Generalized induction of Kazhdan-Lusztig cells

International audienceFollowing Lusztig, we consider a Coxeter group W together with a weight function. Geck showed that the Kazhdan-Lusztig cells of W are compatible with parabolic subgroups. In this paper, we generalize this argument to some subsets of W which may not be parabolic subgroups. We obtain two applications: we show that under specific technical conditions on the parameters, the cells of a certain finite parabolic subgroup of W are cells in the whole group, and we decompose the affine Weyl group G~2 into left and two-sided cells for a whole class of weight functions