Differential operator realization of braid group action on $\imath$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 $\imath$quantum groups $^{\imath}\mathbf U(S)$ of type $\mathrm{AIII}$. 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 $^{\imath}\mathbf U(S)$.