A counter-example to a conjecture of Lusztig

In Lusztig (2003) [7], Lusztig defined two functions a and a′ on a Coxeter group W and conjectured that they are the same. In the present paper, we give a counter-example to Lusztig's conjecture. Then we propose a new conjecture for the description of two-sided cells of W which can be regarded as an improvement of Lusztig's conjecture. Some examples are given to support our conjecture.