  Let $G$ be a Lie group with real semisimple Lie algebra $\mathfrak{g}$.
Further let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be a Cartan
decomposition. The maximal compact subgroup $K \subseteq G$ acts on
$\mathfrak{p}$ via the adjoint representation and the convex hulls of the
resulting orbits are the polar orbitopes. We prove that every polar orbitope is
a spectrahedron by giving an explicit representation. In addition we give a new
proof for the fact that the faces of a polar orbitope are, up to conjugation,
given by the faces of the momentum polytope.

Representation Theory;Algebraic Geometry