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

TL;DR

This paper determines the count of multiplicity-free primitive ideals linked to rigid nilpotent orbits in simple Lie algebras, focusing on the complex and modular cases, with explicit results for E8.

Contribution

It provides a detailed enumeration of such ideals for all rigid nilpotent orbits, including the challenging E8 case, and computes small modules in reduced enveloping algebras.

Findings

Number of multiplicity-free primitive ideals for rigid nilpotent orbits in simple Lie algebras.

Explicit count for the two largest rigid nilpotent orbits in E8.

Calculation of small modules in reduced enveloping algebras over fields of characteristic p>5.

Abstract

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 ${\rm E}_8$. As a corollary we compute the number of small modules in the corresponding reduced enveloping algebras over algebraically closed fields of characteristic $p>5$.