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

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.