Complete reducibility and separability
Authors:
Michael Bate, Benjamin Martin, Gerhard Röhrle and Rudolf Tange
Journal:
Trans. Amer. Math. Soc. 362 (2010), 4283-4311
MSC (2000):
Primary 20G15, 14L24
DOI:
https://doi.org/10.1090/S0002-9947-10-04901-9
Published electronically:
March 4, 2010
MathSciNet review:
2608407
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Let $G$ be a reductive linear algebraic group over an algebraically closed field of characteristic $p > 0$. A subgroup of $G$ is said to be separable in $G$ if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre’s concept of $G$-complete reducibility for subgroups of $G$. A separability hypothesis appears in many general theorems concerning $G$-complete reducibility. We demonstrate that some of these results fail without this hypothesis. On the other hand, we prove that if $G$ is a connected reductive group and $p$ is very good for $G$, then any subgroup of $G$ is separable; we deduce that under these hypotheses on $G$, a subgroup $H$ of $G$ is $G$-completely reducible provided Lie $G$ is semisimple as an $H$-module.
Recently, Guralnick has proved that if $H$ is a reductive subgroup of $G$ and $C$ is a conjugacy class of $G$, then $C\cap H$ is a finite union of $H$-conjugacy classes. For generic $p$ — when certain extra hypotheses hold, including separability — this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig’s deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick’s result is false if one considers conjugacy classes of $n$-tuples of elements from $H$ for $n > 1$.
- Peter Bardsley and R. W. Richardson, Étale slices for algebraic transformation groups in characteristic $p$, Proc. London Math. Soc. (3) 51 (1985), no. 2, 295–317. MR 794118, DOI https://doi.org/10.1112/plms/s3-51.2.295
- Michael Bate, Optimal subgroups and applications to nilpotent elements, Transform. Groups 14 (2009), no. 1, 29–40. MR 2480851, DOI https://doi.org/10.1007/s00031-008-9038-5
- Michael Bate, Benjamin Martin, and Gerhard Röhrle, A geometric approach to complete reducibility, Invent. Math. 161 (2005), no. 1, 177–218. MR 2178661, DOI https://doi.org/10.1007/s00222-004-0425-9
- Michael Bate, Benjamin Martin, and Gerhard Röhrle, Complete reducibility and commuting subgroups, J. Reine Angew. Math. 621 (2008), 213–235. MR 2431255, DOI https://doi.org/10.1515/CRELLE.2008.063
- Janez Bernik, Robert Guralnick, and Mitja Mastnak, Reduction theorems for groups of matrices, Linear Algebra Appl. 383 (2004), 119–126. MR 2073898, DOI https://doi.org/10.1016/j.laa.2003.11.020
- Armand Borel, Properties and linear representations of Chevalley groups, 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. 1–55. MR 0258838
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
- A. Borel and J. De Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200–221 (French). MR 32659, DOI https://doi.org/10.1007/BF02565599
- N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975 (French). MR 0453824
- Robert M. Guralnick, Intersections of conjugacy classes and subgroups of algebraic groups, Proc. Amer. Math. Soc. 135 (2007), no. 3, 689–693. MR 2262864, DOI https://doi.org/10.1090/S0002-9939-06-08544-3
- Gerhard P. Hochschild, Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics, vol. 75, Springer-Verlag, New York-Berlin, 1981. MR 620024
- 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
- James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR 0323842
- James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773
- Jens Carsten Jantzen, Representations of Lie algebras in prime characteristic, Representation theories and algebraic geometry (Montreal, PQ, 1997) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 185–235. Notes by Iain Gordon. MR 1649627
- 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 https://doi.org/10.1090/memo/0674
- Martin W. Liebeck, Benjamin M. S. Martin, and Aner Shalev, On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function, Duke Math. J. 128 (2005), no. 3, 541–557. MR 2145743, DOI https://doi.org/10.1215/S0012-7094-04-12834-9
- 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 https://doi.org/10.1090/memo/0580
