Introduction

The purpose of these lectures is to introduce the audience to the theory of cyclotomic Hecke algebras of type G(m, 1, n). These algebras were introduced by the author and Koike, Broué and Malle independently. As is well known, group rings of Weyl groups allow certain deformation. It is true for Coxeter groups, which are generalization of Weyl groups. These algebras are now known as (Iwahori) Hecke algebras.

Less studied is its generalization to complex reflection groups. As I will explain later, this generalization is not artificial. The deformation of the group ring of the complex reflection group of type G(m, 1, n) is particularly successful. The theory uses many aspects of very modern development of mathematics: Lusztig and Ginzburg's geometric treatment of affine Hecke algebras, Lusztig's theory of canonical bases, Kashiwara's theory of global and crystal bases, and the theory of Fock spaces which arises from the study of solvable lattice models in Kyoto school.

This language of Fock spaces is crucial in the theory of cyclotomic Hecke algebras. I would like to mention a little bit of history about Fock spaces in the context of representation theoretic study of solvable lattice models. For level one Fock spaces, it has origin in Hayashi's work. The Fock space we use is due to Misra and Miwa. For higher levelFock spaces, they appeared in work of Jimbo, Misra, Miwa and Okado, and Takemura and Uglov.