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Zbl 0589.22009
Pedersen, Niels Vigand
On the infinitesimal kernel of irreducible representations of nilpotent Lie groups. (English)
[J] Bull. Soc. Math. Fr. 112, 423-467 (1984). ISSN 0037-9484

Let G be a real connected simply connected nilpotent Lie group, ${\frak g}\sb{{\bbfC}}$ the complexification of the Lie algebra of G and U(${\frak g}\sb{{\bbfC}})$ the enveloping algebra of ${\frak g}\sb{{\bbfC}}$. Let $\pi$ be an irreducible representation of G and $d\pi$ the corresponding representation of U(${\frak g}\sb{{\bbfC}})$. It is known that the coadjoint orbit of $\pi$ can be parametrized by polynomial functions. The author shows that the kernel of $d\pi$ is the right ideal generated by elements of U(${\frak g}\sb{{\bbfC}})$ which are determined by the polynomial functions and the symmetrization map.
[S.Sankaran]

MSC 2000:: *22E27 Representations of nilpotent and solvable Lie groups
17B35 Universal enveloping algebras (Lie algebras)

Keywords: infinitesimal kernel; parametrization by polynomial functions; simply connected nilpotent Lie group; enveloping algebra; irreducible representation; coadjoint orbit; symmetrization map

Cited in: Zbl 0866.22010 Zbl 0788.17008 Zbl 0763.22008

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