  We study ad-nilpotent ideals of a parabolic subalgebra of a simple Lie
algebra. Any such ideal determines an antichain in a set of positive roots of
the simple Lie algebra. We give a necessary and sufficient condition for an
antichain to determine an ad-nilpotent ideal of the parabolic. We write down
all such antichains for the classical simple Lie algebras and in particular
recover the results of D. Peterson. In section 2 of the paper we study the
unique ideal in a parabolic which is irreducible as a module for the reductive
part and give several equivalent statements that are satisfied by the
corresponding subset of roots.
