Construction of canonical coordinates for exponential Lie groups

Abstract

Given an exponential Lie group $G$, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Hölder basis. Thus we obtain a stratification of $\mathfrak {g}^*$ into $G$-invariant algebraic subsets, and for each such subset $\Omega$, an explicit cross-section $\Sigma \subset \Omega$ for coadjoint orbits in $\Omega$, so that each pair $(\Omega , \Sigma )$ behaves predictably under the associated restriction maps on $\mathfrak {g}^*$. The cross-section mapping $\sigma : \Omega \rightarrow \Sigma$ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with $\ell \in \Omega$. For each $\Omega$, algebras $\mathcal E^0(\Omega )$ and $\mathcal E^1(\Omega )$ of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way. Now let $2d > 0$ be the dimension of coadjoint orbits in $\Omega$. An explicit algorithm is given for the construction of complex-valued real analytic functions $\{q_1,q_2, \dots , q_d\}$ and $\{p_1, p_2, \dots , p_d\}$ such that on each coadjoint orbit $\mathcal {O}$ in $\Omega$, the canonical 2-form is given by $\sum dp_k \wedge dq_k$. The functions $\{q_1,q_2, \dots , q_d\}$ belong to $\mathcal E^0(\Omega )$, and the functions $\{p_1, p_2, \dots , p_d\}$ belong to $\mathcal E^1(\Omega )$. The associated geometric polarization on each orbit $\mathcal {O}$ coincides with the complex Vergne polarization, and a global Darboux chart on $\mathcal {O}$ is obtained in a simple way from the coordinate functions $(p_1, \dots , p_d,q_1, \dots , q_d)$ (restricted to $\mathcal {O}$). Finally, the linear evaluation functions $\ell \mapsto \ell (X)$ are shown to be quantizable as well.