Transformation Groups

, Volume 14, Issue 2, pp 417–461 | Cite as

Vinberg’s θ-groups in positive characteristic and Kostant–Weierstrass slices



We generalize the basic results of Vinberg’s θ-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded Lie algebra (in zero or odd positive characteristic), confirming a conjecture of Popov in this case.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Be]
    D. J. Benson, Polynomial Invariants of Finite Groups, London Mathematical Society Lecture Notes, Vol. 190, Cambridge University Press, Cambridge, 1993.MATHGoogle Scholar
  2. [Bo]
    A. Borel, Linear Algebraic Groups, Springer-Verlag, New York, 1991.MATHGoogle Scholar
  3. [Ca]
    R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59.MATHMathSciNetGoogle Scholar
  4. [Ch]
    C. Chevalley, Classification des Groupes Algébriques Semi-Simples, Springer, Berlin, 2005. Collected Works, Vol. 3, edited and with a preface by P. Cartier, with the collaboration of Cartier, A. Grothendieck, and M. Lezard.Google Scholar
  5. [GP]
    I. Gordon, A. Premet, Block representation type of reduced enveloping algebras, Trans. Amer. Math. Soc. 354(4) (2002), 1549–1581.MATHCrossRefMathSciNetGoogle Scholar
  6. [Ha]
    W. J. Haboush, Reductive groups are geometrically reductive, Ann. of Math. (2) 102(1) (1975), 67–83.CrossRefMathSciNetGoogle Scholar
  7. [He]
    A. J. Helminck, Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces, Adv. Math. 71(1) (1988), 21–91.MATHCrossRefMathSciNetGoogle Scholar
  8. [Hu]
    J. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 1975.MATHGoogle Scholar
  9. [Ka]
    N. Kawanaka, Orbits and stabilizers of nilpotent elements of a graded semisimple Lie algebra, J. Fac. Sci. Univ. Tokyo Section IA Math. 34(3) (1987), 573–597.MATHMathSciNetGoogle Scholar
  10. [Ke]
    G. R.Kempf, Instability in invariant theory, Ann. of Math. (2) 108(2) (1978), 299–316.CrossRefMathSciNetGoogle Scholar
  11. [KR]
    B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809.MATHCrossRefMathSciNetGoogle Scholar
  12. [L1]
    P. Levy, Commuting varieties of Lie algebras over fields of prime characteristic, J. Algebra 250(2) (2002), 473–484.MATHCrossRefMathSciNetGoogle Scholar
  13. [L2]
    P. Levy, Involutions of reductive Lie algebras in positive characteristic, Adv. Math. 210(2) (2007), 505–559.MATHCrossRefMathSciNetGoogle Scholar
  14. [MN]
    B. Martin, A. Neeman, The map VV//G need not be separable, Math. Res. Lett. 8(5–6) (2001), 813–817.MATHMathSciNetGoogle Scholar
  15. [MS]
    G. J. McNinch, E. Sommers, Component groups of unipotent centralizers in good characteristic, J. Algebra 260(1) (2003), 323–337. Special issue celebrating the 80th birthday of Robert Steinberg.MATHCrossRefMathSciNetGoogle Scholar
  16. [Pa1]
    Д. И. Панюшев, О пространстве орбит конечных и связных линейных групп, Изв. Акад. Наук СССР, сер. мат. 46(1) (1982), 95–99, 191. Engl. transl.: D. I. Panyushev, On orbit spaces of finite and connected linear groups, Math. USSR-Izv. 20 (1983), 97–101.Google Scholar
  17. [Pa2]
    Д. И. Панюшев, Регулярные злементы в пространствах линейных представлений, II., Изв. Акад. Наук СССР, сер. мат. 49(5) (1985), 979–985, 1120. Engl. transl.: D. I. Panyushev, Regular elements in spaces of linear representations, II, Math. USSR.-Izv. 27 279–284, 1986.Google Scholar
  18. [Pa3]
    D. I. Panyushev, On invariant theory of θ-groups, J. Algebra 283(2) (2005), 655–670.MATHCrossRefMathSciNetGoogle Scholar
