Intertwining operator associated to the complex Dunkl operator of type $G(m,1,N)$
Classical Analysis and ODEs
2013-11-12 v1
Abstract
In this work, we consider the Dunkl complex reflection operators related to the group 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 and the partial derivative operator As application we given an explicit solution of the system:
Cite
@article{arxiv.1311.2295,
title = {Intertwining operator associated to the complex Dunkl operator of type $G(m,1,N)$},
author = {Fethi Bouzeffour and Sami Ghazouani},
journal= {arXiv preprint arXiv:1311.2295},
year = {2013}
}