Quantization, Reduction and Decomposition of Representations

Quantization of a Hamiltonian system with symmetry gives rise to a unitary
representation U of the symmetry group G. It can be decomposed into a direct
sum/integral of irreducible unitary representations of G. On the other hand,
classical reduction describes the structure on the space of orbits of G
contained in the phase space of the system. I will discuss the possibility
of recovering the decomposition of U into irreducible unitary
representations from the data provided by reduction.