Trihedral Soergel bimodules

The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum $mathfrak{sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $mathsf{ADE}$ Dynkin diagrams. Using the quantum Satake correspondence between affine $mathsf{A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $mathfrak{sl}_3$ representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan-Lusztig combinatorics, simple transitive $2$-representations corresponding to tricolored generalized $mathsf{ADE}$ Dynkin diagrams.