type D as a subalgebra.

Then we look for the subalgebra of $\U_q(\mathfrak{sl}_k)$, whose action on the tensor product $\VV^{\otimes m}$ centralizing the action of the Hecke algebra $\mathcal{H}^{1}_{B_m}$ on $\VV^{\otimes m}$. The subalgebra is a coideal subalgebra of the quantum group $\U_q(\mathfrak{sl}_{k})$ of type A, denoted by $\U^{\imath}_q(\mathfrak{sl}_{k})$. Since the Hecke algebra $\mathcal{H}^{1}_{B_m}$ contains $\mc{H}_{D_m}$ as a subalgebra, the actions of $\U^{\imath}_q(\mathfrak{sl}_{k})$ and $\mc{H}_{D_m}$ on the tensor space $\VV^{\otimes n}$ clearly commute.

The coideal subalgebra comes in different forms depending on the parity of $k$. The quantum group $\U_q(\mathfrak{sl}_{k})$ and the coideal subalgebra $\U^{\imath}_q(\mathfrak{sl}_{k})$ form a quantum symmetric pair ([@Ko14]).

Ehrig and Stroppel used the same coideal subalgebra ${\U^{\imath}}_q(\mathfrak{sl}_{k})$ of the quantum group $\U_q(\mathfrak{sl}_{k})$ to study the parabolic category $\mc{O}$ of the Lie algebra $\mathfrak{so}(2m)$ in [@ES13] independently and simultaneously from [@BW13]. They established the commutativity between the actions of ${\U^{\imath}}_q(\mathfrak{sl}_{k})$ and $\mathcal{H}_{D_m}$ on the tensor space $\VV^{\otimes m}$. The actions of the Chevalley type generators of ${\U^{\imath}}_q(\mathfrak{sl}_{k})$ on $\VV^{\otimes m}$ have been identified with the actions of translation functors on the category $\mc{O}$ of the Lie algebra $\mathfrak{so}(2m)$. However, neither the establishment of the $\imath$-canonical basis nor the (re)formulation of the (super) Kazhdan-Lusztig theory of type D was 