Some cells in the weighted Coxeter group (C˜n,ℓ˜2n+1)

The fixed point set of the affine Weyl group (A˜2n+1,S˜) under a certain group automorphism α with α(S˜)=S˜ can be considered as the affine Weyl group (C˜n,S). Then we study the cells of the weighted Coxeter group (C˜n,ℓ˜2n+1) with ℓ˜2n+1 the length function of A˜2n+1. We give an explicit description for all the cells of (C˜n,ℓ˜2n+1) corresponding to the partitions k12n+2−k and (h,2n+2−h) for any 1≤k≤2n+2 and n+1≤h≤2n, and also for all the cells of (C˜3,ℓ˜7).