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Journal of Lie Theory 15 (2005), No. 2, 521--560
Copyright Heldermann Verlag 2005

Canonical Coordinates for Coadjoint Orbits of Completely Solvable Groups

Didier Arnal
Institut de Mathématiques, Université de Bourgogne, CNRS UMR 5584, BP 47870, 21078 Dijon, France

Mabrouk Ben Ammar
Dép. de Mathématiques, Faculté des Sciences, BP 802, 3038 Sfax, Tunisia

Bradley N. Currey
Saint Louis University, Dept. of Mathematics and Computer Science, Saint Louis, MO 63103, U.S.A.

Béchir Dali
Dép. de Mathématiques, Faculté des Sciences, 7021 Zarzouna -- Bizerte, Tunisia

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.

Keywords: Completely solvable Lie groups, parametrization, canonical coordinates.

MSC: 22E25, 22E27, 53D55

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