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 ${\dot{\mathbf{K}}}_n^{\mathfrak{c}}$, ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{ȷı}$, ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{ıȷ}$ and ${\dot{\mathbf{K}}}_{\eta }^{ıı}$, 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 ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{ȷı}$ and ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{ıȷ}$ with compatible monomial, standard and canonical bases.

39 pages, fixed some typos, to appear in Journal of Pure and Applied Algebra

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