AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Centres of centralizers of unipotent elements in simple algebraic groups
About this Title
R. Lawther and D.M. Testerman
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 210, Number 988
ISBNs: 978-0-8218-4769-5 (print); 978-1-4704-0605-9 (online)
DOI: https://doi.org/10.1090/S0065-9266-10-00594-6
Published electronically: July 21, 2010
MSC: Primary 20G15, 20G41
Abstract
Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ whose characteristic is either $0$ or a good prime for $G$, and let $u\in G$ be unipotent. We study the centralizer $C_G(u)$, especially its centre $Z(C_G(u))$. We calculate the Lie algebra of $Z(C_G(u))$, in particular determining its dimension; we prove a succession of theorems of increasing generality, the last of which provides a formula for $\dim Z(C_G(u))$ in terms of the labelled diagram associated to the conjugacy class containing $u$.
We proceed by using the existence of a Springer map to replace $u$ by a nilpotent element lying in the Lie algebra of $G$. The bulk of the work concerns the cases where $G$ is of exceptional type. For these we produce a set of nilpotent orbit representatives $e$ and perform explicit calculations. For each such $e$ we obtain not only the Lie algebra of $Z(C_G(e))$, but in fact the whole upper central series of the Lie algebra of $R_u(C_G(e))$; we write each term of this series explicitly as a direct sum of indecomposable tilting modules for a reductive complement to $R_u(C_G(e))$ in $C_G(e)^\circ$.
- P. Bala and R. W. Carter, Classes of unipotent elements in simple algebraic groups. I, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 3, 401–425. MR 417306, DOI 10.1017/S0305004100052403
- P. Bala and R. W. Carter, Classes of unipotent elements in simple algebraic groups. II, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 1, 1–17. MR 417307, DOI 10.1017/S0305004100052610
- 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
- Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
- Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
- Bomshik Chang, The conjugate classes of Chevalley groups of type $(G_{2})$, J. Algebra 9 (1968), 190–211. MR 227258, DOI 10.1016/0021-8693(68)90020-3
- C. Chevalley, Sur certains groupes simples, Tohoku Math. J. (2) 7 (1955), 14–66 (French). MR 73602, DOI 10.2748/tmj/1178245104
- Charles W. Curtis, Representations of Lie algebras of classical type with applications to linear groups, J. Math. Mech. 9 (1960), 307–326. MR 0110766, DOI 10.1512/iumj.1960.9.59018
- E. B. Dynkin, Selected papers of E. B. Dynkin with commentary, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2000. Edited by A. A. Yushkevich, G. M. Seitz and A. L. Onishchik. MR 1757976
- Peter B. Gilkey and Gary M. Seitz, Some representations of exceptional Lie algebras, Geom. Dedicata 25 (1988), no. 1-3, 407–416. Geometries and groups (Noordwijkerhout, 1986). MR 925845, DOI 10.1007/BF00191935
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
- James E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1343976, DOI 10.1090/surv/043
- 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
- Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032. MR 114875, DOI 10.2307/2372999
- John F. Kurtzke Jr., Centralizers of irregular elements in reductive algebraic groups, Pacific J. Math. 104 (1983), no. 1, 133–154. MR 683733
- R. Lawther, Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra 23 (1995), no. 11, 4125–4156. MR 1351124, DOI 10.1080/00927879508825454
- 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
- Martin W. Liebeck and Gary M. Seitz, The maximal subgroups of positive dimension in exceptional algebraic groups, Mem. Amer. Math. Soc. 169 (2004), no. 802, vi+227. MR 2044850, DOI 10.1090/memo/0802
- M.W. Liebeck and G.M. Seitz, Unipotent and nilpotent classes in algebraic groups and Lie algebras, preprint.
