In the case of a Weyl group \(W\), a partition into families of the irreducible characters has been defined by Lusztig, and Rouquier has shown that these families are in fact blocks of characters of the classical Hecke algebra of \(W\) over an appropriate coefficient ring. In this paper, the authors generalize to the case of complex reflection groups Rouquier’s definition of families of characters, and they show that this notion depends essentially on the choice of the deformation of a cyclotomic algebra.
The first part of the paper is a detailed and clear account of the general block theory, especially of blocks and characters of symmetric algebras over integrally closed Noetherian rings. The second part is devoted to the presentation of the cyclotomic Hecke algebras and their Rouquier blocks.
The authors also describe the Rouquier blocks for all cyclotomic algebras of infinite families of complex reflection groups which generalize infinite families of Weyl groups. The third part of the paper deals with the cyclotomic Ariki-Koike algebras, which are the cyclotomic Hecke algebras of the groups \(C_d\wr{\mathfrak S}_r\), also called of type \(G(d,1,r)\). The final part deals with the cyclotomic Hecke algebras of the complex reflection groups of type \(G(d,d,r)\), which are generalizations of the Weyl groups of type \(D_r\) and of the dihedral groups.
Note that the original difficulty in the generalization of the concept of family consists in the fact that while Lusztig’s definition uses the Kazhdan-Lusztig basis of the Hecke algebra of \(W\), such bases are not available for complex reflection groups. However, the results of this paper suggest that “Kazhdan-Lusztig basis” might exist for reflection groups other than Coxeter groups and for their cyclotomic algebras.