (1978), 217–234.MathSciNetMATHCrossRefGoogle Scholar
  50. [He3]
    W. H. Hesselink: Nilpotency in classical groups over a field of characteristic 2, Math. Z 166 (1979), 165–181.MathSciNetMATHCrossRefGoogle Scholar
  51. [He4]
    W. H. Hesselink: Characters of the nullcone, Math. Ann. 252 (1980), 179–182.MathSciNetMATHCrossRefGoogle Scholar
  52. [Hi]
    V. Hinich: On the singularities of nilpotent orbits, Israel J. Math. 73 (1991), 297–308.MathSciNetMATHCrossRefGoogle Scholar
  53. [HS]
    D. F. Holt, N. Spaltenstein: Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristics, J. Austral. Math. Soc. (A) 38 (1985), 330–350.MathSciNetMATHCrossRefGoogle Scholar
  54. [Ho]
    R. Hotta: On Joseph’s construction of Weyl group representations, Tôhoku Math. J. 36 (1984), 49–74.MathSciNetMATHCrossRefGoogle Scholar
  55. [HoSh]
    R. Hotta, N. Shimomura: The fixed point subvarieties of unipotent transformations on generalized flag varieties and the Green functions. Combinatorial and cohomological treatments centering GLn, Math. Ann. 241 (1979), 193–208.MathSciNetMATHCrossRefGoogle Scholar
  56. [HoS]
    R. Hotta, T. A. Springer: A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math. 41 (1977), 113–127.MathSciNetMATHCrossRefGoogle Scholar
  57. [Hu1]
    J. E. Humphreys: Introduction to Lie Algebras and Representation Theory (Graduate Texts in Math. 9), New York etc. 1972 (Springer).Google Scholar
  58. [Hu2]
    J. E. Humphreys: Linear Algebraic Groups (Graduate Texts in Math. 21), New York etc. 1975 (Springer).Google Scholar
  59. [Hu3]
    J. E. Humphreys: Conjugacy Classes in Semisimple Algebraic Groups (Math. Surveys and Monographs 43), Providence, R.I. 1995 (Amer. Math. Soc).Google Scholar
  60. [Iv]
    B. Iversen: A fixed point formula for action of tori on algebraic varieties, Invent. Math. 16 (1972), 229–236.MathSciNetMATHCrossRefGoogle Scholar
  61. [JK]
    G. James, A. Kerber: The Representation Theory of the Symmetric Group (Encyclopedia of Maths, and its Appl. 16), Reading, Mass. etc. 1981 (Addison-Wesley).Google Scholar
  62. [Ja1]
    J. C. Jantzen: Einhüllende Algebren halbeinfacher Lie-Algebren (Ergebnisse der Math. etc. (3) 3), Berlin etc. 1983 (Springer).Google Scholar
  63. [Ja2]
    J. C. Jantzen: Representations of Algebraic Groups (Pure and Applied Math. 131, Orlando, FL etc. 1987 (Academic).Google Scholar
  64. [Ja3]
    J. C. Jantzen: Representations of Lie algebras in prime characteristic, Notes by Iain Gordon, pp. 185–235 in: A. Broer (ed.), Representation Theories and Algebraic Geometry, Proc. Montreal 1997 (NATO ASI Series C, Vol. 514), Dordrecht etc. 1998 (Kluwer).Google Scholar
  65. [Jo1]
    A. Joseph: On the variety of a highest weight module, J. Algebra 88 (1984), 238–278.MathSciNetMATHCrossRefGoogle Scholar
  66. [Jo2]
    A. Joseph: On the associated variety of a primitive ideal, J. Algebra 93 (1985), 509–523.MathSciNetMATHCrossRefGoogle Scholar
  67. [Jo3]
    A. Joseph: Some ring theoretic techniques and open problems in enveloping algebras, pp. 27–67 in: S. Montgomery, L. Small (eds.), Noncommutative Rings (MSRI Publ. 24) Berlin etc. 1992 (Springer).Google Scholar
  68. [KL]
    D. Kazhdan, G. Lusztig: Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153–215.MathSciNetMATHCrossRefGoogle Scholar
  69. [KR]
    T. Kambayashi, P. Russell: On linearizing algebraic torus actions, J. Pure Appl. Algebra 23 (1982), 243–250.MathSciNetMATHCrossRefGoogle Scholar
