My question is inspired by the analogy of the Berry phase in the spin coherent state representation of a rotator and the Aharonov-Bohm phase of a magnetic monopole (see e.g., Section 1.8.3 in http://www.physics.ubc.ca/~berciu/TEACHING/PHYS503/section1.pdf).
Consider a physical system invariant under a (continuous) group of transformations G. The aim is to define the corresponding quantum mechanical Hilbert space H ``in the richest possbile way'', that is, a representation which has as many state vectors as group elements. In case of G=SO(3) it is the 2-sphere of spin coherent states. The Aharonov-Bohm analogy says that H is a manifold with a connection corresponding to vector potential of some specific kind (in the example of spin it is the vector potential which gives the magnetic field of the monopole). Once we know that connection, can we classify the possible global topologies of representations of G as the angular momentum representations are classified by requiring the uniqueness of the Berry phase of a closed loop?
Can such a manifold be constructed for any G (from a sufficiently interesting class of groups)? My feeling is that this should correspond to some theorem in topolgy/representation theory. I'm very ignorant in these fields of math, and would appreciate a hint on where to look further.