Nilpotent orbits in good characteristic and the Kempf–Rousseau theory

摘要: Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p>0, Image, and suppose that p is good prime for the root system G. In this paper, we give fairly short conceptual proof Pommerening's theorem [Pommerening, J. Algebra 49 (1977) 525–536; 65 (1980) 373–398] which states any nilpotent element in Image Richardson distinguished parabolic subalgebra Lie algebra Levi subgroup As by-product, obtain noncomputational existence transverse slices to G-orbits (for earlier proofs see [Kawanaka, Invent. Math. 84 (1986) 575–616; Premet, Trans. Amer. Soc. 347 (1995) 2961–2988; Spaltenstein, Fac. Sci. Univ. Tokyo Sect. IA 31 (1984) 283–286]). We extend recent results Sommers [Internal. Res. Notices 11 (1998) 539–562] algebras thus providing satisfactory approach computing component groups centralisers elements unipotent Earlier computations these positive characteristics relied, mostly, on work Mizuno [J. 24 525–563; 3 391–459]. Our based theory optimal subgroups G-unstable vectors, also known as Kempf–Rousseau theory, provides substitute Image-theory prominent zero case.