Abstract
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.
References
D. J. Benson, Polynomial Invariants of Finite Groups, London Mathematical Society Lecture Notes, Vol. 190, Cambridge University Press, Cambridge, 1993.
A. Borel, Linear Algebraic Groups, Springer-Verlag, New York, 1991.
R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59.
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.
I. Gordon, A. Premet, Block representation type of reduced enveloping algebras, Trans. Amer. Math. Soc. 354(4) (2002), 1549–1581.
W. J. Haboush, Reductive groups are geometrically reductive, Ann. of Math. (2) 102(1) (1975), 67–83.
A. J. Helminck, Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces, Adv. Math. 71(1) (1988), 21–91.
J. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 1975.
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.
G. R.Kempf, Instability in invariant theory, Ann. of Math. (2) 108(2) (1978), 299–316.
B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809.
P. Levy, Commuting varieties of Lie algebras over fields of prime characteristic, J. Algebra 250(2) (2002), 473–484.
P. Levy, Involutions of reductive Lie algebras in positive characteristic, Adv. Math. 210(2) (2007), 505–559.
B. Martin, A. Neeman, The map V → V//G need not be separable, Math. Res. Lett. 8(5–6) (2001), 813–817.
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.
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.
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.
D. I. Panyushev, On invariant theory of θ-groups, J. Algebra 283(2) (2005), 655–670.
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.
V. L. Popov, Sections in invariant theory, in: The Sophus Lie Memorial Conference (Oslo, 1992), Scandinavian University Press, Oslo, 1994, pp. 315–361.
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.
A. Premet, Complexity of Lie algebra representations and nilpotent elements of the stabilizers of linear forms, Math. Z. 228(2) (1998), 255–282.
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.
A. Premet, R. Tange, Zassenhaus varieties of general linear Lie algebras, J. Algebra 294(1) (2005), 177–195.
R. W. Richardson, Orbits, invariants, and representations associated to involutions of reductive groups, Invent. Math. 66(2) (1982), 287–312.
G. Rousseau, Immeubles sphériques et théorie des invariants, C. R. Acad. Sci. Paris. Sér. A–B 286(5) (1978), A247–A250.
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.
E. Sommers, A generalization of the Bala–Carter theorem for nilpotent orbits, Int. Math. Res. Not. (11) (1998), 539–562.
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.
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.
R. Steinberg, Endomorphisms of Linear algebraic groups, Mem. Amer. Math. Soc. 80, 1968.
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.
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.
T. Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France 102 (1974), 317–333.
D. J. Winter, Fixed points and stable subgroups of algebraic group automorphisms, Proc. Amer. Math. Soc. 18 (1967), 1107–1113.
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Levy, P. Vinberg’s θ-groups in positive characteristic and Kostant–Weierstrass slices. Transformation Groups 14, 417–461 (2009). https://doi.org/10.1007/s00031-009-9056-y
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DOI: https://doi.org/10.1007/s00031-009-9056-y