Topological properties of spaces of projective unitary representations

Let $G$ be a compact and connected Lie group and $PU(\mathcal{H})$ be the group of projective unitary operators on a separable Hilbert space $\mathcal{H}$ endowed with the strong operator topology. We study the space $hom_{st}(G, PU(\mathcal H))$ofcontinuoushomomorphismsfrom$G$to$PU(\mathcal H)$ which are stable, namely the homomorphisms whose induced representation contains each irreducible representation an infinitely number of times. We show that the connected components of $ho{m}_{st}(G,PU(\mathcal{H}))$ are parametrized by the isomorphism classes of $S^1$-central extensions of $G$, and that each connected component has the group $hom(G,S^1)$ for fundamental group and trivial higher homotopy groups. We study the conjugation map $PU(\mathcal H) \to hom_{st}(G, PU(\mathcal H))$,$F \mapsto F\alpha F^{-1}$, we show that it has no local cross sections and we prove that for a map $B\to ho{m}_{st}(G,PU(\mathcal{H}))$ with $B$ paracompact of finite covering dimension, local lifts to $PU(\mathcal H)$ do exist.