two-sided cells in $W$, and, moreover, we have the equality $\mathbf{a}(w)=\mathbf{l}(w)$ provided that $w$ is the longest element in some parabolic subgroup of $W$.

\[propJKL\]

1.  \[propJKL.1\] The map $\mathcal{L}\mapsto \mathcal{L}\cap S$ is a bijection between the set of left cells in $\mathcal{J}$ and $S$.

2.  \[propJKL.2\] For any left cell $\mathcal{L}$ in $\mathcal{J}$, the unique element in $\mathcal{L}\cap S$ is the Duflo involution in $\mathcal{L}$.

3.  \[propJKL.3\] The map $\mathcal{R}\mapsto \mathcal{R}\cap S$ is a bijection between the set of right cells in $\mathcal{J}$ and $S$.

4.  \[propJKL.4\] For any right cell $\mathcal{R}$ in $\mathcal{J}$, the unique element in $\mathcal{R}\cap S$ is the Duflo involution in $\mathcal{R}$.

5.  \[propJKL.5\] An element $w\in W$ such that $w\neq e$ belongs to $\mathcal{J}$ if and only if $w$ has a unique reduced expression.

6.  \[propJKL.6\] An element $w\in W$ belongs to $\mathcal{J}$ if and only if $\mathbf{a}(w)=1$.

If $W$ is a Weyl group, all these results are mentioned in [@Do] with references to [@Lu1; @Lu2; @Lu3]. We note the difference in both the normalizations of the Hecke algebra and the choice of the Kazhdan-Lusztig basis in this paper and in [@Do]. Below, we outline a general argument.

We start by observing that different 