- Martin W. Liebeck and Gary M. Seitz, Variations on a theme of Steinberg, J. Algebra 260 (2003), no. 1, 261–297. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1973585, DOI https://doi.org/10.1016/S0021-8693%2802%2900649-X
- G. Lusztig, On the finiteness of the number of unipotent classes, Invent. Math. 34 (1976), no. 3, 201–213. MR 419635, DOI https://doi.org/10.1007/BF01403067
- Benjamin M. S. Martin, Étale slices for representation varieties in characteristic $p$, Indag. Math. (N.S.) 10 (1999), no. 4, 555–564. MR 1820553, DOI https://doi.org/10.1016/S0019-3577%2800%2987909-4
- Benjamin M. S. Martin, Reductive subgroups of reductive groups in nonzero characteristic, J. Algebra 262 (2003), no. 2, 265–286. MR 1971039, DOI https://doi.org/10.1016/S0021-8693%2803%2900189-3
- Benjamin M. S. Martin, A normal subgroup of a strongly reductive subgroup is strongly reductive, J. Algebra 265 (2003), no. 2, 669–674. MR 1987023, DOI https://doi.org/10.1016/S0021-8693%2803%2900293-X
- 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 https://doi.org/10.1090/S0002-9947-07-04289-4
- R. W. Richardson Jr., Conjugacy classes in Lie algebras and algebraic groups, Ann. of Math. (2) 86 (1967), 1–15. MR 217079, DOI https://doi.org/10.2307/1970359
- R. W. Richardson, On orbits of algebraic groups and Lie groups, Bull. Austral. Math. Soc. 25 (1982), no. 1, 1–28. MR 651417, DOI https://doi.org/10.1017/S0004972700005013
- R. W. Richardson, Conjugacy classes of $n$-tuples in Lie algebras and algebraic groups, Duke Math. J. 57 (1988), no. 1, 1–35. MR 952224, DOI https://doi.org/10.1215/S0012-7094-88-05701-8
- Joseph J. Rotman, An introduction to the theory of groups, 4th ed., Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, 1995. MR 1307623
- Jean-Pierre Serre, Semisimplicity and tensor products of group representations: converse theorems, J. Algebra 194 (1997), no. 2, 496–520. With an appendix by Walter Feit. MR 1467165, DOI https://doi.org/10.1006/jabr.1996.6929
- ---, La notion de complète réductibilité dans les immeubles sphériques et les groupes réductifs, Séminaire au Collège de France, résumé dans \cite[pp. 93–98]tits2, (1997).
- ---, The notion of complete reducibility in group theory, Moursund Lectures, Part II, University of Oregon, 1998, arXiv:math/0305257v1
- Jean-Pierre Serre, Complète réductibilité, Astérisque 299 (2005), Exp. No. 932, viii, 195–217 (French, with French summary). Séminaire Bourbaki. Vol. 2003/2004. MR 2167207
- Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445
- Peter Slodowy, Two notes on a finiteness problem in the representation theory of finite groups, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 331–348. With an appendix by G.-Martin Cram. MR 1635690, DOI https://doi.org/10.1017/S1446788700007254
- T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713
- 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
- Robert Steinberg, Automorphisms of classical Lie algebras, Pacific J. Math. 11 (1961), 1119–1129. MR 143845
- Jacques Tits, Théorie des groupes, Ann. Collège France 87 (1986/87), 89–98 (French). MR 1320558
- E. B. Vinberg, On invariants of a set of matrices, J. Lie Theory 6 (1996), no. 2, 249–269. MR 1424635
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20G15, 14L24
Retrieve articles in all journals with MSC (2000): 20G15, 14L24
Additional Information
Michael Bate
Affiliation:
Christ Church College, Oxford University, Oxford, OX1 1DP, United Kingdom
Address at time of publication:
Department of Mathematics, University of York, York, YO10 5DD, United Kingdom
Email:
bate@maths.ox.ac.uk, meb505@york.ac.uk
Benjamin Martin
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
MR Author ID:
659870
Email:
B.Martin@math.canterbury.ac.nz
Gerhard Röhrle
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
MR Author ID:
329365
Email:
gerhard.roehrle@rub.de
Rudolf Tange
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Address at time of publication:
Department of Mathematics, University of York, York, YO10 5DD, United Kingdom
Email:
rudolf.tange@rub.de, rht502@york.ac.uk
Keywords:
$G$-complete reducibility,
separability,
reductive pairs
Received by editor(s):
March 24, 2008
Received by editor(s) in revised form:
August 12, 2008
Published electronically:
March 4, 2010
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.