Differential operators on locally analytic Shimura varieties

Abstract

We investigate infinite-level Shimura varieties within the framework of analytic stacks of Clausen-Scholze, developing their smooth, completed, locally analytic, and de Rham realizations. We formulate a Grothendieck-Messing-Hodge-Tate period map, and establish a Grothendieck-Messing theory for locally analytic infinite-level Shimura varieties. This theory, combined with a reformulation of Riemann-Hilbert correspondence, implies that the locally analytic infinite-level Shimura variety can be fully reconstructed purely from its perfectoid counterpart and its ${\mathbb{B}}_{\mathrm{dR}}^{+}$-thickening. Building upon this geometric structure, we systematically construct differential operators generalizing those of Pan, and we introduce a Bernstein-Gelfand-Gelfand-Fontaine complex based on dual BGG complexes, conjecturing its automorphic properties. These constructions will be used to establish a locally analytic Jacquet-Langlands correspondence in a companion paper ([Jia26a]).

Submitted version after minor corrections. Comments welcome!