Does anyone know if there exists some natural way to interpret the Kazhdan-Lustiz C-basis in a categorification of the Hecke algebra ? The category of Soergel bimodules categorifies the C'-basis but it doesn't seem clear to me whether this or some other related structure (Rouquier complexes, sheaves...) could be adapted to categorify the C-basis instead ? I am currently working on some elements of the Hecke algebra which seem to have remarkable positivity properties when written in the C-basis (---> (-1) signs occur when writing them in the C'-basis).