EMIS ELibM Electronic Journals Journal of Lie Theory
Vol. 15, No. 2, pp. 521–560 (2005)

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Canonical Coordinates for Coadjoint Orbits of Completely Solvable Groups

Didier Arnal, Mabrouk Ben Ammar, Bradley N. Currey and Béchir Dali

Didier Arnal
Institut de Mathématiques de Bourgogne, Université de Bourgogne
CNRS UMR 5584, BP 47870, F-21078 Dijon Cedex France
Mabrouk Ben Ammar
Département de Mathématiques, Faculté des Sciences de Sfax
BP 802, 3038 Sfax, Tunisie
Bradley N. Currey
Saint Louis University
Department of Mathematics and Computer Science
Saint Louis, MO 63103
Béchir Dali
Département de Mathématiques, Faculté des Sciences de Bizerte
7021 Zarzouna, Bizerte, Tunisie

Abstract: We show that when the methods of Arnal, D. and J. C. Cortet, Representations $*$ des groupes exponentiels, Journal Funct. Anal. {\eightbf92} (1990), 103–135 are combined with the explicit stratification and orbital parameters of Currey, B. N., The structure of the space of co-adjoint orbits of an exponential solvable Lie group, Trans. Amer. Math. Soc. {\eightbf332} (1992), 241–269, and Currey, B. N. 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 ${\cal O}$ 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. \vskip0truemm 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

Classification (MSC2000): 22E25, 22E27, 53D55

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