  70. [KW]
    R. Kiehl, R. Weissauer: Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform (Ergebnisse der Math. etc. (3) 42), Berlin etc. 2001 (Springer).Google Scholar
  71. [Kb]
    T. Kobayashi: Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1997), 229–256.CrossRefGoogle Scholar
  72. [Ko]
    B. Kostant: Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.MathSciNetMATHCrossRefGoogle Scholar
  73. [Kr]
    H. Kraft: Closures of conjugacy classes in G2, J. Algebra 126 (1989), 454–465.MathSciNetMATHCrossRefGoogle Scholar
  74. [KP1]
    H. Kraft, C. Procesi: Closures of conjugacy classes of matrices are normal, Invent. Math. 62(1981), 503–515.MathSciNetMATHCrossRefGoogle Scholar
  75. [KP2]
    H. Kraft, C. Procesi: On the geometry of conjugacy classes in classical groups, Comment. Math. Helvetici 57 (1982), 539–602.MathSciNetMATHCrossRefGoogle Scholar
  76. [KLT]
    S. Kumar, N. Lauritzen, J. F. Thomsen: Frobenius splitting of cotangent bundles of flag varieties, Invent. Math. 136 (1999), 603–621.MathSciNetMATHCrossRefGoogle Scholar
  77. [L]
    S. Lang: Algebra, Reading, Mass. 1969 (Addison-Wesley). [Le] M. van Leeuwen: A Robinson-Schensted Algorithm in the Geometry of Flags for Classical Groups, Thesis, Utrecht 1989.Google Scholar
  78. [Lu1]
    G. Lusztig: Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), 205–272.MathSciNetMATHCrossRefGoogle Scholar
  79. [Lu2]
    G. Lusztig: An induction theorem for Springer’s representations, preprint July 2001.Google Scholar
  80. [LS]
    G. Lusztig, N. Spaltenstein: Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), 41–52.MathSciNetMATHCrossRefGoogle Scholar
  81. [Mc1]
    W. M. McGovern: Rings of regular functions on nilpotent orbits and their covers, Invent. Math. 97 (1989), 209–217.MathSciNetMATHCrossRefGoogle Scholar
  82. [Mc2]
    W. M. McGovern: Dixmier algebras and the orbit method, pp. 397–416 in: A. Connes et al. (eds.) Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Proc. Paris 1989 (Progress in Math. 92), Boston 1990 (Birkhäuser).Google Scholar
  83. [Mc3]
    W. M. McGovern: Completely prime maximal ideals and quantization, Mem. Amer. Math. Soc. 108 (1994), no. 519.Google Scholar
  84. [Mc4]
    W. M. McGovern: Rings of regular functions on nilpotent orbits II: Model algebras and orbits, Commun, in Algebra 22 (1994), 765–772.MathSciNetMATHCrossRefGoogle Scholar
  85. [MK]
    V. B. Mehta, W. van der Kallen: A simultaneous Frobenius splitting for closures of conjugacy classes of nilpotent matrices, Compositio Math. 84 (1992), 211–221.MathSciNetMATHGoogle Scholar
  86. [Me]
    A. Melnikov: Irreducibility of the associated varieties of simple highest weight modules in s[(n), C. R. Acad. Sci. Paris (1) 316 (1993), 53–57.MathSciNetMATHGoogle Scholar
  87. [Mi]
    J. S. Milne: Étale Cohomology (Princeton Mathematical Series 33), Princeton, N. J. 1980 (Princeton Univ.).Google Scholar
  88. [Mo]
    C. Moeglin: Idéaux complétement premiers de l’algébre enveloppante de g[n (C), J. Algebra 106 (1987), 287–366.MathSciNetMATHCrossRefGoogle Scholar
  89. [Mt]
    G. D. Mostow: Fully reducible subgroups of algebraic groups, Amer. Math. J. 78 (1956), 200–221.MathSciNetMATHCrossRefGoogle Scholar
  90. [Mu]
    D. Mumford: Abelian Varieties (Tata Studies in Math. 5) London etc. 1970 (Oxford Univ.).Google Scholar
  91. [Pal]
