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On the variety of a highest weight module. (English) Zbl 0539.17006

Let \({\mathfrak g}\) be a complex semisimple Lie algebra. Let I be a primitive ideal in \(U({\mathfrak g})\). One of the goals of primitive ideal theory for \(U({\mathfrak g})\) is to give some algebro-geometric invariants of I which determine it completely. (This goal is made precise in the Jantzen conjecture, as formulated in the author’s paper in Invent. Math. 56, 191- 213 (1980; Zbl 0446.17006).) One candidate for one of these invariants is the associated variety \({\mathcal V}(I)\), which is a Zariski closed cone contained in the nilpotent cone in \({\mathfrak g}^*\). Work of D. Barbasch and the reviewer [J. Algebra 80, 350-382 (1983; Zbl 0513.22009)] replaced this invariant by an analytically defined one, and proved some of the desired classification results for it.
The goal of this paper is to find a replacement for those analytic ideas, adequate for studying \({\mathcal V}(I)\) itself. Here is the basic idea. Let \({\mathfrak b}\) be a Borel subalgebra of \({\mathfrak g}\). Write \({\mathfrak n}^-\) for the space of linear functionals on \({\mathfrak g}\) vanishing on \({\mathfrak b}\). The closure of the intersection of any nilpotent orbit \({\mathcal O}\) of G with \({\mathfrak n}^-\) has finitely many components \({\mathcal V}_ 1,...,{\mathcal V}_ t\). To each such component, the author attaches a homogeneous polynomial \(p_ i\) in the symmetric algebra of a Cartan subalgebra \({\mathfrak h}\) of \({\mathfrak g}\). This is done by studying the action of \({\mathfrak h}\) on functions on \({\mathcal V}_ i\). He proves that the span of the various \(p_ i\) is stable under the action of the Weyl group and conjectures that the resulting representation is (up to a factor of the sign representation) the one attached to \({\mathcal O}\) by Springer. This fact was subsequently proved by R. Hotta [Tôhoku Math. J., II. Ser. 36, 49-74 (1984)].
On the other hand, suppose M is an irreducible \({\mathfrak b}\)-highest weight module with annihilator I. Then the associated variety \({\mathcal V}(M)\) is a union of several \({\mathcal V}_ i\). (To be precise, we should at this point allow several different \({\mathcal O}'s\) as well; but it turns out that only one is needed.) The author proves a relationship between his polynomials \(p_ i\), and the Goldie rank polynomial for I. (Here again I have been a little imprecise.) All of this sets the stage for the subsequent proof of the Jantzen conjecture, using Hotta’s resuls already mentioned, an equidimensionality result of Gabber, and the Borho-Brylinski theorem that \({\mathcal V}(Ann M)=G\cdot {\mathcal V}(M)\) for highest weight modules.
Reviewer: D.Vogan

MSC:

17B35 Universal enveloping (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E46 Semisimple Lie groups and their representations
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