Differential operators on quantized flag manifolds at roots of unity

TL;DR

This paper studies the structure of differential operators on quantized flag manifolds at roots of unity, showing they form Azumaya algebras and extending results from positive characteristic geometry.

Contribution

It demonstrates that the sheaf of rings defining D-modules on the quantized flag manifold is an Azumaya algebra at roots of unity, linking quantum and classical geometric frameworks.

Findings

Sheaf of rings is an Azumaya algebra over its center.

Restrictions of the sheaf are split Azumaya algebras.

Results extend known positive characteristic D-module theories.

Abstract

The quantized flag manifold, which is a $q$-analogue of the ordinary flag manifold, is realized as a non-commutative scheme, and we can define the category of $D$-modules on it using the framework of non-commutative algebraic geometry; however, when the parameter $q$ is a root of unity, Lusztig's Frobenius morphism allows us to handle $D$-modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold. In this paper we will show that this sheaf of rings on the ordinary flag manifold is an Azumaya algebra over its center. We also show that its restriction to certain subsets are split Azumaya algebras. These are analogues of some results of Bezrukavnikov-Mirkovi\'{c}-Rumynin on $D$-modules on flag manifolds in positive characteristics.