Construction of canonical coordinates for exponential Lie groups

2009

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, a…