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 introducedby 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 level Fock spaces, they appeared in work of Jimbo, Misra, Miwa and Okado, and Takemura and Uglov. We also note that Varagnolo and Vasserot’s version of level one Fock spaces have straight generalization to higher levels and coincide with the Takemura and Uglov’s one. The Fock spaces we use are different from them. But they are essential in the proofs.
Since the cyclotomic Hecke algebras contain the Hecke algebras of type \(A\) and type \(B\) as special cases, the theory of cyclotomic Hecke algebras is also useful to study the modular representation theory of finite classical groups of Lie type.
I shall explain theory of Dipper and James, and its relation to our theory. The relevant Hecke algebras are Hecke algebras of type \(A\). In this case, we have an alternative approach depending on Lusztig’s conjecture on quantum groups, by virtue of Du’s refinement of Jimbo’s Schur-Weyl reciprocity. Even for this rather well studied case, our viewpoint gives a new insight. This viewpoint first appeared in work of Lascoux, Leclerc and Thibon. This Fock space description looks quite different from the Kazhdan-Lusztig combinatorics, since it hides affine Kazhdan-Lusztig polynomials behind the scene. Inspired by this description, Goodman and Wenzl have found a faster algorithm to compute these polynomials. Leclerc and Thibon are key players in the study of this type \(A\) case. I also would like to mention Schiffman and Vasserot’s work here, sinceit makes the relation of canonical bases between modified quantum algebras and quantized Schur algebras very clear.
I will refer to work of Geck, Hiss, and Malle a little if time allows, since we can expect future development in this direction. It is relevant to Hecke algebras of type \(B\). Finally, I will end the lectures with Broué’s famous dream.
Detailed references can be found at the end of these lectures. The first three are for overview, and the rest are selected references for the lectures.
For the entire collection see [Zbl 0980.00028].