In [AL07] and [AL10], Arone and Lesh constructed and studied spectrum level filtrations that interpolate between connective (topological or algebraic) K-theory and the Eilenberg-MacLane spectrum for the integers. In this paper we consider (global) equivariant generalizations of these filtrations and of another closely related class of filtrations, the modified rank filtrations of the K-theory spectra themselves. We lift Arone and Lesh's description of the filtration subquotients to the equivariant context and apply it to compute algebraic filtrations on representation rings that arise on equivariant homotopy groups. It turns out that these representation ring filtrations are considerably easier to express globally than over a fixed compact Lie group. Furthermore, they have formal similarities to the filtration on Burnside rings induced by the symmetric products of spheres, which Schwede computed in [Sch14].

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The aim of this note is to provide a comprehensive treatment of the homotopy theory of Γ-G-spaces for G a finite group. We introduce two level and stable model structures on Γ-G-spaces and exhibit Quillen adjunctions to G-symmetric spectra with respect to a flat level and a stable flat model structure respectively. Then we give a proof that Γ-G-spaces model connective equivariant stable homotopy theory along the lines of the proof in the nonequivariant setting given by Bousfield and Friedlander [2]. Furthermore, we study the smash product of Γ-G-spaces and show that the functor from Γ-G-spaces to G-symmetric spectra commutes with the derived smash product. Finally, we show that there is a good notion of geometric fixed points for Γ-G-spaces. Using the results of Shimakawa [16], we give a proof along the lines of [2] that Γ-G-spaces model connective equivariant stable homotopy theory. Moreover, Γ-Gspaces possess a symmetric monoidal smash product as was shown by Lydakis [6], motivating the question if this equivalence can be realized by a Quillen functor to a symmetric monoidal category of G-spectra which commutes with the derived smash product. This turns out to be true, if one uses the flat model structures on G-symmetric spectra as constructed by Hausmann [4]. Even non-equivariantly, this might be of interest on its own right. In addition, we define a geometric fixed point functor for Γ-G-spaces which has all desirable properties.The structure of the paper is as follows. Sections 2 and 3 contain a brief review of basic facts about G-equivariant homotopy theory and G-symmetric spectra. In particular, we will introduce the flat model structure. In Section 4, we briefly discuss basic definitions and constructions concerning Γ-G-spaces and introduce two level model structures. The projective model structure was employed by Santhanam in [15], too, but we also show how to generalize the strict model structure of [2] to the equivariant setting. In Section 5, we exhibit Quillen pairs between the level model structures on Γ-G-spaces and the flat level model structure on G-symmetric spectra. This requires a characterization of flat cofibrations of G-symmetric spectra which we carry out in the appendix. We also show that spectra obtained from Γ-G-spaces are equivariantly connective and, using the results of [16], we show that very special Γ-G-spaces give rise to GΩ-symmetric spectra up to a level fibrant replacement. After these preparations, we show that the homotopy categories with respect to the level model structures of very special Γ-G-spaces and those connective spectra which are level equivalent to GΩ-spectra are equivalent. In Section 6, we introduce stable equivalences of Γ-G-spaces and the stable model structures on Γ-G-spaces corresponding to the two level model structures. This leads to the equivalence of the homotopy categories of Γ-G-spaces and connective G-symmetric spectra with respect to the stable model structures. Section 7 contains a discussion of the smash product of Γ-G-spaces. Following the non-equi...

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