Integral motivic sheaves and geometric representation theory

K‐theory Soergel bimodules

“…The vanishing of $$\operatorname{Hom}_{\operatorname{DKT}^G(V)}(\mathbb {1}[n],\mathbb {1})$$ for $$n<0$$ allows to define the following weight structure (for an overview over weight structures and weight complex functors for $$\infty$$‐categories, see [19, section 2.1.3]) on $$\operatorname{DKT}^G(V)$$, which exists by [5, Proposition 1.2.3(6)]. Definition Let $$G$$ be a diagonalizable algebraic group and $$V\in \operatorname{Rep}(G)$$.…”

“…between a category of 𝑇-equivariant mixed sheaves, that are locally constant along Bruhat cells, and the category of chain complexes of graded Soergel bimodules. Mixed sheaves D mix (𝑋) are a graded refinement of the category of constructible sheaves D 𝑏 (𝑋) that can be constructed via mixed Hodge modules or mixed 𝓁-adic sheaves, see [2,25], and, most satisfyingly, using mixed motives DM(𝑋), see, for example, [15,19,37,38].…”

“…Mixed sheaves $$\operatorname{D}^{\rm mix}(X)$$ are a graded refinement of the category of constructible sheaves $$\operatorname{D}^b(X)$$ that can be constructed via mixed Hodge modules or mixed $$\ell$$‐adic sheaves, see [2, 25], and, most satisfyingly, using mixed motives $$\operatorname{DM}(X)$$, see, for example, [15, 19, 37, 38].…”

“…Here reduced $$K$$‐motives $$\operatorname{DK}_{\text{r}}$$ should be constructed from $$\operatorname{DK}$$ by removing the higher $$K$$‐theory of the base point, as defined in the context of $$\operatorname{DM}$$ in [19]. In particular, the category $$\operatorname{DK}^{G\times {\mathbb {G}_m}}_{\text{r}}(\operatorname{Gr})$$ should have a combinatorial description in terms of singular $$K$$‐theory Soergel bimodules.…”

K$K$‐Motives and Koszul duality

“…In the Appendix we collect some useful facts about weight structures and $$t$$‐structures. Remark (1)The case of modular coefficients is work in progress joint with Shane Kelly and building on [18] and [19]. (2)The construction of an equivariant (both in the sense of Borel and Bredon) version of $$K$$‐motivic sheaves is work in progress.…”