Abstract:

The fixed point set of the affine Weyl group $(\widetilde{A}_{2n},\widetilde{S})$ under its group automorphism $\alpha$ with $\alpha(\widetilde{S})=\widetilde{S}$ can be seen as the affine Weyl group $(\widetilde{C}_n,S)$. 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 the present paper, by studying the fixed point set of the affine Weyl group $(\widetilde{A}_{2n},\widetilde{S})$ under $\alpha$, we give the description for all the cells of the weighted Coxeter group $(\widetilde{C}_{n},\widetilde{l}_{2n})$ corresponding to the specific partition $\bf{3^32^{n-4}}$. We also prove that each left cell we considered in this paper is left-connected, verifying a conjecture of Lusztig in our case.

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