Macdonald functions associated to complex reflection groups

Let W be the complex reflection group . In the authorʹs previous paper [J. Algebra 245 (2001) 650–694], Hall–Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type Bn, they are closely related to Green polynomials of finite classical groups. In this paper, we introduce a two variables version of the above Hall–Littlewood functions, as a generalization of Macdonald functions associated to symmetric groups. A generalization of Macdonald operators is also constructed, and we characterize such functions by making use of Macdonald operators, assuming a certain conjecture.