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-Holder basis. Thus we obtain a stratification of g * into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section Σ C Ω for coadjoint orbits in Ω, so that each pair (Ω, Σ) behaves predictably under the associated restriction maps on g * . The cross-section mapping σ: Ω → Σ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with ∈ Ω. For each Ω, algebras e 0 (Ω) and e 1 (Ω) 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 Ω. An explicit algorithm is given for the construction of complex-valued real analytic functions {q 1 , q 2 , ..., q d } and {p 1 ,p 2 ,...,p d } such that on each coadjoint orbit O in Ω, the canonical 2-form is given by Σdp k ^ dq k . The functions {q 1 ,q 2 , ..., q d } belong to e 0 (Ω), and the functions {p 1 ,p 2 , ... ,p d } belong to e 1 (Ω). The associated geometric polarization on each orbit O coincides with the complex Vergne polarization, and a global Darboux chart on O is obtained in a simple way from the coordinate functions (p 1 , ... , p d , q 1 , ... , q d ) (restricted to O). Finally, the linear evaluation functions l ↦ l(X) are shown to be quantizable as well.