- Frank Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135–169. MR 1901354, DOI 10.1112/S1461157000000838
- George J. McNinch, Sub-principal homomorphisms in positive characteristic, Math. Z. 244 (2003), no. 2, 433–455. MR 1992546, DOI 10.1007/s00209-003-0508-0
- 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 and Eric Sommers, Component groups of unipotent centralizers in good characteristic, J. Algebra 260 (2003), no. 1, 323–337. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1976698, DOI 10.1016/S0021-8693(02)00661-0
- 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
- Kenzo Mizuno, The conjugate classes of Chevalley groups of type $E_{6}$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 3, 525–563. MR 486170
- Kenzo Mizuno, The conjugate classes of unipotent elements of the Chevalley groups $E_{7}$ and $E_{8}$, Tokyo J. Math. 3 (1980), no. 2, 391–461. MR 605099, DOI 10.3836/tjm/1270473003
- Klaus Pommerening, Über die unipotenten Klassen reduktiver Gruppen, J. Algebra 49 (1977), no. 2, 525–536 (German). MR 480767, DOI 10.1016/0021-8693(77)90256-3
- Klaus Pommerening, Über die unipotenten Klassen reduktiver Gruppen. II, J. Algebra 65 (1980), no. 2, 373–398 (German). MR 585729, DOI 10.1016/0021-8693(80)90226-4
- Alexander Premet, Special transverse slices and their enveloping algebras, Adv. Math. 170 (2002), no. 1, 1–55. With an appendix by Serge Skryabin. MR 1929302, DOI 10.1006/aima.2001.2063
- Alexander Premet, Nilpotent orbits in good characteristic and the Kempf-Rousseau theory, J. Algebra 260 (2003), no. 1, 338–366. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1976699, DOI 10.1016/S0021-8693(02)00662-2
- Alexander Premet, Enveloping algebras of Slodowy slices and the Joseph ideal, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 3, 487–543. MR 2314105, DOI 10.4171/JEMS/86
- Richard Proud, Witt groups and unipotent elements in algebraic groups, Proc. London Math. Soc. (3) 82 (2001), no. 3, 647–675. MR 1816692, DOI 10.1112/plms/82.3.647
- R. Proud, “On centralizers of unipotent elements in algebraic groups”, unpublished manuscript.
- R. W. Richardson Jr., Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London Math. Soc. 6 (1974), 21–24. MR 330311, DOI 10.1112/blms/6.1.21
- Gary M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365, iv+286. MR 888704, DOI 10.1090/memo/0365
- Gary M. Seitz, Unipotent elements, tilting modules, and saturation, Invent. Math. 141 (2000), no. 3, 467–502. MR 1779618, DOI 10.1007/s002220000073
- Gary M. Seitz, Unipotent centralizers in algebraic groups, J. Algebra 279 (2004), no. 1, 226–259. MR 2078397, DOI 10.1016/j.jalgebra.2003.11.003
- Ken-ichi Shinoda, The conjugacy classes of Chevalley groups of type $(F_{4})$ over finite fields of characteristic $2$, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 21 (1974), 133–159. MR 0349863
- Toshiaki Shoji, The conjugacy classes of Chevalley groups of type $(F_{4})$ over finite fields of characteristic $p\not =2$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21 (1974), 1–17. MR 357641
- Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445
- Eric Sommers, A generalization of the Bala-Carter theorem for nilpotent orbits, Internat. Math. Res. Notices 11 (1998), 539–562. MR 1631769, DOI 10.1155/S107379289800035X
- T. A. Springer, A note on centralizers in semi-simple groups, Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math. 28 (1966), 75–77. MR 0194423
- T. A. Springer, The unipotent variety of a semi-simple group, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Oxford Univ. Press, London, 1969, pp. 373–391. MR 0263830
- 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
- Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
- U. Stuhler, Unipotente und nilpotente Klassen in einfachen Gruppen und Liealgebren vom Typ $G_{2}$, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math. 33 (1971), 365–378 (German). MR 0302723
- Donna M. Testerman, A construction of certain maximal subgroups of the algebraic groups $E_6$ and $F_4$, J. Algebra 122 (1989), no. 2, 299–322. MR 999075, DOI 10.1016/0021-8693(89)90218-4
- Donna M. Testerman, $A_1$-type overgroups of elements of order $p$ in semisimple algebraic groups and the associated finite groups, J. Algebra 177 (1995), no. 1, 34–76. MR 1356359, DOI 10.1006/jabr.1995.1285
- Oksana Yakimova, Surprising properties of centralisers in classical Lie algebras, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 903–935 (English, with English and French summaries). MR 2543656