2006.290: Nilpotent orbits in good characteristic and the Kempf-Rousseau theory

2006.290: Alexander Premet (2003) Nilpotent orbits in good characteristic and the Kempf-Rousseau theory. Journal of Algebra, 260 (1). pp. 338-366. ISSN 0021-8669

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DOI: 10.1016/S0021-8693(02)00662-2

Abstract

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p>0, Image, 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 Image 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 Image (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 Image 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 Image-theory prominent in the characteristic zero case.

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