Differential operator realization of braid group action on $\mathrm{ı}$quantum groups

We construct a unique braid group action on modified $q$-Weyl algebra ${\mathbf{A}}_q(S)$. Under this action, we give a realization of the braid group action on quasi-split $\mathrm{ı}$quantum groups ${}^{\mathrm{ı}}\mathbf{U}(S)$ of type $\mathrm{A}\mathrm{I}\mathrm{I}\mathrm{I}$. Furthermore, we directly construct a unique braid group action on polynomial ring $\mathbb{P}$ which is compatible with the braid group action on ${\mathbf{A}}_q(S)$ and ${}^{\mathrm{ı}}\mathbf{U}(S)$.