# Nilpotent orbits in good characteristic and the Kempf-Rousseau theory

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doi:10.1016/S0021-8693(02)00662-2 Journal of Algebra 260 (2003) 338–366 www.elsevier.com/locate/jalgebra Nilpotent orbits in good characteristic and the Kempf–Rousseau theory Alexander Premet Department of Mathematics, University of Manchester, Oxford Road, M13 9PL, UK Received 18 April 2002 Communicated by Robert Guralnick and Gerhard Röhrle Dedicated to Robert Steinberg on his 80th birthday Abstract LetG be a connected reductive algebraic group over an algebraically closed field k of characteristic p > 0, g= LieG, and suppose that p is a good prime for the root system of G. In this paper, we give a fairly short conceptual proof of Pommerening’s theorem [Pommerening, J. Algebra 49 (1977) 525– 536; J. Algebra 65 (1980) 373–398] which states that any nilpotent element in g is Richardson in a distinguished parabolic subalgebra of the Lie algebra of a Levi subgroup of G. As a by-product, we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent G-orbits in g (for earlier proofs of this theorem see [Kawanaka, Invent. Math. 84 (1986) 575–616; Premet, Trans. Amer. Math. Soc. 347 (1995) 2961–2988; Spaltenstein, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984) 283–286]). We extend recent results of Sommers [Internal. Math. Res. Notices 11 (1998) 539–562] to reductive Lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in g and unipotent elements in G. Earlier computations of these groups in positive characteristics relied, mostly, on work of Mizuno [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977) 525–563; Tokyo J. Math. 3 (1980) 391–459]. Our approach is based on the theory of optimal parabolic subgroups for G-unstable vectors, also known as the Kempf–Rousseau theory, which provides a good substitute for the sl(2)-theory prominent in the characteristic zero case. 2003 Elsevier Science (USA). All rights reserved. E-mail addresses: sas

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