Localizations for quiver Hecke algebras III

Let $R$ be a quiver Hecke algebra, and let ${\mathcal{C}}_{w,v}$ be the category of finite-dimensional graded $R$-module categorifying a $q$-deformation of the doubly-invariant algebra ${}^{N^{\mathrm{\prime }}(w)}\mathbb{C}[N{]}^{N(v)}$. In this paper, we prove that the localization ${\tilde{\mathcal{C}}}_{w,v}$ of the category ${\mathcal{C}}_{w,v}$ can be obtained as the localization by right braiders arising from determinantial modules. As its application, we show several interesting properties of the localized category ${\tilde{\mathcal{C}}}_{w,v}$ including the right rigidity.