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Equivalence Classes of Ideals in the Nilradical of a Borel Subalgebra

Published online by Cambridge University Press:  11 January 2016

Eric N. Sommers
Eric N. Sommers*
Affiliation:
University of Massachusetts—AmherstAmherst, MA [email protected]
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Abstract

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An equivalence relation is defined and studied on the set of B-stable ideals in the nilradical of the Lie algebra of a Borel subgroup B. Techniques are developed to compute the equivalence relation and these are carried out in the exceptional groups. There is a natural partial order on equivalence classes coming from inclusion of one ideal in another. A main theorem is that this partial order is a refinement of the closure ordering on nilpotent orbits.

Keywords

17B2020F55
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 183 , 2006 , pp. 161 - 185
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

References

[1] Achar, P., An order reversing duality for conjugacy classes in Lusztig’s canonical quotient, Tranformation Groups, 8 (2003), no. 2, 107–145.Google Scholar
[2] Achar, P. and Sommers, E., Local systems and functions on orbit covers, Representation Theory, 6 (2002), 190–201.Google Scholar
[3] Carter, R., Finite groups of Lie type. Conjugacy classes and complex characters, Wiley Classics Library, John Wiley & Sons, Ltd, Chicester, 1993.Google Scholar
[4] Collingwood, D. H. and McGovern, W. M., Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993.Google Scholar
[5] Concini, C. De, Lusztig, G., and Procesi, C., Zero set of a nilpotent vector field, Journal of the A.M.S., 1 (1988), no. 1, 15–34.Google Scholar
[6] Demazure, M., A very simple proof of Bott’s Theorem, Invent. Math., 33 (1976), 271–272.Google Scholar
[7] Jantzen, J. C., Nilpotent orbits in representation theory, Lie Theory (Boston), Progr. Math., vol. 228, Birkhäauser, Boston (2004), pp. 1–211.Google Scholar
[8] Kawanaka, N., Generalized Gelfand-Graev representations of exceptional simple groups over a finite field I, 84 (1986), Invent. Math., 575–616.CrossRefGoogle Scholar
[9] Kawanaka, N., Orbits and stabilizers of nilpotent elements of a graded semisimple Lie alge bra, J. Fac. Sci. Univ. Tokyo, 34 (1987), 573–597.Google Scholar
[10] Liang, K. and Lu, L., Sheets and rigid covers in the exceptional groups, Chinese Sci. Bull., 43 (1998), no. 20, 1702-1706.CrossRefGoogle Scholar
[11] Lusztig, G., Notes on unipotent classes, Asian J. Math., 1 (1997), 194–207.Google Scholar
[12] Lusztig, G. and Spaltenstein, N., Induced unipotent classes, J. London Math. Soc., 19 (1979), no. 1, 41–52.CrossRefGoogle Scholar
[13] Mizuno, K., The conjugate classes of unipotent elements of the Chevalley groups E7 and E8 , Tokyo J. Math., 3 (1980), no. 2, 391–461.Google Scholar
[14] Panyushev, D. and Röohrle, G., On spherical ideals of Borel subalgebras, Arch. Math. (Basel), 84 (2005), no. 3, 225–232.Google Scholar
[15] Shi, J.-Y., The Kazhdan-Lusztig cells in certain affine Weyl groups, Lecture Notes in Mathematics, no. 1179, Springer-Verlag, 1986.Google Scholar
[16] Sommers, E., A generalization of the Bala-Carter theorem for nilpotent orbits, Internat. Math. Res. Notices (1998), no. 11, 539–562.Google Scholar
[17] Sommers, E., Lusztig’s canonical quotient and generalized duality, J. of Algebra (2001), no. 243, 790–812.Google Scholar
[18] Sommers, E., Normality of nilpotent varieties in E6 , J. Algebra, 270 (2003), 288–306.Google Scholar
[19] Sommers, E., B-stable ideals in the nilradical of a Borel subalgebra, Canadian Math. Bull., 48 (2005), no. 3, 460–472.CrossRefGoogle Scholar
[20] Spaltenstein, N., Classes Unipotentes et Sous-Groupes de Borel, Lecture Notes in Mathematics, no. 946, Springer-Verlag, 1982.Google Scholar
[21] Springer, T. A., Linear Algebraic Groups, 2nd ed., Progress in Mathematics, Birkhäauser, Boston, 1998.Google Scholar
[22] Steinberg, R., Endomorphisms of linear algebraic groups, AMS, Providence, 1968.CrossRefGoogle Scholar