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Zbl 0689.17006
McGovern, William M.
Unipotent representations and Dixmier algebras. (English)
[J] Compos. Math. 69, No.3, 241-276 (1989). ISSN 0010-437X; ISSN 1570-5846/e

Let G be a complex semisimple Lie group and U(${\frak g})$ the universal enveloping algebra of its Lie algebra ${\frak g}$. The author studies U(${\frak g})$-bimodules A which in addition carry the structure (in a compatible way) of a unital associative algebra. He calls such an algebra a Dixmier algebra if the kernel of the corresponding natural map U(${\frak g})\to A$ is a maximal unipotent ideal [see {\it D. Barbasch} and {\it D. A. Vogan}, Ann. Math., II. Ser. 121, 41-110 (1985; Zbl 0582.22007) Def. 5.23 for a precise definition] and A is completely reducible as a bimodule. A is called strongly prime if the product of two nonzero U(${\frak g})$-bisubmodules is nonzero and completely prime if the product of two nonzero elements is nonzero. \par Barbasch and Vogan have shown that one can associate Dixmier algebras to nilpotent orbits in ${\frak g}\sp*$ and parametrize them by certain groups coming with the orbit. The paper under review deals with the question when these Dixmier algebras are strongly prime. In the case that the group associated to the orbit is abelian there is a complete answer. The author uses his results to disprove a conjecture of Vogan which said that there is a bijective correspondence between completely prime Dixmier algebras and ramified covers of orbit closures in ${\frak g}\sp*$.
[J.Hilgert]

MSC 2000:: *17B10 Representations of Lie algebras, algebraic theory
17B35 Universal enveloping algebras (Lie algebras)
22E46 Semi-simple Lie groups and their representations
17B20 Simple and semisimple Lie algebras
22E47 Repres. of Lie and real algebraic groups: algebraic methods

Keywords: orbit method; primitive ideals; complex semisimple Lie group; universal enveloping algebra; Dixmier algebra; strongly prime

Citations: Zbl 0582.22007

Cited in: Zbl 0854.17010

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