Equivariant \(K\)-theory and maps between representation spheres (Q1913414)

Let \(G\) be a compact Lie group, let \(U\), \(W\) be unitary representations of \(G\), and let \(SU\) and \(SW\) be their unit spheres. The author computes their equivariant \(K\)-rings. If \(G\) is abelian and there exists a \(G\)-map \(SU \to SW\), then the author shows that \(\lambda_{-1} W \in (\lambda_{-1} U)\) in \(R(G)\) where \(\lambda_{-1} W\) is the Euler class of \(W\) in \(K_G(pt) = R(G)\) and \((\lambda_{-1} U)\) is the ideal generated by the Euler class \(\lambda_1 U\).