Topological properties of spaces of projective unitary representations

Abstract

LetGbe a compact and connected Lie group andPU(H)be the group of projective unitary operators onan infinite dimensional separable Hilbert spaceHendowed with the strong operator topology. We studythe space homst(G,PU(H))of continuous homomorphisms fromGtoPU(H)which are stable, namelythe homomorphisms whose induced representation contains each irreducible representation an infinitelynumber of times. We show that the connected components of homst(G,PU(H))are parametrized by theisomorphism classes ofS1-central extensions ofG, and that each connected component has the grouphom(G,S1)for fundamental group and trivial higher homotopy groups. We study the conjugation mapPU(H)→homst(G,PU(H)),F →FαF−1, we show that it has no local cross sections and we prove that fora mapB→homst(G,PU(H))withBparacompact of finite paracompact dimension, local lifts toPU(H)do exist.