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

TL;DR

This paper introduces a new class of differential-difference operators linked to root systems, demonstrating their commutativity, constructing an intertwining operator, and identifying eigenfunctions as Kummer functions in one variable.

Contribution

It presents a novel type of Dunkl-type operators with projection terms for orthogonal subsystems in root systems, expanding the theoretical framework of Dunkl operators.

Findings

Operators commute similarly to Dunkl theory

Constructed an intertwining operator between $T_$ and directional derivatives

Kummer functions are eigenfunctions in the one-variable case

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.