Left-connectedness of some left cells in certain Coxeter groups of simply-laced type

Abstract

Let W be an irreducible finite or affine Weyl group of simply-laced type. We show that any w ∈ W with a ( w ) ⩽ 6 satisfies Condition (C): w = x ⋅ w J ⋅ y for some x , y ∈ W and some J ⊆ S with W J finite and ℓ ( w J ) = a ( w ) (see 0.1–0.2 for the notation w J , W J , ℓ ( w ) and a ( w ) ). We also show that if L is a left cell of W all of whose elements satisfy Condition (C), then the distinguished involution d L of W in L satisfies d L = λ ( z −1 , z ) = z ′ − 1 ⋅ w J ⋅ z ′ for any z = w J ⋅ z ′ ∈ E min ( L ) with J = L ( z ) (see 1.6. for the notation λ ( z −1 , z ) , and 0.3. for L ( z ) , E min ( L ) and E ( L ) ), verifying a conjecture of mine in [J.Y. Shi, A survey on the cell theory of affine Weyl groups, Adv. Sci. China Math. 3 (1990) 79–98, Conjecture 8.10] in our case. If E ( L ) = E min ( L ) then we show that the left cell L is left-connected, verifying a conjecture of Lusztig in our case.