On induced completely prime primitive ideals in enveloping algebras of classical Lie algebras

Abstract

A distinguished family of completely prime primitive ideals in the universal enveloping algebra of a reductive Lie algebra ${\mathfrak g}$ over ${\mathbb C}$ are those ideals constructed from one-dimensional representations of finite $W$-algebras. We refer to these ideals as Losev--Premet ideals. For ${\mathfrak g}$ simple of classical type, we prove that for a Losev-Premet ideal $I$ in $U({\mathfrak g})$, there exists a Losev-Premet ideal $I_0$ for a certain Levi subalgebra ${\mathfrak g}_0$ of ${\mathfrak g}$ such that associated variety of $I_0$ is the closure of a rigid nilpotent orbit in ${\mathfrak g}_0$ and $I$ is obtained from $I_0$ by parabolic induction; in turn, this gives a classification of rigid Losev-Premet ideals in $U({\mathfrak g})$. This is deduced from the corresponding statement about one-dimensional representations of finite $W$-algebras.