Hurwitz numbers for reflection groups $G(m,1,n)$

We are extending results from {B-Hurwitz} by building a parallel theory of simple Hurwitz numbers for the reflection groups $G(m,1,n)$. We also study analogs of the cut-and-join operators. An algebraic description as well as a description in terms of ramified covering of Hurwitz numbers is provided. An explicit formula for them in terms of Schur polynomials are provided. In addition the generating function of $G(m,1,n)$-Hurwitz numbers is shown to give rise to m independent variables τ-function of the KP hierarchy. Finally we provide an ELSV-formula type for these new Hurwitz numbers.