Localizations for quiver Hecke algebras II

Abstract

We prove that the localization C∼w$ \widetilde{\mathcal {C}}_w$ of the monoidal category Cw$ \mathcal {C}_w$ is rigid, and the category Cw,v$ \mathcal {C}_{w,v}$ admits a localization via a real commuting family of central objects. For a quiver Hecke algebra R$R$ and an element w$w$ in the Weyl group, the subcategory Cw$ \mathcal {C}_w$ of the category R-gmod$R\text{-}\mathrm{gmod}$ of finite‐dimensional graded R$R$ ‐modules categorifies the quantum unipotent coordinate ring Aq(n(w))$A_q(\mathfrak {n}(w))$ . In the previous paper, we constructed a monoidal category C∼w$ \widetilde{\mathcal {C}}_w$ such that it contains Cw$ \mathcal {C}_w$ and the objects {M(wΛi,Λi)∣i∈I}$\lbrace {{\hspace*{0.6pt}\mathsf {M}}(w\Lambda _i,\Lambda _i)}\mid {i\in I}\rbrace$ corresponding to the frozen variables are invertible. In this paper, we show that there is a monoidal equivalence between the category C∼w$ \widetilde{\mathcal {C}}_w$ and (C∼w−1)rev$(\widetilde{\mathcal {C}}_{w^{-1}})^{\hspace*{0.6pt}\mathrm{rev}}$ . Together with the already known left‐rigidity of C∼w$ \widetilde{\mathcal {C}}_w$ , it follows that the monoidal category C∼w$ \widetilde{\mathcal {C}}_w$ is rigid. If v≼w$v\preccurlyeq w$ in the Bruhat order, there is a subcategory Cw,v$ \mathcal {C}_{w,v}$ of Cw$ \mathcal {C}_w$ that categorifies the doubly‐invariant algebra N′(w)C[N]N(v)$^{N^{\prime }(w)} {\mathbb {C}}[N]^{N(v)}$ . We prove that the family M(wΛi,vΛi)i∈I$\bigl ({\hspace*{0.6pt}\mathsf {M}}(w\Lambda _i,v\Lambda _i)\bigr )_{i\in I}$ of simple R$R$ ‐module forms a real commuting family of graded central objects in the category Cw,v$ \mathcal {C}_{w,v}$ so that there is a localization C∼w,v$ \widetilde{\mathcal {C}}_{w,v}$ of Cw,v$ \mathcal {C}_{w,v}$ in which M(wΛi,vΛi)${\hspace*{0.6pt}\mathsf {M}}(w\Lambda _i,v\Lambda _i)$ are invertible. Since the localization of the algebra N′(w)C[N]N(v)$^{N^{\prime }(w)} {\mathbb {C}}[N]^{N(v)}$ by the family of the isomorphism classes of M(wΛi,vΛi)${\hspace*{0.6pt}\mathsf {M}}(w\Lambda _i,v\Lambda _i)$ is isomorphic to the coordinate ring C[Rw,v]${\mathbb {C}}[R_{w,v}]$ of the open Richardson variety associated with w$w$ and v$v$ , the localization C∼w,v$ \widetilde{\mathcal {C}}_{w,v}$ categorifies the coordinate ring C[Rw,v]${\mathbb {C}}[R_{w,v}]$ .