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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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References:

[1] A. BEILINSON et J. BERNSTEIN , Localisation de g-modules (C. R. Acad. Sci. Paris, t. 292, 1981 , p. 15-18). MR 82k:14015 | Zbl 0476.14019 · Zbl 0476.14019
[2] A. BEILINSON et J. BERNSTEIN , A generalisation of Casselman’s Submodule Theorem (Progress in Math., 40, Birkaüser, 1983 , p. 35-52). MR 85e:22024 | Zbl 0526.22013 · Zbl 0526.22013
[3] A. BEILINSON et J. BERNSTEIN , A Proof of Jantzen Conjectures (preprint). · Zbl 0790.22007
[4] Y. BENOIST , Les modules simples sphériques d’une algèbre de Lie nilpotente (Compos. Math., vol. 73, 1990 , p. 295-327). Numdam | MR 91g:17010 | Zbl 0711.17007 · Zbl 0711.17007
[5] P. BERNAT , N. CONZE , M. DUFLO , M. LEVY-NAHMAS , M. RAIS , P. RENOUARD et M. VERGNE , Représentations des groupes de Lie résolubles (Mon. Soc. Math. Fr., Dunod, Paris, 1972 ). Zbl 0248.22012 · Zbl 0248.22012
[6] A. BOREL et al., Algebraic D-modules , Acad. Press, 1987 . MR 89g:32014 | Zbl 0642.32001 · Zbl 0642.32001
[7] J. DIXMIER , Algèbre enveloppante , Gauthier-Villars, Paris, 1974 . MR 58 #16803a · Zbl 0146.26102
[8] H. FUJIWARA , Représentations monomiales des groupes de Lie nilpotents (Pacific Journ. Math., vol. 127, 1986 , p. 329-352). Article | MR 89c:22015 | Zbl 0588.22008 · Zbl 0588.22008
[9] V. GINZBURG , Sympletic Geometry and Representations (Funct. An. and Appl., vol. 17, 1983 , p. 225-227). MR 85b:58052 | Zbl 0529.58010 · Zbl 0529.58010
[10] M. KASHIWARA , Representation Theory and D-modules on Flag Varieties (Astérisque, vol. 173-174, 1989 , p. 55-109). MR 90k:17029 | Zbl 0705.22010 · Zbl 0705.22010
[11] A. KIRILLOV , Représentations unitaires des groupes de Lie nilpotents (Uspek. Math. Nauk., vol. 17, 1962 , p. 57-110). MR 25 #5396 | Zbl 0106.25001 · Zbl 0106.25001
[12] A. KIRILLOV , Éléments de la théorie des représentations , Ed. M.I.R., 1974 . MR 52 #14134
[13] G. LION , Intégrales d’entrelacement sur des groupes de Lie nilpotents et indice de Maslov , LN 587, Springer, 1977 , p. 160-176. MR 58 #6068 | Zbl 0391.22008 · Zbl 0391.22008
[14] D. MILICIC , D-modules and representation theory , Livre en préparation.
[15] R. PENNEY , Abstract Plancherel Theorems and a Frobenius Reciprocity Theorem (Journ. Funct. Anal., vol. 18, 1975 , p. 177-190). MR 56 #3191 | Zbl 0305.22016 · Zbl 0305.22016
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