Intertwining operator associated to the complex Dunkl operator of type $G(m,1,N)$

Abstract

In this work, we consider the Dunkl complex reflection operators related to the group $G(m,1,N)$ in the complex plane \begin{align*} T_i=\frac{\partial}{\partial z_i}+k_0\sum_{j\neq i}\sum_{r=0}^{m-1}\frac{1-s_i^{-r}(i,j)s_i^r} {z_i-\varepsilon^r z_j}+\sum_{j=1}^{m-1}k_j\sum_{r=0}^{m-1}\frac{\varepsilon^{-rj}s_i^r}{z_i}, \,\,1\leq i\leq N. \end{align*} We first review the theory of Dunkl operators for complex reflection groups we recall some results related to the hyper--Bessel functions, which are solutions of a higher order differential equation. Secondly, we construct a new explicit intertwining operator between the operator $T_i$ and the partial derivative operator $\frac{\partial}{\partial x_i}.$ As application we given an explicit solution of the system: $$ T_if(x)=\kappa\lambda_i f(x),\,\,\, f(0)=1.$$