  In this paper, we introduce a new differential-difference operator $T_\xi$
$(\xi \in \mathbb{R}^N)$ by using projections associated to orthogonal
subsystems in root systems. Similarly to Dunkl theory, we show that these
operators commute and we construct an intertwining operator between $T_\xi$ and
the directional derivative $\partial_\xi$. In the case of one variable, we
prove that the Kummer functions are eigenfunctions of this operator.
