The cells of the affine Weyl group C˜n in a certain quasi-split case

The affine Weyl group (C˜n,S) can be realized as the fixed point set of the affine Weyl group (A˜2n−1,S˜) under a certain group automorphism α with α(S˜)=S˜. Let ℓ˜ be the length function of A˜2n−1. We study the cells of the weighted Coxeter group (C˜n,ℓ˜). The main results of the paper are to give an explicit description for all the cells of (C˜n,ℓ˜) corresponding to the partitions k12n−k and h212n−h−2 for all 1⩽k⩽2n and 2⩽h⩽2n−2, and also for all the cells of (C˜3,ℓ˜).