The cells in the weighted Coxeter group (C˜n,ℓ˜m)

The affine Weyl group (C˜n,S) can be realized as the fixed point set of the affine Weyl group (A˜m,S˜m), m∈{2n−1,2n,2n+1}m∈{2n−1,2n,2n+1}, under a certain group automorphism αm,nαm,n. Let ℓ˜m be the length function of A˜m. The aim of the present paper is to give a combinatorial description for all the left cells of A˜m which have non-empty intersection with C˜n. Then we use this description to deduce some formulae for the number of left cells of the weighted Coxeter group (C˜n,ℓ˜m) in the set EλEλ associated to any partition λ   of m+1m+1.