Localizations for quiver Hecke algebras III

Let R be a quiver Hecke algebra, and let \(\mathscr {C}_{w,v}\) ${\mathcal{C}}_{w,v}$ be the category of finite-dimensional graded R-module categorifying a q-deformation of the doubly-invariant algebra \(^{N'(w)} {\mathbb {C}}[N] ^{N(v)} \) ${}^{N^{\prime }(w)}\mathbb{C}{[N]}^{N(v)}$ . In this paper, we prove that the localization \(\widetilde{\mathscr {C}}_{w,v}\) ${\tilde{\mathcal{C}}}_{w,v}$ of the category \(\mathscr {C}_{w,v}\) ${\mathcal{C}}_{w,v}$ 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 \(\widetilde{\mathscr {C}}_{w,v} \) ${\tilde{\mathcal{C}}}_{w,v}$ including the right rigidity.