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 \mathfrak {g} over C \mathbb {C} are those ideals constructed from one-dimensional representations of finite W W -algebras. We refer to these ideals as Losev–Premet ideals . For g \mathfrak {g} simple of classical type, we prove that for a Losev–Premet ideal I I in U ( g ) U(\mathfrak {g}) , there exists a Losev–Premet ideal I 0 I_0 for a certain Levi subalgebra g 0 \mathfrak {g}_0 of g \mathfrak {g} such that the associated variety of I 0 I_0 is the closure of a rigid nilpotent orbit in g 0 \mathfrak {g}_0 and I I is obtained from I 0 I_0 by parabolic induction; in turn, this gives a classification of rigid Losev–Premet ideals in U ( g ) U(\mathfrak {g}) . This is deduced from the corresponding statement about one-dimensional representations of finite W