Why do we always classify states under covering group representations instead of the group itself? For example see the following picture I lifted from 'Symmetry in physics' by Gross
So in the first example, why not classify states under SO(3) instead of SU(2)?
EDIT: From a physicists perspective, I know enough that we consider projective representations of groups in quantum mechanics because those are the most general thing that keeps $ |\langle \phi|\psi\rangle|^2 $ invariant. But Gross above seems to indicate that representations of the covering group are a better way to go. Furthermore, covering groups are always simply connected, to where-as groups with projective representations are in general not simply connected - is this property of (universal) covering groups related to why we use them in these sorts of cases?