Abstract: Let \alpha be a group automorphism of the affine Weyl group (\widetilde{A}_{2n},\widetilde{S}) with \alpha(\widetilde{S})=\widetilde{S}. Affine Weyl group(\widetilde{C}_n,S) can be seen as the fixed point set of the affine Weyl group (\widetilde{A}_{2n},\widetilde{S}) under its group automorphism \alpha. The restriction to \widetilde{C}_n of the length function \widetilde{l}_{2n} on \widetilde{A}_{2n} can be seen as a weight function on \widetilde{C}_n. In this paper, we
give the description for all the left and two-sided cells of the specific weighted Coxeter group (\widetilde{C}_3,\widetilde{l}_6) and prove that each left cell in (\widetilde{C}_3,\widetilde{l}_6) is left-connected.

Key words: affine Weyl group, weighted Coxeter group, quasi-split case, partitions of n, left cells