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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 $K$-theory.
For this, we introduce and study categories
$\operatorname{DK}_{\mathcal{S}}(X)$ of $\mathcal{S}$-constructible $K$-motivic
sheaves on varieties $X$ with an affine stratification $\mathcal{S}$. We show
that there is a natural and geometric functor, called Beilinson realisation,
from $\mathcal{S}$-constructible mixed sheaves
$\operatorname{D}^{mix}_{\mathcal{S}}(X)$ to
$\operatorname{DK}_{\mathcal{S}}(X)$.
We then show that Koszul duality intertwines the Betti realisation and
Beilinson realisation functors and descends to an equivalence of constructible
sheaves and constructible $K$-motivic sheaves on Langlands dual flag varieties.