Computing Kazhdan–Lusztig cells for unequal parameters

Following Lusztig, we consider a Coxeter group W together with a weight function L. This gives rise to the pre-order relation L and the corresponding partition of W into left cells. We introduce an equivalence relation on weight functions such that, in particular, L is constant on equivalent classes. We shall work this out explicitly for W of type F4 and check that several of Lusztigʹs conjectures concerning left cells with unequal parameters hold in this case, even for those parameters which do not admit a geometric interpretation. The proofs involve some explicit computations using CHEVIE.