The two-color Soergel calculus

Abstract

We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group$W$. The (two-colored) Temperley–Lieb category is embedded inside this category as the degree$0$morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones–Wenzl projectors. When$W$is infinite, the parameter$q$of the Temperley–Lieb algebra may be generic, yielding a quantum version of the geometric Satake equivalence for$\mathfrak{sl}_{2}$. When$W$is finite,$q$must be specialized to an appropriate root of unity, and the negligible Jones–Wenzl projector yields the Soergel bimodule for the longest element of$W$.