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Let (W,S) be a Coxeter system, let G be a finite solvable group of automorphisms of (W, S) and let ϕ be a weight function which is invariant under G. Let ϕ G denote the weight function on WG obtained by restriction from ϕ. The aim of this paper is to compare the a-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated with (W, ϕ) and to (WG ,ϕ G), provided that Lusztig's Conjectures hold.doi:10.1017/s0027763000009752 fatcat:b2vqs6b6w5forisswxjf3hx7xq