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Transformation Groups

, Volume 15, Issue 4, pp 937–964 | Cite as

Levi decompositions of a linear algebraic group

  • George J. Mcninch
Article

Abstract

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G.

Let \( \mathcal{A} \) be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth \( \mathcal{A} \) -group schemes \( \mathcal{P} \) whose generic fibers \( {{\mathcal{P}} \left/ {K} \right.} \) coincide with G; these are known as parahoric group schemes. The special fiber \( {{\mathcal{P}} \left/ {K} \right.} \) of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that \( {{\mathcal{P}} \left/ {K} \right.} \) has a Levi factor, and that any two Levi factors of \( {{\mathcal{P}} \left/ {K} \right.} \) are geometrically conjugate.

Keywords

Algebraic Group Parabolic Subgroup Group Scheme Maximal Torus Borel Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Bor 91]
    A. Borel, Linear Algebraic Groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.Google Scholar
  2. [BS68]
    A. Borel, T. A. Springer, Rationality properties of linear algebraic groups, II, Tôhoku Math. J. (2) 20 (1968), 443–497.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [BoTi 65]
    A. Borel, J. Tits, Groupes réductifs, Publ. Math. Inst. Hautes Études Sci. 27 (1965), 55–151. Russian transl.: Ж. Титс, Редукmцвные груnnы, сб Математика 11:1 (167), 43–111 и 11:2 (1967), 3–31.Google Scholar
  4. [Bou 02]
    N. Bourbaki, Lie Groups and Lie algebras, Chaps. 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by A. Pressley.Google Scholar
  5. [Bro 94]
    K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original.Google Scholar
  6. [BrTi 72]
    F. Bruhat, J. Tits, Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [BrTi 84]
    F. Bruhat, J. Tits, Groupes réductifs sur un corps local, II, Schémas en groupes, Existence d'une donnée radicielle valuée, Publ. Math. Inst. Hautes Études Sci. 60 (1984), 197–376.CrossRefMathSciNetGoogle Scholar
  8. [CGP 10]
    B. Conrad, O. Gabber, G. Prasad, Pseudo-Reductive Groups, New Mathematical Monographs, vol. 17, Cambridge University Press, Cambridge, 2010.Google Scholar
  9. [DeGa 70]
    M. Demazure, P. Gabriel, Groupes Algébriques, Tome I: Géométrie Algébrique, Généralités, Groupes Commutatifs, Masson, 1970.Google Scholar
  10. [Hum 67]
    J. E. Humphreys, Existence of Levi factors in certain algebraic groups, Pacific J. Math. 23 (1967), 543–546.zbMATHMathSciNetGoogle Scholar
  11. [Jac 79]
    N. Jacobson, Lie Algebras, Dover, New York, 1979. Republication of the 1962 original. Russian transl.: Н. Джекобсон, Алвебры Лц, Мир, М., 1964.Google Scholar
  12. [Jan 03]
    J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, Amer. Math. Soc., Providence, RI, 2003.Google Scholar
  13. [Jan 97]
    J. C. Jantzen, Low-dimensional representations of reductive groups are semi-simple, in: Algebraic Groups and Lie Groups, Australian Mathematical Society Lecture Series, Vol. 9, Cambridge University Press, Cambridge, 1997, pp. 255–266.Google Scholar
  14. [McN 03]
    G. J. McNinch, Faithful representations of SL2 over truncated Witt vectors, J. Algebra 265 (2003), no. 2, 606–618.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [McN 98]
    G. J. McNinch, Dimensional criteria for semisimplicity of representations, Proc. London Math. Soc. (3) 76 (1998), no. 1, 95–149.CrossRefMathSciNetGoogle Scholar
  16. [Oes 84]
    J. Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique p, Invent. Math. 78 (1984), no. 1, 13–88.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Pra 01]
    G. Prasad, Galois-fixed points in the Bruhat-Tits building of a reductive group, Bull. Soc. Math. France 129 (2001), no. 2, 169–174.zbMATHMathSciNetGoogle Scholar
  18. [PR 84]
    G. Prasad, M. S. Raghunathan, Topological central extensions of semisimple groups over local fields, Ann. of Math. (2) 119 (1984), no. 1, 143–201.CrossRefMathSciNetGoogle Scholar
  19. [Ros 63]
    M. Rosenlicht, Questions of rationality for solvable algebraic groups over non-perfect fields, Ann. Mat. Pura Appl. (4) 61 (1963), 97–120.zbMATHMathSciNetGoogle Scholar
  20. [Rou 77]
    G. Rousseau, Immeubles des groupes réductifs sur les corps locaux, Thèse, Université de Paris-Sud (1977).Google Scholar
  21. [Ser 79]
    J.-P. Serre, Local Fields, Graduate Texts in Mathematics, Vol. 67, Springer-Verlag, New York, 1979. Translated from the French by M. J. Greenberg.Google Scholar
  22. [Ser 97]
    J.-P. Serre, Galois Cohomology, Springer-Verlag, Berlin, 1997. Translated from the French by P. Ion and revised by the author. Russian transl.: Ж.-П. Серр, Ковомоловцц Галуа, Мир, М., 1968.Google Scholar
  23. [Ser 88]
    J.-P. Serre, Algebraic Groups and Class Fields, Graduate Texts in Mathematics, Vol. 117, Springer-Verlag, New York, 1988. Translated from the French. Russian transl.: Ж. Серр, Алгебрацческце груnnы ц nоля классов, Мир, М., 1968.Google Scholar
  24. [Spr 98]
    T. A. Springer, Linear Algebraic Groups, 2nd ed., Progress in Mathematics, Vol. 9, Birkhäuser, Boston, 1998.Google Scholar
  25. [Tit 77]
    J. Tits, Reductive groups over local fields, in: Automorphic Forms, Representations and L-Functions (Proc. Sympos. Pure Math., Oregon State University, Corvallis, OR, 1977), Part 1, Proc. Sympos. Pure Math., Vol. XXXIII, Amer. Math. Soc., Providence, RI, 1979, pp. 29–69.Google Scholar
  26. [Wei 94]
    C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsTufts UniversityMedfordUSA

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