Localizations for quiver Hecke algebras II

We prove that the localization (C) over tilde (w). of the monoidal category C-w is rigid, and the category C-w,C-v admits a localization via a real commuting family of central objects. For a quiver Hecke algebra R and an element in the Weyl group, the subcategory C-w of the category R-gmod of finite-dimensional graded R-modules categorifies the quantum unipotent coordinate ringc A(q)(n(w)). In the previous paper, we constructed a monoidal category (C) over tilde (w) such that it contains C-w and the objects {M(omega Lambda(t), Lambda(t)) vertical bar i is an element of I-.} corresponding to the frozen variables are invertible. In this paper, we show that there is a monoidal equivalence between the category (C) over tilde (w) and ((C) over tilde (w-1))(rev). Together with the already known left-rigidity of (C) over tilde (w), it follows that the monoidal category (C) over tilde (w) is rigid v less than or similar to w in the Bruhat order, there is a subcategory C-w,C-v of C-w that categorifies the doubly-invariant algebra (N)'C-(w)[N](N(nu)). We prove that the family (M(omega Lambda(t), u Lambda(t))(i is an element of I) of simple R-module forms a real commuting family of graded central objects in the category C-w,C-v so that there is a localization (C) over tilde (w,v) of C-w,C-v in which {M(omega Lambda(t), u Lambda(t)) are invertible. Since the localization of the algebra (N)'C-(w)[N](N(nu)) by the family of the isomorphism classes of (M(omega Lambda(t), v Lambda(t)) is isomorphic to the coordinate ring C[R-w,R-v] of the open Richardson variety associated with w and v, the localization (C) over tilde (w,v) categorifies the coordinate ring C[R-w,R-v].