Localization for quantum groups at a root of unity

Abstract

In the paper Quantum flag varieties, equivariant quantum D \mathcal {D} -modules, and localization of Quantum groups, Backelin and Kremnizer defined categories of equivariant quantum O q \mathcal {O}_q -modules and D q \mathcal {D}_q -modules on the quantum flag variety of G G . We proved that the Beilinson-Bernstein localization theorem holds at a generic q q . Here we prove that a derived version of this theorem holds at the root of unity case. Namely, the global section functor gives a derived equivalence between categories of U q U_q -modules and D q \mathcal {D}_q -modules on the quantum flag variety. For this we first prove that D q \mathcal {D}_q is an Azumaya algebra over a dense subset of the cotangent bundle T ⋆ X T^\star X of the classical (char 0 0 ) flag variety X X . This way we get a derived equivalence between representations of U q U_q and certain O T ⋆ X \mathcal {O}_{T^\star X} -modules. In the paper Localization for a semi-simple Lie algebra in prime characteristic, by Bezrukavnikov, Mirkovic, and Rumynin, similar results were obtained for a Lie algebra g p \mathfrak {g}_p in char p p . Hence, representations of g p \mathfrak {g}_p and of U q U_q (when q q is a p p ’th root of unity) are related via the cotangent bundles T ⋆ X T^\star X in char 0 0 and in char p p , respectively.