Sandwich elements and the Richardson property

Abstract

A restricted Lie algebra L over an algebraically closed field k of characteristic p>0 is said to possess the Richardson property if there exists a finite dimensional faithful restricted L-module V with associated representation ρ:L→gl(V) such that gl(V)=ρ(L)⊕R where R is a subspace of gl(V) such that [ρ(L),R]⊆R. In [Pre99], the author conjectured that if L has the Richardson property with respect to an irreducible restricted L-module V then there exists a reductive algebraic group G over k such that L≅Lie(G) as restricted Lie algebras. In this note we confirm this conjecture under the assumption that p>3. This assumption is needed since our proof relies in a crucial way on the classification of Lie algebras without strong degeneration obtained in [Pre87a] for p>5 and in [Pre86] for p=5.