available there.

It turns out the action of $\U^{\imath}_q(\mathfrak{sl}_{2r+1})$ (that is $k=2r+1$, being an odd number) on $(\VV^*)^{\otimes n}$, tensor product of the restricted dual $\VV^*$ of the natural representation $\VV$ of $\U_q(\mathfrak{sl}_{2r+1})$, centralizes the action of $\mc{H}^q_{B_n}$ ($p=q$) on $(\VV^*)^{\otimes n}$. It is actually more suitable to consider $\mc{H}^q_{B_n}$ as the Hecke algebra $\mc{H}_{C_n}$ of type C (of equal parameters) due to the connection with the BGG category $\mc{O}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$ (in particular, the specialization $\mathfrak{sp}(2n)$ when $m =0$).

In [@BW13] we considered the Hecke algebra $\mathcal{H}^{q}_{B_m}$ with equal parameters $p =q$. It was showed there that certain coideal subalgebra of the quantum group $\U_q(\mathfrak{sl}_{k})$ of type A forms double centralizers with $\mathcal{H}^{q}_{B_m}$ when acting on the tensor space $\VV^{\otimes m}$. In this paper, we consider the Hecke algebra $\mathcal{H}^{1}_{B_m}$ with $p =1$. Thus the (different) centralizing coideal subalgebra $\U^{\imath}_q(\mathfrak{sl}_{k})$ of $\U_q(\mathfrak{sl}_{k})$ consider in this paper is of different parameters than the one considered in [@BW13], where the choice of the parameters in $\U^{\imath}_q(\mathfrak{sl}_{k})$ corresponds to the choice of the parameter $p$ in the two parameters Hecke algebra $\mathcal{H}^{p}_{B_m}$.

The construction of $\imath$-canonical bases developed in [@BW13] applies to the coideal subalgebras with different parameters without 