Families of the characters of the cyclotomic Hecke algebras of G(de,e,r)

Motivated by the work of Lusztig on the families of Irr(W) for the Weyl group W of any type, especially for the types Br and Dr, we described in the article [M. Broué and S. Kim, Familles de caractères des algèbres de Hecke cyclotomiques, Adv. Math. 172 (2002) 53–136] the families of the characters of G(e,1,r) and G(e,e,r), which are the blocks of the cyclotomic Hecke algebras over an appropriate ring called “Rouquier ring.” The results coincide with the results of Lusztig on the Weyl groups of types Br and Dr when specializing the parameters of the Hecke algebras. In this paper, generalizing this work to the whole series of G(de,e,r) which is a normal subgroup of G(de,1,r) of index e, we extend the results of Broué and Kim, op. cit.