Generalized induction of Kazhdan-Lusztig cells

TL;DR

This paper generalizes the concept of Kazhdan-Lusztig cells in Coxeter groups, extending their compatibility beyond parabolic subgroups and applying this to affine Weyl groups for various weight functions.

Contribution

It introduces a generalized approach to Kazhdan-Lusztig cells, not limited to parabolic subgroups, and applies it to decompose affine Weyl groups into cells.

Findings

Cells of certain finite parabolic subgroups are also cells in the whole group under specific conditions.

Decomposition of affine Weyl group G2 into left and two-sided cells for various weight functions.

Abstract

Following 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 $\tilde{G}_{2}$ into left and two-sided cells for a whole class of weight functions.