Intermediate extensions and crystalline distribution algebras

Abstract

Let $G$ be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible modules over the crystalline distribution algebra of $G$ in terms of overconvergent isocrystals on locally closed subspaces in the flag variety of $G$. We treat the case of ${\mathrm{SL}}_2$ as an example.