Soergel calculus for monodromic Hecke categories

Abstract: We introduce two 2-categories which categorify the monodromic Hecke algebra. The first is algebraic in nature and generalizes Abe's theory of Soergel bimodules. The second is a diagrammatic category defined via generators and relations which generalizes the Elias-Williamson diagrammatic calculus. As our first main result, we prove that these algebraic and diagrammatic categorifications are equivalent, extending an earlier theorem of Abe. Furthermore, we relate these new categorifications to a third categorification via parity sheaves which was previously studied by the author. More precisely, we provide a monodromic analogue of a theorem of Riche and Williamson to show that the diagrammatic category is equivalent to the monodromic Hecke category of parity sheaves associated to a reductive group. Finally, we show that these monodromic Hecke categories can be described by unipotent Hecke categories associated to endoscopic Coxeter groups.