The two-color Soergel calculus

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 sl₂ . 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 .