We identify the group of homomorphisms $\operatorname{ Hom}_{ \mathcal{ GF}}(F,\mathbf{ RU}_{ \mathbb Q})$ in the category of ($\operatorname{ fin}$)-global functors to the rationalization of the unitary representation ring functor and deduce that the higher $\operatorname{ Ext}$-groups $\operatorname{ Ext}^n_{ \mathcal{ GF}}(F,\mathbf{ RU}_{ \mathbb Q})$, $n\geq 2$ have to vanish. This leads to a rational splitting of the ($\operatorname{ fin}$)-global equivariant $K$-theory spectrum into a sum of Eilenberg-MacLane spectra. Interpreted in terms of cohomology theories, it means that the equivariant Chern character is compatible with restrictions along all group homomorphisms.
Categories: math.AT math.RT