Localization of quantum biequivariant D-modules and q-W algebras

Abstract: We present a biequivariant version of Kremnizer-Tanisaki localization theorem for quantum D-modules. We also obtain an equivalence between a category of finitely generated equivariant modules over a quantum group and a category of finitely generated modules over a q-W algebra defined in arXiv:1011.2431. This equivalence can be regarded as an equivariant quantum group version of Skryabin equivalence. The biequivariant localization theorem for quantum D-modules together with the equivariant quantum group version of Skryabin equivalence yield an equivalence between a certain category of quantum biequivariant D-modules and a category of finitely generated modules over a q-W algebra.