Split and Quasi-Split Cases

Abstract

First, we recall briefly how the category of Soergel bimodules categorifies the Hecke algebra and that, through this categorification, indecomposable Soergel bimodules correspond to the Kazhdan–Lusztig basis (Elias–Williamson Theorem). We explain how this can be extended to the quasi-split case, following Lusztig. We then show how these results on the category of Soergel bimodules allow one to show that Lusztig’s Conjectures (P1), (P2), ..., (P14) hold in the split and the quasi-split case, and that (P15) holds in the split case.