Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System

Abstract

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.