Abstract: The fixed point set of the affine Weyl group
($\widetilde{A}_{2n},\widetilde{S}$) under a certain group
automorphism $\alpha$ with $\alpha\,(\widetilde{S}) = \widetilde{S}$
can be considered as the affine Weyl group ($\widetilde{C}_n,S$).
Then the left and two-sided cells of the weighted Coxeter group
($\widetilde{C}_n,\widetilde{\ell}$), where $\widetilde{\ell}$ is
the length function of $\widetilde{A}_{2n}$, can be given an
explicit description  by studying the fixed point set of the affine
Weyl group ($\widetilde{A}_{2n},\widetilde{S}$) under $\alpha$. We
describe the cells of ($\widetilde{C}_n,\widetilde{\ell}$)
corresponding to the partitions
$\textbf{k}\textbf{1}^{\textbf{2n+1-k}}$ with $1\leqslant k
\leqslant 2n+1$ and $(2n-1,2)$.

Key words: affine Weyl groups, left cells, quasi-split case, weighted Coxeter group