Isaac Newton Institute for Mathematical Sciences

Complex reflection groups and their associated braid groups and Hecke algebras

Author: M Broue (Institut Henri Poincaré )

Abstract

Weyl groups are finite groups acting as reflection groups on rational vector spaces. These rational reflection groups appear as the ``skeleton'' of many important mathematical objects, like algebraic groups, Hecke algebras, Artin--Tits braid groups, etc. By extension of the base field, Weyl groups may be viewed as particular cases of finite complex reflection groups, which have been characterized by Shephard--Todd and Chevalley. It has been recently discovered that complex reflection groups (not only Weyl groups) play a key role in the structure as well as in the representation theory of finite reductive groups. In the meantime, it has appeared that many of the known properties of Weyl groups can be generalized to complex reflection groups -- although in most cases new methods have to be found. We shall give a survey of what is known in that direction, from properties of the complex reflection groups themselves to properties of their associated braid groups, and the various Hecke algebras (among which the ``cyclotomic Hecke algebras'') which they give rise to.