For finite abelian G with Sylow p-subgroup N_p, the KU_G/p-local sphere equals homotopy fixed points of a p-complete KO_{N_p}-module and a wedge of equivariant Morava K-theory spheres, with computed Z-graded and RO(G)-graded homotopy Mackey functors.
arXiv preprint arXiv:2411.00421,
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On the equivariant $KU_G$-local sphere for finite abelian groups
For finite abelian G with Sylow p-subgroup N_p, the KU_G/p-local sphere equals homotopy fixed points of a p-complete KO_{N_p}-module and a wedge of equivariant Morava K-theory spheres, with computed Z-graded and RO(G)-graded homotopy Mackey functors.