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Transactions of the American Mathematical Society

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The structure of the space of coadjoint orbits of an exponential solvable Lie group


Author: Bradley N. Currey
Journal: Trans. Amer. Math. Soc. 332 (1992), 241-269
MSC: Primary 22E25; Secondary 22E15, 22E27
DOI: https://doi.org/10.1090/S0002-9947-1992-1046014-2
MathSciNet review: 1046014
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Abstract: In this paper we address the problem of describing in explicit algebraic terms the collective structure of the coadjoint orbits of a connected, simply connected exponential solvable Lie group $ G$. We construct a partition $ \wp $ of the dual $ {\mathfrak{g}^{\ast} }$ of the Lie algebra $ \mathfrak{g}$ of $ G$ into finitely many $ \operatorname{Ad}^{\ast} (G)$-invariant algebraic sets with the following properties. For each $ \Omega \in \wp $, there is a subset $ \Sigma $ of $ \Omega $ which is a cross-section for the coadjoint orbits in $ \Omega $ and such that the natural mapping $ \Omega /\operatorname{Ad}^{\ast} (G) \to \Sigma $ is bicontinuous. Each $ \Sigma $ is the image of an analytic $ \operatorname{Ad}^{\ast}(G)$-invariant function $ P$ on $ \Omega $ and is an algebraic subset of $ {\mathfrak{g}^{\ast}}$. The partition $ \wp $ has a total ordering such that for each $ \Omega \in \wp $, $ \cup \{ \Omega \prime:\Omega \prime \leq \Omega \} $ is Zariski open. For each $ \Omega $ there is a cone $ W \subset {\mathfrak{g}^{\ast} }$, such that $ \Omega $ is naturally a fiber bundle over $ \Sigma $ with fiber $ W$ and projection $ P$. There is a covering of $ \Sigma $ by finitely many Zariski open subsets $ O$ such that in each $ O$, there is an explicit local trivialization $ \Theta :{P^{ - 1}}(O) \to W \times O$. Finally, we show that if $ \Omega $ is the minimal element of $ \wp $ (containing the generic orbits), then its cross-section $ \Sigma $ is a differentiable submanifold of $ {\mathfrak{g}^{\ast} }$. It follows that there is a dense open subset $ U$ of $ G\hat \emptyset $ such that $ U$ has the structure of a differentiable manifold and $ G\widehat\emptyset \sim U$ has Plancherel measure zero.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1046014-2
Article copyright: © Copyright 1992 American Mathematical Society

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