Certain imprimitive reflection groups and their generic versions

The present paper is concerned with the connection between the imprimitive reflection groups $G(m,m,n)$, $m\in \mathbb{N}$, and the affine Weyl group $\widetilde {A}_{n-1}$. We show that $\widetilde {A}_{n-1}$ is a generic version of the groups $G(m,m,n)$, $m\in \mathbb{N}$. We introduce some new presentations of these groups which are shown to have some group-theoretic advantages. Then we define the Hecke algebras of these groups and of their braid versions, each in two ways according to two presentations. Finally we give a new description for the affine root system $\overline{\Phi }$ of $\widetilde {A}_{n-1}$ such that the action of $\widetilde {A}_{n-1}$ on $\overline{\Phi }$ is compatible with that of $G(m,m,n)$ on its root system in some sense.