Locally analytic representations in mixed characteristic via stacks

Abstract

For any $p$-adic Lie group $G$, we construct a group stack $G^{\mathrm{la}}$ over the $p$-adic branch of Clausen--Scholze's stack of norms such that quasicoherent sheaves on its classifying stack $\ast /G^{\mathrm{la}}$ 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 $\ast /G^{\mathrm{la}}$ and establish Poincaré duality for cohomology of locally analytic representations in mixed characteristic.

arXiv comment40 pages. Comments welcome! v2: Corrected the definition of the stack of norms. v3: Minor corrections