The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits

In this paper we describe the number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits in finite-dimensional simple Lie algebras. Thanks to the results obtained earlier we need to solve the problem for the two largest rigid nilpotent orbits in Lie algebras of type ${\mathrm{E}}_8$. As a corollary we compute the number of small modules in the corresponding reduced enveloping algebras over algebraically closed fields of characteristic $p>5$.