Equivariant motives and geometric representation theory. (with an appendix by F. Hörmann and M. Wendt)

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 étale motives over a finite field or the homotopical stable algebraic derivator arising from 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 algebras and modules.

K$K$‐Motives and Koszul duality

“…We note that there is a different way of proving the last theorem using a formalism called 'tilting', see [33] and [32]. We prefer the weight complex functor, since it also exists without the pointwise purity assumption.…”

Revisiting mixed geometry

“…Recently, using the theory of motives, Soergel, Virk, and Wendt [SW18, SVW18] constructed a streamlined theory that shares many nice features with the theory developed in this paper. At a technical level, the main difference between the two approaches is that the results of [SW18,SVW18] are formulated using the theory of motives whereas ours make use of mixed ℓ-adic sheaves. On the one hand, their results are much more sophisticated and can be applied to other cohomology theories, such as K-theory, [Ebe19,ES21a].…”