Canonical Coordinates for Coadjoint Orbits of Completely Solvable Groups

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

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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