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

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_{q}L\mathfrak{g}$ the corresponding quantum affine algebra. We construct a functor ${}^{\theta }\mathsf{F}$ between finite-dimensional modules over a quantum symmetric pair of affine type $U_q\mathfrak{k}\subset U_{q}L\mathfrak{g}$ 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 $U_{q}L\mathfrak{g}$-modules. By construction, ${}^{\theta }\mathsf{F}$ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, ${}^{\theta }\mathsf{F}$ is a functor of module categories. Relying on a suitable isomorphism à la Brundan-Kleshchev-Rouquier, we prove that ${}^{\theta }\mathsf{F}$ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type $\mathsf{AIII}$.

Final version. Substantial revision: new examples of enhanced J-quivers of Dynkin type have been added in Section 6; the material of the former Sections 6,7, and 11 from v2 has been removed and it will be included in a separate paper. 50 pages

References in corpus (4)

• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Universal K-matrices for quantum Kac-Moody algebras
• Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types
• Representations of orientifold Khovanov-Lauda-Rouquier algebras and the Enomoto-Kashiwara algebra