Equivariant vector bundles over classifying spaces for proper actions

Let G be an infinite discrete group and let (E) under barG be a classifying space for proper actions of G. Every G-equivariant vector bundle over (E) under barG gives rise to a compatible collection of representations of the finite subgroups of G. We give the first examples of groups G with a cocompact classifying space for proper actions (E) under barG admitting a compatible collection of representations of the finite subgroups of G that does not come from a G-equivariant (virtual) vector bundle over (E) under barG. This implies that the Atiyah-Hirzebruch spectral sequence computing the G-equivariant topological K-theory of (E) under barG has nonzero differentials. On the other hand, we show that for right-angled Coxeter groups this spectral sequence always collapses at the second page and compute the K-theory of the classifying space of a right-angled Coxeter group