A criterion for an ideal to be induced
Let U(\({\mathfrak g})\) denote the enveloping algebra of a complex semisimple Lie algebra. In the study of the primitive spectrum of U(\({\mathfrak g})\), an important question is to determine those primitive ideals which are completely prime. One such family is provided by the so-called induced ideals which are the annihilators of modules induced from a one- dimensional module of a parabolic subalgebra \({\mathfrak p}.\) Much is known about induced ideals [see for example \textit{N. Conze- Berline} and \textit{M. Duflo}, Compos. Math. 34, 307-336 (1977; Zbl 0389.22016)]; \textit{M. Duflo}, Ann. Math., II. Ser. 105, 107-120 (1977; Zbl 0346.17011); \textit{O. Gabber} and \textit{A. Joseph}, Compos. Math. 43, 107-131 (1981; Zbl 0461.17004)]. Let \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak g}\) and W the Weyl group for the pair (\({\mathfrak g},{\mathfrak h})\). Let Z(\({\mathfrak g})\) denote the centre of U(\({\mathfrak g})\). It is known that if \(\lambda\in {\mathfrak h}^*\) then, under the Harish-Chandra isomorphism, there corresponds a maximal ideal Z(\(\lambda)\) of Z(\({\mathfrak g})\) and the ideal \(I(\lambda)=U({\mathfrak g})Z(\lambda)\) is primitive and is an induced ideal with respect to a Borel subalgebra \({\mathfrak b}\); moreover every primitive ideal contains some I(\(\lambda)\) [see \textit{J. Dixmier}, Algèbres enveloppantes (Gauthier-Villars, Paris 1974; Zbl 0308.17007) {\S}{\S} 7.4, 8.4]. R. K. Gupta has observed that except for ``special'' \(\lambda\in {\mathfrak h}^*\), any ideal induced with respect to \({\mathfrak p}\supsetneqq {\mathfrak b}\) is generated over U(\({\mathfrak g})\) by I(\(\lambda)\) and one copy of the adjoint representation, and the aim of this present paper is to prove a converse of Gupta's theorem and to strengthen it. It is shown that in almost all cases (i.e. more generally than in Gupta's theorem) an ideal generated over U(\({\mathfrak g})\) by I(\(\lambda)\) and one copy of the adjoint representation is an induced ideal with respect to some \({\mathfrak p}\supsetneqq {\mathfrak b}\), and every induced ideal is so obtained. Moreover, a description is given of what happens in the exceptional cases and how these arise.
scientific article; zbMATH DE number 4027709
scientific article