simple reflections have different left (right) descent sets and hence are not right (left) equivalent.

Further, if $w\in W$ has more than one reduced expression, then any such reduced expression can be obtained from any other by means of the braid relations, see [@Bo §IV.1.5]. This means that, with respect to the Bruhat order, $w$ is bigger than or equal to the longest element in some parabolic subgroup of $W$ of rank two. As longest element in parabolic subgroup of $W$ of rank two are not in $\mathcal{J}$ (since the value of $\mathbf{a}$ on such elements is strictly bigger than $1=\mathbf{a}(s)$, where $s\in S$), we get $w\not\in\mathcal{J}$.

By induction with respect to the rank of $W$, one checks that any two elements of the form $xs$ and $ys$ (resp. $sx$ and $sy$) which, moreover, have unique reduced expressions, belong to the same left (resp. right) cell. Indeed, the basis of the induction is the rank two case which is standard. The induction step follows from the relation $\theta_s\theta_t\theta_s=\theta_{sts}\oplus \theta_s$ which is true for any pair $s$, $t$ of non-commuting simple reflections. This, combined with the observation in the previous paragraph, yields , , and .

That simple reflections are Duflo 