Locally analytic representations in mixed characteristic via stacks

For any $p$-adic Lie group $G$, we construct a group stack $G^{\mathrm{l}\mathrm{a}}$ over the $p$-adic branch of Clausen–Scholze’s stack of norms such that quasicoherent sheaves on its classifying stack $\ast \mathrm{/}{G}^{\mathrm{l}\mathrm{a}}$ recover locally analytic representations of $G$ after base change to any suitable Tate Huber pair, even in positive or mixed characteristic. We also show that smooth ${\mathbb{F}}_p$-representations of $G$ embed fully faithfully into sheaves on the mod $p$ fibre of $\ast \mathrm{/}{G}^{\mathrm{l}\mathrm{a}}$ and establish Poincaré duality for cohomology of locally analytic representations in mixed characteristic.