Localizations for quiver Hecke algebras II

We prove that the localization (Formula presented.) of the monoidal category (Formula presented.) is rigid, and the category (Formula presented.) admits a localization via a real commuting family of central objects. For a quiver Hecke algebra (Formula presented.) and an element (Formula presented.) in the Weyl group, the subcategory (Formula presented.) of the category (Formula presented.) of finite-dimensional graded (Formula presented.) -modules categorifies the quantum unipotent coordinate ring (Formula presented.). In the previous paper, we constructed a monoidal category (Formula presented.) such that it contains (Formula presented.) and the objects (Formula presented.) corresponding to the frozen variables are invertible. In this paper, we show that there is a monoidal equivalence between the category (Formula presented.) and (Formula presented.). Together with the already known left-rigidity of (Formula presented.), it follows that the monoidal category (Formula presented.) is rigid. If (Formula presented.) in the Bruhat order, there is a subcategory (Formula presented.) of (Formula presented.) that categorifies the doubly-invariant algebra (Formula presented.). We prove that the family (Formula presented.) of simple (Formula presented.) -module forms a real commuting family of graded central objects in the category (Formula presented.) so that there is a localization (Formula presented.) of (Formula presented.) in which (Formula presented.) are invertible. Since the localization of the algebra (Formula presented.) by the family of the isomorphism classes of (Formula presented.) is isomorphic to the coordinate ring (Formula presented.) of the open Richardson variety associated with (Formula presented.) and (Formula presented.), the localization (Formula presented.) categorifies the coordinate ring (Formula presented.).