A Beilinson-Bernstein Theorem for analytic quantum groups

TL;DR

This paper develops a $p$-adic analytic framework for quantum groups, establishing an equivalence between categories of twisted $D$-modules on quantum flag varieties and modules over completed quantum groups, extending classical Beilinson-Bernstein results.

Contribution

It introduces a $p$-adic analytic analogue of quantum flag varieties and proves a Beilinson-Bernstein type equivalence for regular dominant weights in this setting.

Findings

Established an equivalence of categories between twisted $D$-modules and quantum group modules.

Defined a category of $ ext{lambda}$-twisted $D$-modules on the analytic quantum flag variety.

Identified Banach comodules over the quantum Borel algebra with topologically integrable modules.

Abstract

We introduce a $p$-adic analytic analogue of Backelin and Kremnizer's construction of the quantum flag variety of a semisimple algebraic group, when $q$ is not a root of unity and $| q-1|<1$. We then define a category of $\lambda$-twisted $D$-modules on this analytic quantum flag variety. We show that when $\lambda$ is regular and dominant and when the characteristic of the residue field does not divide the order of the Weyl group, the global section functor gives an equivalence of categories between the coherent $\lambda$-twisted $D$-modules and the category of finitely generated modules over $\widehat{U_q^\lambda}$, where the latter is a completion of the ad-finite part of the quantum group with central character corresponding to $\lambda$. Along the way, we also show that Banach comodules over the Banach completion $\widehat{\mathcal{O}_q(B)}$ of the quantum coordinate algebra of the Borel can be naturally identified with certain topologically integrable modules.