D^\dagger-affinity of formal models of flag varieties

TL;DR

This paper establishes D-affinity for formal models of flag varieties over p-adic fields and connects arithmetic D-modules with locally analytic G-representations, advancing the understanding of p-adic geometric representation theory.

Contribution

It introduces a new equivalence between coadmissible arithmetic D-modules and admissible locally L-analytic G-representations with trivial infinitesimal character.

Findings

Formal models of flag varieties are D-affine for certain sheaves of arithmetic differential operators.

The category of coadmissible G-equivariant arithmetic D-modules is anti-equivalent to admissible locally L-analytic G-representations.

Explicit computations of equivariant arithmetic D-modules for specific representations.

Abstract

Let G be the group of L-rational points of a connected split reductive group over a finite extension L of Q_p. We show that formal models of the algebraic flag variety X of G are D-affine for certain sheaves of arithmetic differential operators. We then introduce the category of coadmissible G-equivariant arithmetic D-modules on the system of formal models of X and prove that it is anti-equivalent to the category of admissible locally L-analytic G-representations with trivial infinitesimal character. We compute the equivariant arithmetic D-modules of certain classes of representations.