Rational extensions of the representation ring global functor and a splitting of global equivariant $K$-theory

TL;DR

This paper studies the structure of global functors related to equivariant K-theory, showing a rational splitting of the spectrum into simpler components and confirming the compatibility of the equivariant Chern character with group restrictions.

Contribution

It identifies the homomorphisms in the category of global functors to the rationalized representation ring and proves a rational splitting of global equivariant K-theory spectrum.

Findings

Higher Ext groups vanish for n ≥ 2.

Global equivariant K-theory splits into Eilenberg-MacLane spectra rationally.

Equivariant Chern character is compatible with all group homomorphisms.

Abstract

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.