Differential operators on quantized flag manifolds at roots of unity

The quantized flag manifold, which is a qq-analogue of the ordinary flag manifold, is realized as a non-commutative scheme, and we can define the category of DD-modules on it using the framework of non-commutative algebraic geometry; however, when the parameter qq is a root of unity, Lusztig’s Frobenius morphism allows us to handle DD-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–Mirković–Rumynin on DD-modules on flag manifolds in positive characteristics.