Semicontinuity properties of Kazhdan-Lusztig cells
  Computations in small Coxeter groups or dihedral groups suggest that the partition into Kazhdan-Lusztig cells with unequal parameters should obey to some semicontinuity phenomenon (as the parameters vary). The aim of this paper is to provide a rigorous theoretical background for supporting this intuition that will allow to state several precise conjectures. 
C\'edric Bonnaf\'e (LM-Besan\c{c}on)