Trihedral Soergel bimodules

Abstract

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.

61 pages, many colored figures, revised version, comments welcome, to appear in Fund. Math