Deformation theory for prismatic <i>G</i>-displays

Abstract 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.