Irreducibility in algebraic groups and regular unipotent elements

Testerman, Donna; Zalesski, Alexandre

doi:10.1090/S0002-9939-2012-11898-2

Irreducibility in algebraic groups and regular unipotent elements
HTML articles powered by AMS MathViewer

by Donna Testerman and Alexandre Zalesski PDF

Proc. Amer. Math. Soc. 141 (2013), 13-28 Request permission

Abstract:

We study (connected) reductive subgroups $G$ of a reductive algebraic group $H$, where $G$ contains a regular unipotent element of $H$. The main result states that $G$ cannot lie in a proper parabolic subgroup of $H$. This result is new even in the classical case $H = \mathrm {SL}(n,F)$, the special linear group over an algebraically closed field, where a regular unipotent element is one whose Jordan normal form consists of a single block. In previous work, Saxl and Seitz (1997) determined the maximal closed positive-dimensional (not necessarily connected) subgroups of simple algebraic groups containing regular unipotent elements. Combining their work with our main result, we classify all reductive subgroups of a simple algebraic group $H$ which contain a regular unipotent element.

References

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
A. Borel and J. Tits, Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math. 12 (1971), 95–104 (French). MR 294349, DOI 10.1007/BF01404653
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
R. Brauer and C. Nesbitt, On the modular characters of groups, Ann. of Math. (2) 42 (1941), 556–590. MR 4042, DOI 10.2307/1968918
Hikoe Enomoto, The characters of the finite Chevalley group $G_{2}(q),q=3^{f}$, Japan. J. Math. (N.S.) 2 (1976), no. 2, 191–248. MR 437628, DOI 10.4099/math1924.2.191
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592, DOI 10.1090/surv/040.1
James E. Humphreys, Modular representations of finite groups of Lie type, London Mathematical Society Lecture Note Series, vol. 326, Cambridge University Press, Cambridge, 2006. MR 2199819
Jens Carsten Jantzen, Low-dimensional representations of reductive groups are semisimple, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 255–266. MR 1635685
Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
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
George J. McNinch, Dimensional criteria for semisimplicity of representations, Proc. London Math. Soc. (3) 76 (1998), no. 1, 95–149. MR 1476899, DOI 10.1112/S0024611598000045
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
Richard Proud, Jan Saxl, and Donna Testerman, Subgroups of type $A_1$ containing a fixed unipotent element in an algebraic group, J. Algebra 231 (2000), no. 1, 53–66. MR 1779592, DOI 10.1006/jabr.2000.8353
J. Saxl, Overgroups of special elements in simple algebraic groups and finite groups of Lie type, Algebraic groups and their representations (Cambridge, 1997) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 291–300. MR 1670776
Jan Saxl and Gary M. Seitz, Subgroups of algebraic groups containing regular unipotent elements, J. London Math. Soc. (2) 55 (1997), no. 2, 370–386. MR 1438641, DOI 10.1112/S0024610797004808
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
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, Maximal subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 90 (1991), no. 441, iv+197. MR 1048074, DOI 10.1090/memo/0441
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, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49–80. MR 180554
Irina D. Suprunenko, Irreducible representations of simple algebraic groups containing matrices with big Jordan blocks, Proc. London Math. Soc. (3) 71 (1995), no. 2, 281–332. MR 1337469, DOI 10.1112/plms/s3-71.2.281
Donna M. Testerman, Irreducible subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 75 (1988), no. 390, iv+190. MR 961210, DOI 10.1090/memo/0390
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