  Following Lusztig, we consider a Coxeter group $W$ together with a weight
function $L$. This gives rise to the pre-order relation $\leq_{L}$ and the
corresponding partition of $W$ into left cells. We introduce an equivalence
relation on weight functions such that, in particular, $\leq_{L}$ is constant
on equivalent classes. We shall work this out explicitly for $W$ of type $F_4$
and check that several of Lusztig's 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 {\sf CHEVIE}.

Representation Theory