Complex reflection groups, braid groups, Hecke algebras

Abstract

Presentations a la Coxeter are given for all irreducible nite com plex re ection groups They provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite Coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra Hecke algebras We introduce monodromy representations of the braid groups which factorize through the Hecke algebras extending results of Cherednik Opdam Kohno and others Summary Introduction Complex re ection groups and their presentations A Background from complex re ection groups B Presentations Braid groups and their diagrams A Generalities about hyperplane complements B Generalities about the braid groups C The braid diagrams Proofs of the main theorems for the braid groups B de e r A Notation and prerequisites B Computation of B de e r and of its center for d C Computation of B e e r and of its center Hecke algebras A Background from di erential equations and monodromy B A family of monodromy representations of the braid group C Hecke algebras Diagrams and tables Appendix Generators of the monodromy around a divisor Appendix tables to Mathematics Subject Classi cation Primary G We thank Jean Michel Peter Orlik Pierre Vogel for useful conversations and the Isaac Newton Institute for its hospitality while the last version of this manuscript was written up The second named author gratefully acknowledges nancial support by the Fondation Alexander von Humboldt for his stays in Paris Michel Brou e Gunter Malle Raphael Rouquier