Affine flag varieties and quantum symmetric pairs, II. Multiplication formula

We establish a multiplication formula for a tridiagonal standard basis element in the idempotented coideal subalgebras of quantum affine ${\mathfrak{g}\mathfrak{l}}_n$ arising from the geometry of affine partial flag varieties of type $C$. We apply this formula to obtain the stabilization algebras ${\dot{\mathbf{K}}}_n^{\mathfrak{c}}$, $\dot{\mathbf K}^{\jmath \imath}{\mathfrak{n}}$,$\dot{\mathbf K}^{\imath \jmath}{\mathfrak{n}}$and$\dot{\mathbf K}^{\imath \imath}{\eta}$, which are idempotented coideal subalgebras of quantum affine ${\mathfrak{g}\mathfrak{l}}_n$. The symmetry in the formula leads to an isomorphism of the idempotented coideal subalgebras $\dot{\mathbf K}^{\jmath \imath}{\mathfrak{n}}$and$\dot{\mathbf K}^{\imath \jmath}_{\mathfrak{n}}$ with compatible monomial, standard and canonical bases.