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Zbl 0717.17015
Benoist, Yves
Modules simples sur une algèbre de Lie nilpotente contenant un vecteur propre pour une sous-algèbre. (Simple modules of a nilpotent Lie algebra having an eigenvector for a subalgebra). (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 23, No. 3, 495-517 (1990). ISSN 0012-9593

Let ${\frak g}$ be a finite dimensional nilpotent Lie algebra over ${\bbfC}$, let U(${\frak g})$ be its enveloping algebra, and let I be a primitive ideal in U(${\frak g})$. The coadjoint orbit $\Omega\subset {\frak g}\sp*$ corresponding to I is a symplectic variety. Let $f\in {\frak g}\sp*$, and let ${\frak k}$ be a subalgebra of ${\frak g}$ such that f([${\frak k},{\frak k}])=0$, and define ${\frak k}\sp f=\{X-f(X) \vert$ $X\in {\frak k}\}$. For each irreducible component $\Lambda$ of $Z=\Omega \cap (f+{\frak k}\sp{\perp})$ which is Lagrangian, the author constructs a simple ${\frak g}$-module $M\sb{\Lambda}$ such that $Ann(M\sb{\Lambda})=I$ and $\{m\in M\sb{\Lambda} \vert$ ${\frak k}\sp fm=0\}\ne 0$. The construction of $M\sb{\Lambda}$ is one of the key points of the paper; it is defined in terms of ${\cal D}$-modules and involves a choice of polarisation which is shown not to affect the construction. It is proved that the length of $M=U({\frak g})/I+U({\frak g}){\frak k}\sp f$ is finite if and only if Z itself is Lagrangian. In this case M is semisimple, with its simple components being the $M\sb{\Lambda}$ where $\Lambda$ is as above: there is an open question as to the geometric meaning of the multiplicity of $M\sb{\Lambda}$ in M being precisely 1. \par The paper is an elegant blend of ancient and modern: it shows how the ${\cal D}$-module point of view sheds new light on the venerable topic of enveloping algebras of nilpotent Lie algebras (and leads to some new and interesting questions).
[S.P.Smith]

MSC 2000:: *17B35 Universal enveloping algebras (Lie algebras)
32C38 Sheaves of differential operators (analytic spaces)
16S30 Universal enveloping algebras of Lie algebras (associative)
17B30 Solvable, nilpotent Lie algebras

Keywords: simple modules; nilpotent Lie algebra; enveloping algebra; primitive ideal; ${\cal D}$-modules

Cited in: Zbl 0959.18004

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