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On Cocharacters Associated to Nilpotent Elements of Reductive Groups

Published online by Cambridge University Press:  11 January 2016

Russell Fowler  and
Gerhard Röhrle
Russell Fowler
Affiliation:
School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK
Gerhard Röhrle
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany, [email protected]
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Abstract

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Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J. C. Jantzen.

Keywords

20G1514L3017B50
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 190 , 2008 , pp. 105 - 128
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

References

[1] Bardsley, P. and Richardson, R. W., Etale slices for algebraic transformation groups in characteristic p, Proc. London Math. Soc., (3), 51 (1985), no. 2, 295–317.Google Scholar
[2] Borel, A., Linear Algebraic Groups, Graduate Texts in Mathematics, 126, Springer-Verlag, 1991.Google Scholar
[3] Borel, A. and Siebenthal, J. de, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helvet., 23 (1949), 200–221.Google Scholar
[4] Bourbaki, N., Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1975.Google Scholar
[5] Carter, R. W., Finite groups of Lie type. Conjugacy classes and complex characters, Pure and Applied Mathematics, New York, 1985.Google Scholar
[6] Humphreys, J. E., Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, 43, American Mathematical Society, Providence, RI, 1995.Google Scholar
[7] Jantzen, J. C., Nilpotent Orbits in Representation Theory, Lie Theory, Lie Algebras and Representations (J.-P. Anker and B. Orsted, eds.), Progress in Math., vol. 228, Birkhäuser Boston, 2004.Google Scholar
[8] Kempf, G. R., Instability in Invariant Theory, Ann. Math., 108 (1978), 299–316.Google Scholar
[9] Lawther, R., Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra, 23 (1995), no. 11, 4125–4156.Google Scholar
[10] McNinch, G., Nilpotent orbits over ground fields of good characteristic, Math. Ann., 329 (2004), no. 1, 49–85.Google Scholar
[11] McNinch, G., Optimal SL(2)-homomorphisms, Comment. Math. Helv., 80 (2005), no. 2, 391–426.CrossRefGoogle Scholar
[12] McNinch, G. and Sommers, E., Component groups of unipotent centralizers in good characteristic, Special issue celebrating the 80th birthday of Robert Steinberg, J. Algebra, 260 (2003), no. 1, 323–337.CrossRefGoogle Scholar
[13] Mostow, G. D., Fully reducible subgroups of algebraic groups, Amer. Math. J., 78 (1956), 200–221.CrossRefGoogle Scholar
[14] Nagata, M., Complete reducibility of rational representations of a matric group, J. Math. Kyoto University, 1 (1961), 87–99.Google Scholar
[15] Premet, A., An analogue of the Jacobson-Morozov Theorem for Lie algebras of reductive groups of good characteristics, Trans. Amer. Math. Soc., 347 (1995), 2961–2988.Google Scholar
[16] Premet, A., Nilpotent orbits in good characteristic and the Kempf-Rousseau theory, Special issue celebrating the 80th birthday of Robert Steinberg, J. Algebra, 260 (2003), no. 1, 338–366.CrossRefGoogle Scholar
[17] Richardson, R. W., On orbits of algebraic groups and Lie groups, Bull. Austral. Math. Soc., 25 (1982), no. 1, 1–28.Google Scholar
[18] Rousseau, G., Immeubles sphériques et théorie des invariants, C. R. Acad. Sci. Paris, 286 (1987), 247–250.Google Scholar
[19] Seitz, G. M., Maximal subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc., 441, 1991.Google Scholar
[20] Seitz, G. M., Unipotent elements, tilting modules, and saturation, Invent. Math., 141 (2000), no. 3, 467–502.Google Scholar
[21] Slodowy, P., Theorie der optimalen Einparameteruntergruppen, Algebraische Transformationsgruppen und Invariantentheorie (H. Kraft, P. Slodowy and T. A. Springer, eds.), DMV Seminar, 13, Birkhäuser Verlag, Basel, 1989.Google Scholar
[22] Springer, T. A., Linear Algebraic Groups, Second edition, Progress in Mathematics, 9, Birkhäuser Boston, Inc., Boston, MA, 1998.Google Scholar
[23] Steinberg, R., Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc., 80, 1968.Google Scholar
[24] Steinberg, R., Notes on Chevalley Groups, Yale University, New Haven (1968).Google Scholar
[25] Springer, T. A. and Steinberg, R., Conjugacy classes, Seminar on algebraic groups and related finite groups, Lecture Notes in Mathematics, 131, Springer-Verlag, Heidelberg, 1970, pp. 167–266.Google Scholar
[26] Testerman, D. M., A construction of certain maximal subgroups of the algebraic groups E6 and F4 , J. Algebra, 122 (1989), no. 2, 299–322.Google Scholar