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

## TL;DR

This paper develops a multiplication formula for certain basis elements in quantum affine algebras related to affine flag varieties, leading to new stabilization algebras and isomorphisms of coideal subalgebras with compatible bases.

## Contribution

It introduces a multiplication formula for standard basis elements in quantum affine coideal subalgebras and constructs new stabilization algebras with isomorphic structures.

## Key findings

- Established a multiplication formula for standard basis elements.
- Constructed stabilization algebras from affine flag variety geometry.
- Proved isomorphisms between coideal subalgebras with compatible bases.

## 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.