    D. I. Panyushev: Rationality of singularities and the Gorenstein property for nilpotent Lie algebras, Funct. Anal. Appl. 25 (1991), 225–226, translated from:Функц. анализ и его прил 25:3 (1991), 76-78.MathSciNetMATHCrossRefGoogle Scholar
  92. [Pa2]
    D. I. Panyushev: Complexity and nilpotent orbits, Manuscripta Math. 83 (1994), 223–237MathSciNetMATHCrossRefGoogle Scholar
  93. [Pa3]
    D. I. Panyushev: On spherical nilpotent orbits and beyond, Ann. Inst. Fourier (Grenoble) 49 (1999), 1453–1476.MathSciNetMATHCrossRefGoogle Scholar
  94. [Ps]
    B. J. Parshall: Cohomology of algebraic groups, pp. 233–248 in: P. Fong (ed.), The Arcata Conference on Representations of Finite Groups, Proc. Arcata 1986 (Proc. Symposia Pure Math. 47:1), Providence, R.I., 1987 (Amer. Math. Soc.).Google Scholar
  95. [Pol]
    K. Pommerening: Über die unipotenten Klassen reduktiver Gruppen, J. Algebra 49 (1977), 525–536.MathSciNetMATHCrossRefGoogle Scholar
  96. [Po2]
    K. Pommerening: Über die unipotenten Klassen reduktiver Gruppen II, J. Algebra 65 (1980), 373–398.MathSciNetMATHCrossRefGoogle Scholar
  97. [Pr]
    A. Premet: An analogue of the Jacobson-Morozov theorem for Lie algebras of reductive groups of good characteristics, Trans. Amer. Math. Soc. 347 (1995), 2961–2988.MathSciNetMATHCrossRefGoogle Scholar
  98. [RR]
    S. Ramanan, A. Ramanathan: Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), 217–224.MathSciNetMATHCrossRefGoogle Scholar
  99. [SV]
    W. Schmid, K. Vilonen: Characteristic cycles and wave front cycles of representations of reductive Lie groups, Ann. of Math. (2) 151 (2000), 1071–1118.MathSciNetMATHCrossRefGoogle Scholar
  100. [SGA1]
    Séminaire de Géométrie Algébrique du Bois Marie 1960/61, dirigé par A. Grothendieck, (Lecture Notes in Math. 224), Berlin etc. 1971 (Springer).Google Scholar
  101. [SGA4]
    Séminaire de Géométrie Algébrique du Bois Marie 1963-64, dirigé par M. Artin, A. Grothendieck, J. L. Verdier, Tome 3 (Lecture Notes in Math. 305), Berlin etc. 1973 (Springer).Google Scholar
  102. [SGA5]
    Séminaire de Géométrie Algébrique du Bois Marie 1965-66, dirigé par A. Grothendieck (Lecture Notes in Math. 589), Berlin etc. 1977 (Springer).Google Scholar
  103. [Se]
    J.-P. Serre: Espaces fibrés algébriques, exposé 1 in: Séminaire C. Chevalley, 2e année: 1958, Anneaux de Chow et applications, Paris 1958.Google Scholar
  104. [Sh]
    I. R. Shafarevich: Basic Algebraic Geometry (Grundlehren d. math. Wiss. 213, Berlin etc. 1974 (Springer).Google Scholar
  105. [Sho]
    T. Shoji: Geometry of orbits and Springer correspondence, Astérisque 168 (1988), 61–140.MathSciNetGoogle Scholar
  106. [S1]
    P. Slodowy: Simple Singularities and Simple Algebraic Groups (Lecture Notes in Math. 815 Berlin etc. 1980 (Springer).Google Scholar
  107. [ST]
    E. Sommers, P. Trapa: The adjoint representation in rings of functions, Represent. Theory 1 (1997), 182–189.MathSciNetMATHCrossRefGoogle Scholar
  108. [Spa1]
    N. Spaltenstein: The fixed point set of a unipotent transformation on the flag manifold, Indag. Math. 38 (1976), 452–456.MathSciNetGoogle Scholar
  109. [Spa2]
    N. Spaltenstein: Classes Unipotentes et Sous-groupes de Borel (Lecture Notes in Math. 946, Berlin etc. 1982 (Springer).Google Scholar
  110. [Spa3]
    N. Spaltenstein: On unipotent and nilpotent elements of groups of type E 6 J-London Math. Soc. (2) 27 (1983), 413–420.MathSciNetMATHCrossRefGoogle Scholar
  111. [Spa4]
