  Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak
g=\mathfrak k \oplus \mathfrak p$ be a Cartan decomposition of its Lie algebra.
Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak k$ and
let $\tau$ be an irreducible representation of $G$ on a complex vector space
$V$. Let $h$ be a Hermitian scalar product on $V$ such that $\tau(G)$ is
compatible with respect to $\mathrm{U}(V,h)^{\mathbb C}$. We denote by
$\mu_{\mathfrak p}:\mathbb P(V) \longrightarrow \mathfrak p$ the $G$-gradient
map and by $\mathcal O$ the unique closed orbit of $G$ in $\mathbb P(V)$, which
is a $K$-orbit, contained in the unique closed orbit of the Zariski closure of
$\tau(G)$ in $\mathrm{U}(V,h)^{\mathbb C}$. We prove that up to equivalence the
set of irreducible representations of parabolic subgroups of $G$ induced by
$\tau$ are completely determined by the facial structure of the polar orbitope
$\mathcal E=\mathrm{conv}(\mu_{\mathfrak p} (\mathcal O))$. Moreover, any
parabolic subgroup of $G$ admits a unique closed orbit which is well-adapted to
$\mathcal O$ and $\mu_{\mathfrak p}$ respectively. These results are new also
in the complex reductive case. The connection between $\mathcal E$ and $\tau$
provides a geometrical description of the Satake compactifications without root
data. In this context the properties of the Bourguignon-Li-Yau map are also
investigated. Given a measure $\gamma$ on $\mathcal O$, we construct a map
$\Psi_\gamma$ from the Satake compactification of $G/K$ associated to $\tau$
and $\mathcal E$. If $\gamma$ is a $K$-invariant measure then $\Psi_\gamma$ is
an homeomorphism of the Satake compactification and $\mathcal E$. Finally, we
prove that for a large class of measures the map $\Psi_\gamma$ is surjective.
