Bohr-Sommerfeld-Heisenberg Quantization of the 2-dimensional Harmonic Oscillator

We perform a Bohr-Sommerfeld-Heisenberg quantization of the 2-dimensional harmonic oscillator, and obtain a reducible unitary representation of SU(2). Each energy level carries an irreducible unitary representation. This leads to a decomposition of the representation of SU(2), obtained by quantization of the harmonic oscillator, into a direct sum of irreductible unitary representations. Classical reduction of the energy level, corresponding to an irreducible unitary representation, gives the corresponding coadjoint orbit. However, the representation obtained by Bohr-Sommerfeld-Heisenberg quantization of the coadjoint orbit gives a unitary representation of SU(2) that is the direct sum of two irreducible representations.