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

Abstract: We establish a multiplication formula for a tridiagonal standard basis element in the idempotented coideal subalgebras of quantum affine $\mathfrak{gl}n$ arising from the geometry of affine partial flag varieties of type $C$. We apply this formula to obtain the stabilization algebras $\dot{\mathbf K}^{\mathfrak{c}}_n$, $\dot{\mathbf K}^{\jmath \imath}{\mathfrak{n}}$, $\dot{\mathbf K}^{\imath \jmath}{\mathfrak{n}}$ and $\dot{\mathbf K}^{\imath \imath}{\eta}$, which are idempotented coideal subalgebras of quantum affine $\mathfrak{gl}n$. 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.