## 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)Zbl 0717.17015

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

### MSC:

 17B35 Universal enveloping (super)algebras 32C38 Sheaves of differential operators and their modules, $$D$$-modules 16S30 Universal enveloping algebras of Lie algebras 17B30 Solvable, nilpotent (super)algebras
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