A naive approach to genuine $G$-spectra and cyclotomic spectra

Abstract

For any compact Lie group $G$, we give a description of genuine $G$-spectra in terms of the naive equivariant spectra underlying their geometric fixedpoints. We use this to give an analogous description of cyclotomic spectra in terms of naive $T$-spectra (where $T$ denotes the circle group), generalizing Nikolaus--Scholze's recent work in the eventually-connective case. We also give an explicit formula for the homotopy invariants of the cyclotomic structure on a cyclotomic spectrum in these terms.