# Multiplet Classification of Reducible Verma Modules over the $G_2$ Algebra

## TL;DR

This paper classifies reducible Verma modules over the split real form of the $G_2$ algebra and identifies singular vectors, facilitating the construction of invariant differential operators.

## Contribution

It provides a systematic classification of reducible Verma modules and singular vectors for the $G_{2(2)}$ algebra, advancing the understanding of invariant differential operators.

## Key findings

- Classification of reducible Verma modules for $G_{2(2)}$
- Identification of singular vectors between modules
- Foundation for constructing invariant differential operators

## Abstract

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $G_{2(2)}$ which is split real form of $G_2$. We give the classification of reducible Verma modules $G_2$. We give also the singular vectors between these modules, thus setting the stage for construction of the invariant differential operators over $G_{2(2)}$.