Geodesic
Canonical Coordinates for Coadjoint Orbits of Completely Solvable Groups
Journal of Lie Theory, Tome 15 (2005) no. 2, pp. 521-560

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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.
Classification : 22E25, 22E27, 53D55
Mots-clés : Completely solvable Lie groups, parametrization, canonical coordinates 
D. Arnal; M. Ben Ammar; B. N. Currey; B. Dali. Canonical Coordinates for Coadjoint Orbits of Completely Solvable Groups. Journal of Lie Theory, Tome 15 (2005) no. 2, pp. 521-560. http://geodesic.mathdoc.fr/item/JOLT_2005_15_2_a9/
@article{JOLT_2005_15_2_a9,
     author = {D. Arnal and M. Ben Ammar and B. N. Currey and B. Dali},
     title = {Canonical {Coordinates} for {Coadjoint} {Orbits} of {Completely} {Solvable} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {521--560},
     year = {2005},
     volume = {15},
     number = {2},
     zbl = {1074.22003},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2005_15_2_a9/}
}
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