Prismatic $G$-displays and descent theory

For a smooth affine group scheme $G$ over the ring of $p$-adic integers $\mathbb{Z}_p$ and a cocharacter $\mu$ of $G$, we study $G$-$\mu$-displays over the prismatic site of Bhatt-Scholze. In particular, we obtain several descent results for them. If $G=\mathrm{GL}_n$, then our $G$-$\mu$-displays can be thought of as Breuil-Kisin modules with some additional conditions. The relation between our $G$-$\mu$-displays and prismatic $F$-gauges introduced by Drinfeld and Bhatt-Lurie is also discussed. In fact, our results are formulated and proved for smooth affine group schemes over the ring of integers $\mathcal{O}_E$ of any finite extension $E$ of $\mathbb{Q}_p$ by using $\mathcal{O}_E$-prisms, which are $\mathcal{O}_E$-analogues of prisms.