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

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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.