Journal of Lie Theory

We show that when the methods of D. Arnal and J. C. Cortet ["Representations * des groupes exponentiels", Journal Funct. Anal. 92 (1990) 103--135] are combined with the explicit stratification and orbital parameters of B. N. Currey ["The structure of the space of co-adjoint orbits of an exponential solvable Lie group", Trans. Amer. Math. Soc. 332 (1992) 241--269], and B. N. Currey and R. C. Penney ["The structure of the space of co-adjoint orbits of a completely solvable Lie group", Michigan Math. J. 36 (1989), 309--320], the result is a construction of explicit analytic canonical coordinates for any coadjoint orbit O of a completely solvable Lie group. For each layer in the stratification, the canonical coordinates and the orbital cross-section together constitute an analytic parametrization for the layer.
Finally, we quantize the minimal open layer with the Moyal star product and prove that the coordinate functions are in a convenient completion of spaces of polynomial functions on g*, for a metric topology naturally related to the star product.