On the smoothness of centralizers in reductive groups
HTML articles powered by AMS MathViewer
- by Sebastian Herpel PDF
- Trans. Amer. Math. Soc. 365 (2013), 3753-3774 Request permission
Abstract:
Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$. In a recent paper, Bate, Martin, Röhrle and Tange show that every (smooth) subgroup of $G$ is separable provided that the characteristic of $k$ is very good for $G$. Here separability of a subgroup means that its scheme-theoretic centralizer in $G$ is smooth. Serre suggested extending this result to arbitrary, possibly non-smooth, subgroup schemes of $G$. The aim of this paper is to prove this more general result. Moreover, we provide a condition on the characteristic of $k$ that is necessary and sufficient for the smoothness of all centralizers in $G$. We finally relate this condition to other standard hypotheses on connected reductive groups.References
- Michael Bate, Benjamin Martin, and Gerhard Röhrle, A geometric approach to complete reducibility, Invent. Math. 161 (2005), no. 1, 177–218. MR 2178661, DOI 10.1007/s00222-004-0425-9
- Michael Bate, Benjamin Martin, Gerhard Röhrle, and Rudolf Tange, Complete reducibility and separability, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4283–4311. MR 2608407, DOI 10.1090/S0002-9947-10-04901-9
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
- M. Demazure and A. Grothendieck. Schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). (Lecture Notes in Math., 151–153). Springer-Verlag, Berlin, 1970.
- Simon M. Goodwin, On generation of the root lattice by roots, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 41–45. MR 2296389, DOI 10.1017/S0305004106009595
- G. M. D. Hogeweij, Almost-classical Lie algebras. I, II, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 4, 441–452, 453–460. MR 683531
- Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
- Jens Carsten Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211. MR 2042689
- Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779, DOI 10.1090/coll/044
- R. Lawther and D. M. Testerman, $A_1$ subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 141 (1999), no. 674, viii+131. MR 1466951, DOI 10.1090/memo/0674
- Martin W. Liebeck and Gary M. Seitz, Reductive subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 121 (1996), no. 580, vi+111. MR 1329942, DOI 10.1090/memo/0580
- Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
- George J. McNinch, Optimal $\textrm {SL}(2)$-homomorphisms, Comment. Math. Helv. 80 (2005), no. 2, 391–426. MR 2142248, DOI 10.4171/CMH/19
- George J. McNinch, The centralizer of a nilpotent section, Nagoya Math. J. 190 (2008), 129–181. MR 2423832, DOI 10.1017/S0027763000009594
- George J. McNinch and Donna M. Testerman, Completely reducible $\rm SL(2)$-homomorphisms, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4489–4510. MR 2309195, DOI 10.1090/S0002-9947-07-04289-4
- George J. McNinch and Donna M. Testerman, Nilpotent centralizers and Springer isomorphisms, J. Pure Appl. Algebra 213 (2009), no. 7, 1346–1363. MR 2497582, DOI 10.1016/j.jpaa.2008.12.007
- R. W. Richardson Jr., Conjugacy classes in Lie algebras and algebraic groups, Ann. of Math. (2) 86 (1967), 1–15. MR 217079, DOI 10.2307/1970359
- R. W. Richardson, Affine coset spaces of reductive algebraic groups, Bull. London Math. Soc. 9 (1977), no. 1, 38–41. MR 437549, DOI 10.1112/blms/9.1.38
- T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
- T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192
- William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117
Additional Information
- Sebastian Herpel
- Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
- Address at time of publication: Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
- Email: sebastian.herpel@rub.de
- Received by editor(s): March 29, 2011
- Received by editor(s) in revised form: November 4, 2011
- Published electronically: December 12, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3753-3774
- MSC (2010): Primary 20G15
- DOI: https://doi.org/10.1090/S0002-9947-2012-05745-X
- MathSciNet review: 3042602