Equivalences of the form $\Sigma^{V} X\simeq\Sigma^{W} X$ in equivariant stable homotopy theory

We study equivalences of the form \Sigma^{V}X\simeq\Sigma^{W}X , where G is a compact Lie group, X is a G -spectrum, and V and W are G -representations. These equivalences encode a periodicity phenomenon in G -equivariant homotopy theory which generalizes the classical James periodicity for G=C_{2} .In the case where X=C(a_{\lambda}) is the cofiber of an Euler class, we construct an \mathit{RO}(G) -graded J -homomorphism J\colon\pi_{\lambda}\mathit{KO}_{G}\rightarrow\pi_{\star}^{G} C(a_{\lambda})^{\times} which gives control over these periodicities. It also produces infinite periodic families in the G -equivariant stable stems. We illustrate this with several explicit examples.More generally, our work gives information about \mathit{RO}(G) -graded units in equivariant stable cohomotopy rings. We apply this to construct universal periodicities and differentials in the G -homotopy fixed point spectral sequence, and other equivariant Atiyah–Hirzebruch spectral sequences.