Triply Extended Group of Translations of $\mathbb{R}^{4}$ as Defining Group of NCQM: relation to various gauges

The role of the triply extended group of translations of $\mathbb{R}^{4}$, as the defining group of two dimensional noncommutative quantum mechanics (NCQM), has been studied in \cite{ncqmjmp}. In this paper, we revisit the coadjoint orbit structure and various irreducible representations of the group associated with them. The two irreducible representations corresponding to the Landau and symmetric gauges are found to arise from the underlying defining group. The group structure of the transformations, preserving the commutation relations of NCQM, has been studied along with specific examples. Finally, the relationship of a certain family of UIRs of the underlying defining group with a family of deformed complex Hermite polynomials has been explored .