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 LosevPremet 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. This is deduced from the corresponding statement about one-dimensional representations of finite W-algebras.