The geometric Satake theory gives the following equivalence of tensor categories

$$ \mathrm{Rep}_G \xrightarrow \cong \mathrm{Perv}_{\hat G(k[[t]])}(\Gr_{\hat G}) $$

between the category of algebraic representations of a reductive group $G$ and certain category of equivariant perverse sheaves on the affine Grassmannian of the dual group of $G$ (defined over an algebraically closed field $k$). We consider the case when $k$ has characteristic $p>0$, and extend this equivalence to a natural morphism

$$ \mathrm{Coh}_{fr}([G/\mathrm{Ad}(G)]) \to \mathrm{Perv}(\mathrm{Sht}_{\hat G}) $$

from the category of locally free coherent sheaves on the stack $[G/ \mathrm{Ad}(G)]$, to the category of perverse sheaves on the moduli of local shtukas for the dual group $\hat G$. If time permits, I will discuss the motivation of such a construction coming from the geometry of Shimura varieties. This is a joint work with Xinwen Zhu.