Local Models For The Moduli Stacks of Global $G$-Shtukas

In this article we develop the theory of local models for the moduli stacks of global $G$-shtukas, the function field analogs for Shimura varieties. Here $G$ is a smooth affine group scheme over a smooth projective curve. As the first approach, we relate the local geometry of these moduli stacks to the geometry of Schubert varieties inside global affine Grassmannian, only by means of global methods. Alternatively, our second approach uses the relation between the deformation theory of global $G$-shtukas and associated local $P$-shtukas at certain characteristic places. Regarding the analogy between function fields and number fields, the first (resp. second) approach corresponds to the Beilinson-Drinfeld-Gaitsgory (resp. Rapoport-Zink) local model for (PEL-)Shimura varieties. As an application, we prove the flatness of these moduli stacks over their reflex rings, for tamely ramified group $G$. Furthermore, we introduce the Kottwitz-Rapoport stratification on these moduli stacks and discuss the intersection cohomology of the special fiber.