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

We establish a multiplication formula for a tridiagonal standard basis element in the idempotent version, i.e., the Lusztig form, of the coideal subalgebras of quantum affine gl _n arising from the geometry of affine partial flag varieties of type C. We apply this formula to obtain the stabilization algebras K˙ _n ^c , K˙ _n ^ȷı , K˙ _n ^ıȷ and K˙ _η ^ıı , which are idempotented coideal subalgebras of quantum affine gl _n . The symmetry in the formula leads to an isomorphism of the idempotented coideal subalgebras K˙ _n ^ȷı and K˙ _n ^ıȷ with compatible monomial, standard and canonical bases.