Conformally Covariant Bi-Differential Operators on a Simple Real Jordan Algebra

TL;DR

This paper constructs a family of bi-differential operators on simple real Jordan algebras that are covariant under conformal group actions, generalizing classical Rankin-Cohen brackets.

Contribution

It introduces conformally covariant bi-differential operators on simple real Jordan algebras, extending classical Rankin-Cohen brackets to a broader algebraic setting.

Findings

Operators are covariant under the conformal group.

Generalization of Rankin-Cohen brackets to Jordan algebras.

Provides explicit construction of these operators.

Abstract

For a simple real Jordan algebra $V,$ a family of bi-differential operators from $\mathcal{C}^\infty(V\times V)$ to $\mathcal{C}^\infty(V)$ is constructed. These operators are covariant under the rational action of the conformal group of $V.$ They generalize the classical {\em Rankin-Cohen} brackets (case $V=\mathbb{R}$).