Invariant convex sets in polar representations

Understand

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.