## 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

### Citations:

Zbl 0446.17006; Zbl 0513.22009
Full Text:

### References:

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