Equivariant motives and geometric representation theory. (with an appendix by F. H\"ormann and M. Wendt)

TL;DR

This paper develops a comprehensive framework for equivariant mixed Tate motives, establishing their formal properties, equivalences of definitions, and applications to categorification and representation theory.

Contribution

It introduces equivalent definitions of equivariant motives, constructs a six-functor formalism, and demonstrates weight structures and tilting results in this setting.

Findings

Equivariant mixed Tate motives have equivalent definitions via simplicial Borel construction and algebraic approximations.

A six-functor formalism for these motives is established.

Weight structures and tilting results are proven for certain categories of equivariant motives.

Abstract

We consider categories of equivariant mixed Tate motives, where equivariant is understood in the sense of Borel. We give the two usual definitions of equivariant motives, via the simplicial Borel construction and via algebraic approximations of it. The definitions turn out to be equivalent and give rise to a full six-functor formalism. For rational \'etale motives over a finite field or the homotopical stable algebraic derivator arising from the semisimplified Hodge realization, the equivariant mixed Tate motives provide a graded version of the equivariant derived category. We show that, in sufficiently nice and clean cases, these categories admit weight structures; moreover, a tilting result holds which identifies the category of equivariant mixed Tate motives with the bounded homotopy category of the heart of its weight structure. This can be seen as a formality result for equivariant derived categories. We also discuss convolution functors on equivariant mixed Tate motives, and consequences for the categorification of the Hecke algebra and some of its modules.