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


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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