A simple proof of Pommerening's theorem. (Q950227)

Let \(G\) be a connected reductive algebraic group over an algebraically closed field of good characteristic \(p>0\). Pommerening's theorem asserts that any `distinguished' nilpotent element in the Lie algebra \(\mathfrak g\) of \(G\) is a Richardson element for a `distinguished' parabolic subgroup of \(G\). The aim of this paper is to give a unified proof that further simplifies the uniform proof of \textit{A. Premet} [J. Algebra 260, No. 1, 338-366 (2003; Zbl 1020.20031)]. The proof still uses Kempf-Rousseau theory but it avoids Bala-Carter theory as well as consideration of finite reductive groups. In good characteristic Bala-Carter theory is implied by Pommerening's theorem. The author also simplifies Premet's proof of the existence of good transverse slices to the nilpotent \(\text{Ad}(G)\) orbits in \(\mathfrak g\).