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

A distinguished family of completely prime primitive ideals in the universal enveloping algebra of a reductive Lie algebra 𝔤 over ℂ are those ideals constructed from one-dimensional representations of finite W-algebras. We refer to these ideals as Losev--Premet ideals. For 𝔤 simple of classical type, we prove that for a Losev-Premet ideal I in U(𝔤), there exists a Losev-Premet ideal I₀ for a certain Levi subalgebra 𝔤0 of 𝔤 such that associated variety of I₀ is the closure of a rigid nilpotent orbit in 𝔤0 and I is obtained from I₀ by parabolic induction; in turn, this gives a classification of rigid Losev-Premet ideals in U(𝔤). This is deduced from the corresponding statement about one-dimensional representations of finite W-algebras.