Structure and representation of cyclotomic Hecke algebras

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The proposed research concerns the structure and representation theory of an important class of finite dimensional algebras, the so-called cyclotomic Hecke algebras. These algebras were introduced fifteen years ago as deformations of group algebras of complex reflection groups. They are analogues of the Iwahori–Hecke algebras which play a central role in the representation theory of finite groups of Lie type. It has become apparent in recent years that many properties of Iwahori–Hecke algebras have counterparts in the theory of cyclotomic Hecke algebras. Often, though, new approaches have to be found for their proof, different from the one for Iwahori–Hecke algebras. The present project aims at continuing this fruitful approach by investigating analogues of two recent constructions for Weyl groups. The first is the new construction by M. Geck has of Lusztig’s algebra J in the case of Coxeter groups, which seems to lend itself to a generalization to cyclotomic Hecke algebras. Together with a suitably modified concept of W-graph for the explicit construction of irreducible representations this should lead to a better understanding of these algebras. The second starting point is Fiebig’s combinatorial notion of sheaves on moment graphs.