Minimal and maximal elements in Kazhdan-Lusztig double sided cells of Sn and a Robinson-Schensted correspondance in simply laced Coxeter groups

Let W be a Coxeter group. We generalise the Knuth plactic relations to W and we show that their equivalence classes decompose the Kazhdan-Lusztig cells. In the case of symmetric groups, we show that the set of elements of minimal length in a double sided cell is the set of elements of maximal length in conjugated parabolic (i.e. Young) subgroups. We also give an interpretation of this set with tableau, using the Robinson-Schensted correspondance. We show also that the set of elements of maximal length in a two sided cell is the set of longest minimal right coset representatives of conjuagted parabolic subgroups. In the case where W is simply laced, we generalise the RobinsonSchensted correspondance to W .