Some cells in the weighted Coxeter group (C̃n,ℓ̃2n)

The affine Coxeter group ([Formula: see text]) can be realized as the fixed point set of the affine Coxeter group ([Formula: see text]) under a certain group automorphism [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the length function of [Formula: see text]. Then the left and two-sided cells of the weighted Coxeter group ([Formula: see text]) can be described explicitly as subsets of ([Formula: see text]). We study the cells of ([Formula: see text]) in the set [Formula: see text] with [Formula: see text] for any [Formula: see text] with [Formula: see text]. Our main result is to show that [Formula: see text] is a two-sided cell of [Formula: see text] which is two-sided connected and that any left cell of [Formula: see text] in [Formula: see text] is left-connected.

On Certain Distinguished Involutions in the Weyl Group of Type Bn

Let (W,S) be a Weyl group of type Bn. In this paper, we find certain nontrivial distinguished involutions in the two-sided cell Ωt of W with a-value [Formula: see text] for 1≤ 2t ≤ n and n even (resp., [Formula: see text] for 1≤ 2t < n and n odd). Besides, for 1≤ i1< ⋯ < it≤ n-1, let Li1⋯ it be a left cell of W with a-value t2 and R(Li1⋯ it) ={si1,…,sit}. There are only two involutions y(i1,i1)⋯ y(it-1,it-1) sit and y(i1,i1)⋯ y(it,it) in the left cell Li1⋯ it. We prove that the former (resp., latter) is a distinguished involution in the left cell Li1⋯ it for t odd (resp., even).