Satake-Furstenberg compactifications and gradient map

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