The starting point is the 3-dimensional Heisenberg group $H_3$. The non-degenerate unirreps of $H_3$ are realized on $L^{2}(\mathbb{R})$. Each such representation corresponds to a nonzero value of $\frac{1}{\hbar}$. One knows that each such irreducible representation is in 1-1 correspondence with a 2-dimensional plane parallel to the X-Y plane in the dual of the lie algebra $\mathfrak{h}_{3}$ (which is $\mathbb{R}^{3}$) of the Lie group $H_3$ corresponding to a nonzero value of z (which stands for $\frac{1}{\hbar}$) by Orbit method pioneered by Kirillov. And then $z=0$ or equivalently $\frac{1}{\hbar}=0$ corresponds to the degenerate 1-dimensional unirrep of $H_3$ where the 2 noncentral generators are mapped to nonzero complex numbers and hence commute with each other. These degenerate unirreps of $H_3$ correspond to 0-dimensional coadjoint orbits which are points of the X-Y plane of the dual of $\mathbb{R}^{3}$. It seems like the classical limit is achieved at $\frac{1}{\hbar}=0$ and not as $\hbar\rightarrow 0$. What am I missing?
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