Computing Vergne Polarizing Subalgebras

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According to the orbit method, the construction of a unitary irreducible representation of a nilpotent Lie group requires a precise computation of some polarizing subalgebra subordinated to a linear functional in the linear dual of the corresponding Lie algebra. This important step is generally challenging from a computational viewpoint. In this paper, we provide an algorithmic approach to the construction of the well-known Vergne polarizing subalgebras. The algorithms presented in this paper are specifically designed so that they can be implemented in computer algebra systems. We also show there are instances where Vergne’s construction could be refined for the sake of efficiency. Finally, we adapt our refined procedure to free nilpotent finite-dimensional Lie algebras of step-two to obtain simple and precise descriptions of Vergne polarizing algebras corresponding to all linear functionals in a dense open subset of the linear dual of the corresponding Lie algebra. Also, a program written for Mathematica is presented at the end of the paper.

Original Citation

Oussa, V. (2015). Computing Vergne Polarizing Subalgebras. Linear and Multilinear Algebra, 63(3), 578-585. https://doi.org/10.1080/03081087.2014.880434