On Hecke Algebras

Abstract

We will start this chapter by giving the definition and the classification of complex reflection groups. We will also define the braid group and the pure braid group associated to a complex reflection group. We will then introduce the generic Hecke algebra of a complex reflection group, which is a quotient of the group algebra of the associated braid group defined over a Laurent polynomial in a finite number of indeterminates. Under certain assumptions, which have been verified for all but a finite number of cases, we prove (Theorem 28) that the generic Hecke algebras of complex reflection groups are essential. Therefore, all results obtained in Chapter 3 apply to the case of the generic Hecke algebras. A cyclotomic Hecke algebra is obtained from the generic Hecke algebra via a cyclotomic specialization (Definition 26). We prove (Theorem 30) that any cyclotomic specialization is essentially an adapted morphism. Thus, we can use Theorem 21 in order to obtain the Rouquier blocks of a cyclotomic Hecke algebra (i.e., its blocks over the Rouquier ring, defined in Section 4.4), which are a substitute for the families of characters that can be applied to all complex reflection groups. We will see that the Rouquier blocks have the property of semi-continuity, thus depending only on some “essential” hyperplanes for the group, which are determined by the generic Hecke algebra. The theory developed in this chapter will allow us to determine the Rouquier blocks of the cyclotomic Hecke algebras of all (irreducible) complex reflection groups in the next chapter.