    N. Spaltenstein: Nilpotent classes in Lie algebras of type F 4 over fields of characteristic 2, J. Fac. Sci. Univ. Tokyo (IA Math.) 30 (1984), 517–524.MathSciNetMATHGoogle Scholar
  112. [Spa5]
    N. Spaltenstein: Existence of good transversal slices to nilpotent orbits in good characteristics, J. Fac. Sci. Univ. Tokyo (IA Math.) 31 (1984), 283–286.MathSciNetMATHGoogle Scholar
  113. [Sp1]
    T. A. Springer: Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207.MathSciNetMATHCrossRefGoogle Scholar
  114. [Sp]
    T. A. Springer: Linear Algebraic Groups (Progress in Math. 9), 2nd ed., Boston etc. 1998 (Birkhäuser).Google Scholar
  115. [SpSt]
    T. A. Springer, R. Steinberg: Conjugacy Classes, pp. 167–266 in: A. Borel et al.: Seminar on Algebraic Groups and Related Finite Groups (Lecture Notes in Math. 131), Berlin etc. 1970 (Springer)Google Scholar
  116. [Stu]
    U. Stuhler: Unipotente und nilpotente Klassen in einfachen Gruppen und Liealgebren vom Typ G 2, Indag. Math. 33 (1971), 365–378.MathSciNetGoogle Scholar
  117. [Ta]
    T. Tanisaki: Characteristic varieties of highest weight modules and primitive quotients, pp. 1–30 in: K. Okamoto, T. Oshima (eds.), Representations of Lie Groups, Proc. Kyoto/Hiroshima 1986 (Advanced Studies in Pure Math. 14), Tokyo 1988 (Kinokuniya).Google Scholar
  118. [Th]
    J. F. Thomsen: Normality of certain nilpotent varieties in positive characteristic, J. Algebra 227 (2000), 595–613.MathSciNetMATHCrossRefGoogle Scholar
  119. [Ve]
    F. D. Veldkamp: The center of the universal enveloping algebra of a Lie algebra in characteristic p, Ann. scient. Éc. Norm. Sup. (4) 5 (1972), 217–240.MathSciNetMATHGoogle Scholar
  120. [VK]
    É B. Vinberg, B. N. Kimel’f el’d: Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups, Funct. Anal. Appl. 12 (1978), 168–174, translated from: Однородные области на флаговых многообразияхи сферические подгруппы полупростых групп Ли, Функц. анализ и его прил. 12:3 (1978), 12-19.CrossRefGoogle Scholar
  121. [VP]
    É. B. Vinberg, V. L. Popov: On a class of quasihomogeneous affine varieties, Math. USSR (Izvestya) 6 (1972), 743–758, translated from: Об одном классе аффинных квазиоднородных многообразий, Изв. Акад. Наук СССР (Серия Матем.) 36 (1972), 749-764.CrossRefGoogle Scholar
  122. [Vo1]
    D. A. Vogan, Jr.: The orbit method and primitive ideals for semisimple Lie algebras, pp. 281–316 in: D. J. Britten et al. (eds.), Lie Algebras and Related Topics, Proc. Windsor, Ont., 1984 (CMS Conf. Proc. 5), Providence, R.I. 1986 (Amer. Math. Soc).Google Scholar
  123. [Vo2]
    D. A. Vogan, Jr.: Dixmier algebras, sheets, and representations theory, pp. 333–395 in: A. Connes et al. (eds.), Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Proc. Paris 1989 (Progress in Math. 92), Boston 1990 (Birkhäuser).Google Scholar
  124. [Vo3]
    D. A. Vogan, Jr.: Associated varieties and unipotent representations, pp. 315–388 in: W. Barker, P. Sally (eds.), Harmonic Analysis on Reductive Groups, Proc. Brunswick, ME, 1989 (Progress in Math. 101), Boston 1991 (Birkhäuser).Google Scholar
  125. [ZS]
    O. Zariski, P. Samuel: Commutative Algebra, Volume I/II (Graduate Texts in Math. 28/29, New York etc. (Springer).Google Scholar

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© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Jens Carsten Jantzen
    • 1
  1. 1.Department of Mathematics Ny MunkegadeAarhus CDenmark

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