Invariant convex sets in polar representations

TL;DR

This paper investigates the structure of invariant convex sets in polar representations of compact Lie groups, revealing how their face structures relate to those of convex subsets in maximal abelian subalgebras, with applications to momentum maps.

Contribution

It establishes a complete characterization of the face structure of invariant convex sets in polar representations via their intersections with maximal abelian subalgebras, including conditions for exposed faces.

Findings

Face structure of invariant convex sets is determined by that of their intersection with a maximal abelian subalgebra.

Exposed faces of the convex set correspond to exposed faces of the intersection.

Results apply to the convex hull of the image of a restricted momentum map.

Abstract

We study a compact invariant convex set $E$ in a polar representation of a compact Lie group. Polar rapresentations are given by the adjoint action of $K$ on $\mathfrak{p}$, where $K$ is a maximal compact subgroup of a real semisimple Lie group $G$ with Lie algebra $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$. If $\mathfrak{a} \subset \mathfrak{p}$ is a maximal abelian subalgebra, then $P=E\cap \mathfrak{a}$ is a convex set in $\mathfrak{a}$. We prove that up to conjugacy the face structure of $E$ is completely determined by that of $P$ and that a face of $E$ is exposed if and only if the corresponding face of $P$ is exposed. We apply these results to the convex hull of the image of a restricted momentum map.