Higher-order quantization on a Lie group

In this paper we prove the irreducibility of representations associated with higher-order polarizations on a connected Lie group with a U(1)-principal bundle structure. The representation technique here used is formulated on the basis of a group quantization formalism previously introduced which generalizes the Kostant-Kirillov co-adjoint orbits method for connected Lie groups as well as the Borel-Weyl-Bott representation algorithm for semisimple groups. We illustrate the fundamentals of the group approach with the help of the symmetry of the harmonic oscillator and the need for a higher-order polarization with the Schr\"odinger group which constitutes the simplest example of an anomalous symmetry