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On the orbit method for a solvable Lie algebra. (Sur la méthode des orbites pour une algèbre de Lie résoluble.) (French) Zbl 0917.17005

Let \(\mathfrak g\) be a finite-dimensional completely solvable Lie algebra over a field of characteristic 0. Let \(U(\mathfrak g)\) be its enveloping algebra. In [Proc. Noncommutative Harmonic Analysis, Marseille-Luminy 1978, Lect. Notes Math. 728, 42-63 (1979; Zbl 0409.22003)], J. Dixmier has shown how to attach a primitive ideal in \(U(\mathfrak g)\) to every coadjoint orbit in \(\mathfrak g^*\) without using polarizations, if \(\mathfrak g\) is nilpotent. He also sketched how to do this in the solvable case. In this paper the author gives an alternative construction of the ideal attached to a coadjoint orbit in the solvable case. This construction, like Dixmier’s, is heavily influenced by Duflo’s work on biinvariant differential operators on Lie groups.

MSC:

17B30 Solvable, nilpotent (super)algebras
17B35 Universal enveloping (super)algebras

Citations:

Zbl 0409.22003
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References:

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