Central Characters of G_{NC}, Darboux Normalization, and the Kinematical Inequivalence of NCQM and QM

We analyze generalized Bopp shifts and Darboux normalization in two-dimensional noncommutative quantum mechanics (NCQM) from the viewpoint of the unitary representation theory of the kinematical symmetry group \(G_{\mathrm{NC}}\). This group is a step-two nilpotent Lie group with three-dimensional center, and the regular part of its unitary dual \(\widehat{G_{\mathrm{NC}}}\) is labelled by central characters \((\hbar,\vartheta,B_{\mathrm{in}})\). Ordinary two-dimensional quantum mechanics (QM) appears inside \(\widehat{G_{\mathrm{NC}}}\) as the family of Weyl-Heisenberg representations inflated along the quotient \(G_{\mathrm{NC}}\rightarrow G_{\mathrm{WH}}\), with central character \((\hbar,0,0)\). We prove that a generic nondegenerate NCQM sector \((\hbar_0,\vartheta_0,B_0)\), with \(\hbar_0,\vartheta_0,B_0\neq 0\) and \(\hbar_0-B_0\vartheta_0\neq 0\), is not unitarily equivalent to the ordinary QM sector \((\hbar_0,0,0)\) as a \(G_{\mathrm{NC}}\)-representation. Consequently, generalized Bopp shifts and Darboux normalizations, although they can produce auxiliary operator quadruples satisfying canonical commutation relations, do not establish kinematical equivalence of the corresponding sectors. We further explain that the apparent identification arises only after passing to a Darboux-normalized or coarse star-product description, where the original \(G_{\mathrm{NC}}\)-central-character label is no longer part of the data. The result clarifies the relation between operator-level Darboux normalization, deformation-quantization equivalence, and representation-theoretic equivalence in NCQM.