THE CELLS OF THE AFFINE WEYL GROUP C̃n IN A CERTAIN QUASI-SPLIT CASE

The affine Weyl group ( e Cn, S) can be realized as the fixed point set of the affine Weyl group ( e A2n−1, e S) under a certain group automorphism α with α( S) = e S. Let e ` be the length function of e A2n−1. We study the cells of the weighted Coxeter group ( e Cn, e `). The main results of the paper are to give an explicit description for all the cells of ( e Cn, e `) corresponding to the partitions k12n−k and h212n−h−2 for all 1 6 k 6 2n and 2 6 h 6 2n − 2, and also for all the cells of ( e C3, e `). §0. Introduction. 0.1. In his book [8], Lusztig introduced a weighted Coxeter group (W,L), which is, by definition, a Coxeter system (W,S) together with a weight function L : W −→ Z. He proposed a bundle of conjectures, intending to generalize many results on cells of W in the equal parameter case to the unequal parameters case. The most successful part for such a generalization is when (W,L) is in a certain quasi-split case, that is, W can be realized as the fixed point set of a finite or an affine Coxeter system (W̃ , S̃) under a group automorphism α with α(S̃) = S̃, the weight function L is the restriction to W of the length function ̃̀ of W̃ (see [8, Chapter 16], [6], [2], [4]).