(Co)Minuscule Hecke categories

We determine the p-Kazhdan-Lusztig bases for antispherical (co)minuscule Hecke categories in all characteristics, and for spherical (co)minuscule Hecke categories in good characteristic. This is achieved using geometric and diagrammatic methods. The 2-Kazhdan-Lusztig bases of antispherical cominuscule Hecke categories exhibit extremely pathological behaviour. The notions of p-small resolutions and p-tight elements are introduced and conjecturally explain this behaviour.