Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$, $\mathfrak{n}$ be the nilradical of $\mathfrak{b}$, and $U(\mathfrak{n})$ be the universal enveloping algebra of $\mathfrak{n}$. We study primitive ideals of $U(\mathfrak{n})$. Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center $Z(\mathfrak{n})$ of $U(\mathfrak{n})$. We present an explicit characterization of the centrally generated primitive ideals of $U(\mathfrak{n})$ in terms of the Dixmier map and the Kostant cascade in the case when $\mathfrak{g}$ is a simple algebra of exceptional type. (For classical simple Lie algebras, a similar characterization was obtained by Ivan Penkov and the first author.) As a corollary, we establish a classification of centrally generated primitive ideals of $U(\mathfrak{n})$ for an arbitrary semisimple algebra $\mathfrak{g}$.