On Complex Reflection Groups and Their Associated Braid Groups

Presentations \\ a la Coxeter" are given for all (irreducible) nite complex reeec-tion groups. They provide presentations for the corresponding generalized braid groups (still conjectural in some cases) which allow us to generalize some of the known properties of nite Coxeter groups (center of the braid group, construction of Hecke algebras). 1. Background from complex reflection groups For all the results quoted here, we refer the reader to the classical literature on complex reeections groups, such as Bou], Ch], Co], ShTo], Sp], and also to the more recent and fundamental work on the subject by Orlik, Solomon and Terao (see OrSo], OrTe]). Let V be a complex vector space of dimension r. A pseudo{reeection of GL(V) is a non trivial element of GL(V) which acts trivially on a hyperplane, called the reeecting hyperplane of. Let G be a nite subgroup of GL(V) generated by pseudo{reeections. The pair (V; G) is called a \complex reeection group". A parabolic subgroup of G is by deenition the subgroup of elements of G which act trivially on a subspace of V. By a theorem of Steinberg ((St], Theorem 1.5), a parabolic subgroup is generated by pseudo{reeections. We denote by A the set of reeecting hyperplanes of G, and we set N := jAj. For H 2 A, we denote by G H the xator (pointwise stabilizer) of H (a minimal parabolic subgroup of G), and we set e H := jG H j. We denote by N the number of pseudo-reeections in G. This is a summary of work in progress. The nal version of this paper will be submitted for publication elsewhere. We thank Jean Michel for useful conversations and for his help in preparing this manuscript.