Integral u-deformed involution modules

Let (W,S) be a Coxeter system and ⁎ an automorphism of W with order ≤2 and S ^⁎ =S. Lusztig and Vogan ([20], [23]) have introduced a u-deformed version M _u of Kottwitz's involution module over the Iwahori–Hecke algebra H _u (W) with Hecke parameter u ² , where u is an indeterminate. Lusztig has proved that M _u is isomorphic to the left H _u (W)-submodule of Hˆ _u generated by X _∅ :=∑ _{w ^⁎ =w∈W} u ^−ℓ(w) T _w , where Hˆ _u is the vector space consisting of all formal (possibly infinite) sums ∑ _x∈W c _x T _x (c _x ∈Q(u) for each x). He speculated that one can extend this by replacing u with any λ∈C∖{0,1,−1}. In this paper, we give a positive answer to his speculation for any λ∈K∖{0,1,−1} and any W, where K is an arbitrary ground field.