Differential operators on quantized flag manifolds at roots of unity II

TL;DR

This paper proposes a conjectural derived equivalence between modules over a quantized enveloping algebra at roots of unity and twisted D-modules on the quantized flag manifold, extending Beilinson-Bernstein theory.

Contribution

It formulates a conjecture relating quantum group modules and D-modules on quantized flag manifolds at roots of unity, reducing the proof to global sections of differential operators.

Findings

Reduction of proof to global sections of differential operators

Reformulation of the conjecture via induction functor

Establishment of a Beilinson-Bernstein type equivalence at roots of unity

Abstract

We formulate a Beilinson-Bernstein type derived equivalence for a quantized enveloping algebra at a root of 1 as a conjecture. It says that there exists a derived equivalence between the category of modules over a quantized enveloping algebra at a root of 1 with fixed regular Harish-Chandra central character and the category of certain twisted $D$-modules on the corresponding quantized flag manifold. We show that the proof is reduced to a statement about the (derived) global sections of the ring of differential operators on the quantized flag manifold. We also give a reformulation of the conjecture in terms of the (derived) induction functor.