Invariant differential operators for the Jacobi algebra $ {\cal G}_2$

TL;DR

This paper explicitly constructs intertwining differential operators for the Jacobi algebra ${\cal G}_2$ using singular vectors of Verma modules, detailing function spaces and representation actions involved.

Contribution

It provides a novel explicit construction of intertwining differential operators for ${\cal G}_2$ based on previously developed singular vectors of Verma modules.

Findings

Explicit formulas for intertwining operators are derived.

Function spaces suitable for these operators are constructed.

Two versions of representation actions are presented.

Abstract

In the present paper we construct explicitly the intertwining differential operators for the Jacobi algebra ${\cal G}_2.$ For the construction we use the singular vectors of the Verma modules over ${\cal G}_2$ which we have constructed earlier. We construct the function spaces on which the operators act. We display two versions of the left (representation) action and the right action. The latter is inserted in the singular vectors to provide the intertwining differential operators.