Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_qL\mathfrak{g}$ the corresponding quantum affine algebra. We construct a functor ${}^{\theta}{\sf F}$ between finite-dimensional modules over a quantum symmetric pair of affine type $U_q\mathfrak{k}\subset U_qL{\mathfrak{g}}$ and an orientifold KLR algebra arising from a framed quiver with a contravariant involution, providing a boundary analogue of Kang-Kashiwara-Kim-Oh generalized Schur-Weyl duality. With respect to their construction, our combinatorial model is further enriched with the poles of a trigonometric K-matrix intertwining the action of $U_q\mathfrak{k}$ on finite-dimensional $U_qL{\mathfrak{g}}$-modules. By construction, ${}^{\theta}{\sf F}$ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, ${}^{\theta}{\sf F}$ is a functor of module categories. Relying on a suitable isomorphism \`a la Brundan-Kleshchev-Rouquier, we prove that ${}^{\theta}{\sf F}$ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type $\sf AIII$.