# Convexity properties of gradient maps associated to real reductive representations

## TL;DR

This paper explores the convexity properties of gradient maps linked to real reductive Lie group actions, providing explicit computations, generalizations, and a new proof of the Hilbert-Mumford criterion.

## Contribution

It explicitly computes the gradient map images for Abelian groups, generalizes previous results, and offers a novel proof of the Hilbert-Mumford criterion for real reductive groups.

## Key findings

- Explicit gradient map images for Abelian groups
- Generalization of Kac and Peterson's results
- New proof of the Hilbert-Mumford criterion

## Abstract

Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson. A similar result is proved for the gradient map associated to the natural $G$ action on P(V). We also investigate the convex hull of the image of the gradient map restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert-Mumford criterion for real reductive Lie groups avoiding any algebraic result