Projections from Furstenberg boundaries onto maximal flats and barycenter maps

Let $G$ be a semisimple connected Lie group of non-compact type with finite center. Let $K<G$ be a maximal compact subgroup and $P<G$ be a minimal parabolic subgroup. For any pair $(F,x)$, where $F$ is a maximal flat in $G/K$ and $x \in G/P$ is opposite to the Weyl chambers determined by $F$, we define a projection $\Phi(F, x) \in F$ which is continuous and $G$-equivariant. Furthermore, if $q \geq 3$, we exhibit a $G$-equivariant continuous map defined on an open subset of full measure of the space of $q$-tuples of $(G/P)^q$ with image in $G/K$. When $G$ is the orientation preserving isometries of real hyperbolic space and $q = 3$, we recover the geometric barycenter of the corresponding ideal triangle. All our proofs are constructive.