Nilpotent orbits in good characteristic and the Kempf–Rousseau theory

AbstractLet G 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