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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Borel, Linear Algebraic Groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.
A. Borel, T. A. Springer, Rationality properties of linear algebraic groups, II, Tôhoku Math. J. (2) 20 (1968), 443–497.
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.
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.
K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original.
F. Bruhat, J. Tits, Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251.
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.
B. Conrad, O. Gabber, G. Prasad, Pseudo-Reductive Groups, New Mathematical Monographs, vol. 17, Cambridge University Press, Cambridge, 2010.
M. Demazure, P. Gabriel, Groupes Algébriques, Tome I: Géométrie Algébrique, Généralités, Groupes Commutatifs, Masson, 1970.
J. E. Humphreys, Existence of Levi factors in certain algebraic groups, Pacific J. Math. 23 (1967), 543–546.
N. Jacobson, Lie Algebras, Dover, New York, 1979. Republication of the 1962 original. Russian transl.: Н. Джекобсон, Алвебры Лц, Мир, М., 1964.
J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, Amer. Math. Soc., Providence, RI, 2003.
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.
G. J. McNinch, Faithful representations of SL2 over truncated Witt vectors, J. Algebra 265 (2003), no. 2, 606–618.
G. J. McNinch, Dimensional criteria for semisimplicity of representations, Proc. London Math. Soc. (3) 76 (1998), no. 1, 95–149.
J. Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique p, Invent. Math. 78 (1984), no. 1, 13–88.
G. Prasad, Galois-fixed points in the Bruhat-Tits building of a reductive group, Bull. Soc. Math. France 129 (2001), no. 2, 169–174.
G. Prasad, M. S. Raghunathan, Topological central extensions of semisimple groups over local fields, Ann. of Math. (2) 119 (1984), no. 1, 143–201.
M. Rosenlicht, Questions of rationality for solvable algebraic groups over non-perfect fields, Ann. Mat. Pura Appl. (4) 61 (1963), 97–120.
G. Rousseau, Immeubles des groupes réductifs sur les corps locaux, Thèse, Université de Paris-Sud (1977).
J.-P. Serre, Local Fields, Graduate Texts in Mathematics, Vol. 67, Springer-Verlag, New York, 1979. Translated from the French by M. J. Greenberg.
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.
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.
T. A. Springer, Linear Algebraic Groups, 2nd ed., Progress in Mathematics, Vol. 9, Birkhäuser, Boston, 1998.
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.
C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Vladimir Morozov and to his contributions to mathematics
Research of the author was supported in part by the US NSA award H98230-08-1-0110.
Rights and permissions
About this article
Cite this article
Mcninch, G.J. Levi decompositions of a linear algebraic group. Transformation Groups 15, 937–964 (2010). https://doi.org/10.1007/s00031-010-9111-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-010-9111-8