A simple proof of Pommerening's theorem

Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p⩾0. Assume that p is good for G. Pommerening's theorem [K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, J. Algebra 49 (1977) 525–536; K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, II, J. Algebra 65 (1980) 373–398] asserts that any distinguished nilpotent element in the Lie algebra g of G is a Richardson element for a distinguished parabolic subgroup of G. This theorem implies the Bala–Carter theorem in good characteristic. In this paper we give a short proof of Pommerening's theorem, which is a further simplification of Premet's first uniform proof [A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory, J. Algebra 260 (2003) 338–366]. We also simplify Premet's proof of the existence theorem for good transverse slices to the nilpotent Ad(G)-orbits in g.