2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141342We extend the results of Cellini-Papi on the characterizations of nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple Lie algebra. These characterizations are given in terms of elements of the affine Weyl group and faces of alcoves. In the case of a parabolic subalgebra of a classical Lie algebra, we give formulas for the number of these ideals.Proof of 4.1 corrected and generalized to all types. Further references and remarks addedRepresentation Theory17B20text