K$K$‐Motives and Koszul duality

Abstract

We construct an ungraded version of Beilinson–Ginzburg–Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of K$K$‐theory. For this, we introduce and study categories DKS(X)$\hbox{DK}_\mathcal {S}(X)$ of S$\mathcal {S}$‐constructible K$K$‐motivic sheaves on varieties X$X$ with an affine stratification. We show that there is a natural and geometric functor, called Beilinson realisation, from S$\mathcal {S}$‐constructible mixed sheaves DSmix(X)$\hbox{D}^{mix}_\mathcal {S}(X)$ to DKS(X)$\hbox{DK}_\mathcal {S}(X)$. We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible K$K$‐motivic sheaves on Langlands dual flag varieties.

K‐theory Soergel bimodules

“…Remarkably, Koszul duality intertwines the Tate-twist and shift functor (1) [2] with the Tate twist (1). This motivated our construction of a nonmixed/ungraded Koszul duality for flag varieties, see [13],…”

“…Remarkably, Koszul duality intertwines the Tate‐twist and shift functor $$(1)[2]$$ with the Tate twist $$(1)$$. This motivated our construction of a nonmixed/ungraded Koszul duality for flag varieties, see [13], $$$$\begin{equation*} \operatorname{DK}_{(B)}(X)\stackrel{\sim }{\rightarrow }\operatorname{D}_{(B)}(X^{\vee }), \end{equation*}$$$$relating $$K$$‐motives to constructible sheaves: $$K$$‐motives admit a phenomenon called Bott periodicity which implies that $$(1)[2]$$ is the identity functor, while the Tate twist $$(1)$$ acts trivially on (nonmixed) constructible sheaves.…”