Deformation theory for prismatic $G$-displays

For a smooth affine group scheme $G$ over the ring of $p$-adic integers and a cocharacter $\mu$ of $G$, we develop the deformation theory for $G$-$\mu$-displays over the prismatic site of Bhatt-Scholze, and discuss how our deformation theory can be interpreted in terms of prismatic $F$-gauges introduced by Drinfeld and Bhatt-Lurie. As an application, we prove the local representability and the formal smoothness of integral local Shimura varieties with hyperspecial level structure. We also revisit and extend some classification results of $p$-divisible groups.