# The Model Orbit in $G_2$

## TL;DR

This paper decomposes the ring of regular functions on a specific nilpotent orbit of the complex G_2 group, confirming McGovern's predictions and establishing the unitarizability of a related representation.

## Contribution

It provides a detailed decomposition of the regular functions on a nilpotent orbit in G_2 and verifies McGovern's conjecture about the associated representation.

## Key findings

- Confirmed the decomposition of the ring of regular functions on the orbit.
- Proved the unitarizability of McGovern's proposed representation.
- Validated the orbit's representation properties against prior predictions.

## Abstract

In this article, we decompose the ring of regular functions on the nilpotent orbit of dimension 8 for the complex $G_2$ in which every irreducible representation of $G_2$ appears exactly once. This confirms the predication of McGovern and we have shown that his proposed representation attaching to this orbit is unitarizable.