# On conjugacy of stabilizers of reductive group actions

## TL;DR

This paper demonstrates that a key result in reductive group actions, previously established by Wallach, is actually a special case of more general classical theorems, simplifying its proof.

## Contribution

It shows that Wallach's principal orbit type theorems and the Kempf--Ness Theorem are special cases of classical Richardson and Luna theorems, providing a shorter proof.

## Key findings

- Wallach's main result is a special case of classical theorems.
- A short argument suffices to derive Wallach's result from classical theorems.
- The paper clarifies the relationship between modern and classical results in reductive group actions.

## Abstract

It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf--Ness Theorem, arXiv:1811.07195v1 (17 Nov 2018) is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems.