Vector-valued covariant differential operators for the M\"obius transformation

TL;DR

This paper introduces vector-valued covariant differential operators linked to Gegenbauer polynomials, which serve as symmetry breaking operators for conformal Lie group pairs, revealing new functional identities and geometric insights.

Contribution

It constructs a new family of differential operators of arbitrary order associated with Gegenbauer polynomials that act as symmetry breaking operators for specific Lie group pairs.

Findings

Derived functional identities for vector-valued functions and their inversions.

Established differential operators as symmetry breaking operators for (SL(2,C), SL(2,R)).

Connected differential operators to conformal geometry and Lie group representations.

Abstract

We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials. These differential operators are symmetry breaking for the pair of Lie groups $(SL(2,\mathbb C), SL(2,\mathbb R))$ that arise from conformal geometry.