  19. [Po1]
    В. Л. Попов, Представления со свободным модулем ковариантов, Функц. анализ и его прил. 10(3) (1976), 91–92. Engl. transl.: V. L. Popov, Representations with a free module of covariants, Functional Anal. Appl. 10 (1976), no. 3, 242–244.Google Scholar
  20. [Po2]
    V. L. Popov, Sections in invariant theory, in: The Sophus Lie Memorial Conference (Oslo, 1992), Scandinavian University Press, Oslo, 1994, pp. 315–361.Google Scholar
  21. [PV]
    Э. Б. Винберг, В. Л. Попов, Теория инвариантов, Итоги науки и техн., Совр. пробл. матем., Фунд. направл., ВИНИТИ, Москва, т. 55, 1989, стр. 137–309. Engl. transl.: V. L. Popov, E. B. Vinberg, Invariant Theory, in: Algebraic Geometry IV, Encycl. Math. Sci., Vol. 55, Springer-Verlag, Heidelberg, 1994, pp. 123–284.Google Scholar
  22. [Pr1]
    A. Premet, Complexity of Lie algebra representations and nilpotent elements of the stabilizers of linear forms, Math. Z. 228(2) (1998), 255–282.MATHCrossRefMathSciNetGoogle Scholar
  23. [Pr2]
    A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory, J. Algebra 260(1) (2003), 338–366. Special issue celebrating the 80th birthday of Robert Steinberg.MATHCrossRefMathSciNetGoogle Scholar
  24. [PT]
    A. Premet, R. Tange, Zassenhaus varieties of general linear Lie algebras, J. Algebra 294(1) (2005), 177–195.MATHCrossRefMathSciNetGoogle Scholar
  25. [Ri]
    R. W. Richardson, Orbits, invariants, and representations associated to involutions of reductive groups, Invent. Math. 66(2) (1982), 287–312.MATHCrossRefMathSciNetGoogle Scholar
  26. [Ro]
    G. Rousseau, Immeubles sphériques et théorie des invariants, C. R. Acad. Sci. Paris. Sér. A–B 286(5) (1978), A247–A250.MATHMathSciNetGoogle Scholar
  27. [Sl]
    P. Slodowy, Die Theorie der optimalen Einparameteruntergruppen für instabile Vektoren, in: Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., Vol. 13, Birkhäuser, Basel, 1989, pp. 115–131.Google Scholar
  28. [So]
    E. Sommers, A generalization of the Bala–Carter theorem for nilpotent orbits, Int. Math. Res. Not. (11) (1998), 539–562.CrossRefMathSciNetGoogle Scholar
  29. [Sp]
    T. A. Springer, The classification of involutions of simple algebraic groups, J. Fac. Sci. Univ. Tokyo Section IA Math. 34(3) (1987), 655–670. 1987.MATHMathSciNetGoogle Scholar
  30. [SpSt]
    T. A. Springer, R. Steinberg, Conjugacy classes, in: Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, NJ, 1968/69), Lecture Notes in Mathematics, Vol. 131, Springer-Verlag, Berlin, 1970, pp. 167–266.Google Scholar
  31. [St]
    R. Steinberg, Endomorphisms of Linear algebraic groups, Mem. Amer. Math. Soc. 80, 1968.Google Scholar
  32. [Ve]
    F. D. Veldkamp, The center of the universal enveloping algebra of a Lie algebra in characteristic p, Ann. Sci. École Norm. Sup. (4) 5 (1972), 217–240.MATHMathSciNetGoogle Scholar
  33. [Vi]
    Э. Б. Винберг, Группа Вейля градуированной алгебры Ли, Изв. Акад. Наук СССР, сер. мат. 40(3) (1976), 488–526, 709. Engl. transl.: È. B. Vinberg, The Weyl group of a graded Lie algebra, Math. USSR-Izv. 10 (1977), 463–495.Google Scholar
  34. [Vu]
    T. Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France 102 (1974), 317–333.MATHMathSciNetGoogle Scholar
  35. [W]
    D. J. Winter, Fixed points and stable subgroups of algebraic group automorphisms, Proc. Amer. Math. Soc. 18 (1967), 1107–1113.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.EPFL SB IGATBâtiment BCHLausanneSwitzerland

Personalised recommendations