Finite reflection groups and the Dunkl-Laplace differential-difference operators in conformal geometry

TL;DR

This paper introduces a Dunkl operator-based approach to conformally invariant differential operators on spheres, extending classical constructions via a deformation of the ambient metric method.

Contribution

It develops a new framework for conformally invariant operators using Dunkl theory, linking reflection groups with conformal geometry.

Findings

Derived Dunkl-Laplace conformally invariant operators

Extended Fefferman-Graham ambient metric construction

Provided explicit formulas for operators in the Dunkl setting

Abstract

For a finite reflection subgroup $G\leq O(n+1,1,\mR)$ of the conformal group of the sphere with standard conformal structure $(S^n,[g_0])$, we geometrically derive differential-difference Dunkl version of the series of conformally invariant differential operators with symbols given by powers of Laplace operator. The construction can be regarded as a deformation of the Fefferman-Graham ambient metric construction of GJMS operators.