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

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⋅wJ⋅y for some x,y∈W and some J⊆S with WJ finite and ℓ(wJ)=a(w) (see 0.1–0.2 for the notation wJ, WJ, 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 dL of W in L satisfies for any z=wJ⋅z′∈Emin(L) with J=L(z) (see 1.6. for the notation λ(z−1,z), and 0.3. for L(z), Emin(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)=Emin(L) then we show that the left cell L is left-connected, verifying a conjecture of Lusztig in our case.