A geometric take on Kostant's Convexity Theorem

TL;DR

This paper introduces a metric criterion for convexity in orbit spaces of compact Lie group actions, generalizing Kostant's Convexity Theorem through the concept of fat sections and submetries.

Contribution

It extends Kostant's Convexity Theorem to a broader setting involving submetries and introduces the notion of fat sections, unifying various geometric structures.

Findings

A metric criterion for convex pre-images in orbit spaces.

Generalization of Kostant's Convexity Theorem to submetries with fat sections.

Examples showing the limits of this generalization.

Abstract

Given a compact Lie group $G$ and an orthogonal $G$-representation $V$, we give a purely metric criterion for a closed subset of the orbit space $V/G$ to have convex pre-image in $V$. In fact, this also holds with the natural quotient map $V\to V/G$ replaced with an arbitrary submetry $V\to X$. In this context, we introduce a notion of "fat section" which generalizes polar representations, representations of non-trivial copolarity, and isoparametric foliations. We show that Kostant's Convexity Theorem partially generalizes from polar representations to submetries with a fat section, and give examples illustrating that it does not fully generalize to this situation.