Triviality of equivariant vector bundles for finite abelian groups (Q1977172)

It is shown that if \(G\) is a finite group and \(V\) is the representation space of a diagonalizable representation of \(G\), then all \(G\)-vector bundles over \(V\) are trivial. A \(G\)-vector bundle is a vector bundle \(\pi:E\to V\) where \(G\) acts on \(E\) and \(V\), \(\pi\) is equivariant, and for any \(g\in G\), the action of \(g\) on \(\pi^{-1}(v)\to g\pi^{-1}(v)\) is linear for all \(v\in V\). \textit{C. De Concini} and \textit{F. Fagnani} [Math. Control Signals Syst. 6, 307-321 (1993; Zbl 0802.93029)] showed that if \(G\) is abelian and the action of \(G\) on \(V\) is generated by pseudo-reflections, then the result holds. \textit{M. Masuda, L. Moser-Jauslin} and \textit{T. Petrie} [Topology 35, 329-334 (1996; Zbl 0884.14007)], showed that if \(G\) is abelian and reductive, but not necessarily finite, then all \(G\)-vector bundles over a representation space are trivial. The result of the present article generalizes the case concerning finite groups.