Revisiting mixed geometry

Abstract

We provide a uniform construction of “mixed versions” or “graded lifts” in the sense of Beilinson–Ginzburg–Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our new theory associates to each Artin stack of finite type $\mathcal{Y}$ over ${\overline{\mathbb{F}}}_q$ a symmetric monoidal $\mathsf{DG}$-category ${\mathsf{Shv}}_{\mathsf{gr},c}(\mathcal{Y})$ of constructible graded sheaves on $\mathcal{Y}$ along with the six-functor formalism, a perverse $t$-structure, and a weight (or co-$t$-)structure in the sense of Bondarko and Pauksztello, compatible with the six-functor formalism, perverse $t$-structures, and Frobenius weights on the category of (mixed) $\ell$-adic sheaves. Classically, mixed versions were only constructed in very special cases due to the non-semisimplicity of Frobenius. Our construction sidesteps this issue by semisimplifying the Frobenius action itself. However, the category ${\mathsf{Shv}}_{\mathsf{gr},c}(\mathcal{Y})$ agrees with those previously constructed when they are available. For example, for any reductive group $G$ with a fixed pair $T\subset B$ of a maximal torus and a Borel subgroup, we have an equivalence of monoidal $\mathsf{DG}$ weight categories ${\mathsf{Shv}}_{\mathsf{gr},c}(B\backslash G/B)\simeq {\mathsf{Ch}}^b({\mathsf{SBim}}_W)$, where ${\mathsf{Ch}}^b({\mathsf{SBim}}_W)$ is the monoidal $\mathsf{DG}$-category of bounded chain complexes of Soergel bimodules and $W$ is the Weyl group of $G$.