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Sur la classification des idéaux primitifs des algèbres enveloppantes. (French) Zbl 0549.17007

Let G be a connected algebraic group over \({\mathbb{C}}\), \({\mathfrak g}\) its Lie algebra, and \(U({\mathfrak g})\) the enveloping algebra of \({\mathfrak g}\). Write Prim \(U({\mathfrak g})\) for the set of primitive ideals (that is, annihilators of irreducible modules) in \(U({\mathfrak g})\). A basic problem is to give a somewhat geometric description of Prim \(U({\mathfrak g})\). The main obstructions to a nice answer arise from the case when G is reductive; so M. Duflo proposed [Acta Math. 149, 153-213 (1982; Zbl 0529.22011)] to give simply a nice geometric reduction to the reductive case. He defined a set of parameters (f,\(\xi)\); and for each such parameter, a primitive ideal \(I_{f,\xi}\). He also showed that every primitive ideal is of the form \(I_{f,\xi}\). The group G acts on the parameter set, and the ideal \(I_{f,\xi}\) does not change under this action. Duflo conjectured, but did not prove, that two parameters define the same primitive ideal only if they are conjugate by G. That conjecture is proved here.
The basic tool (as in Duflo’s work) is the authors’ version of the ”Mackey machine” for primitive ideals, which shows how to use any ideal in \({\mathfrak g}\) to exhibit any ideal in \(U({\mathfrak g})\) as induced. In light of their earlier work on this, the new ingredient is a uniqueness assertion for the inducing ideal. The proof of this uniqueness provides a partial inverse for Duflo’s construction of ideals, and a number of interesting consequences are deduced in the last part of the paper. In particular, there is a partition of Prim \(U({\mathfrak g})\) parametrized by the G orbits on \({\mathfrak g}^*\) admitting solvable polarizations. This generalizes the partition by infinitesimal character in the reductive case, and has analogous properties.
Reviewer: D.Vogan

MSC:

17B35 Universal enveloping (super)algebras
17B45 Lie algebras of linear algebraic groups
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16Dxx Modules, bimodules and ideals in associative algebras

Citations:

Zbl 0529.22011
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References:

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