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 $W$ .